Nothing Special   »   [go: up one dir, main page]

You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 
Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (118)

Search Parameters:
Keywords = peristaltic flow

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
19 pages, 2110 KiB  
Review
Mechanobiological Approach for Intestinal Mucosal Immunology
by Hyeyun Kim, Se-Hui Lee and Jin-Young Yang
Biology 2025, 14(2), 110; https://doi.org/10.3390/biology14020110 - 22 Jan 2025
Viewed by 599
Abstract
The intestinal area is composed of diverse cell types that harmonize gut homeostasis, which is influenced by both endogenous and exogenous factors. Notably, the environment of the intestine is exposed to several types of mechanical forces, including shear stress generated by fluid flow, [...] Read more.
The intestinal area is composed of diverse cell types that harmonize gut homeostasis, which is influenced by both endogenous and exogenous factors. Notably, the environment of the intestine is exposed to several types of mechanical forces, including shear stress generated by fluid flow, compression and stretch generated by luminal contents and peristaltic waves of the intestine, and stiffness attributed to the extracellular matrix. These forces play critical roles in the regulation of cell proliferation, differentiation, and migration. Many efforts have been made to simulate the actual intestinal environment in vitro. The three-dimensional organoid culture system has emerged as a powerful tool for studying the mechanism of the intestinal epithelial barrier, mimicking rapidly renewing epithelium from intestinal stem cells (ISCs) in vivo. However, many aspects of how mechanical forces, such as shear stress, stiffness, compression, and stretch forces, influence the intestinal area remain unresolved. Here, we review the recent studies elucidating the impact of mechanical forces on intestinal immunity, interaction with the gut microbiome, and intestinal diseases. Full article
(This article belongs to the Special Issue Mechanobiology 2.0)
Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Types of mechanical strains in the intestine. The fluid flow and digestion leads to shear stress (<b>I</b>), compression (<b>III</b>), and stretch (<b>IV</b>). Stiffness (<b>II</b>) represents the cross-linking status between ECM components. These mechanical strains influence ISC behavior in various ways. The arrows indicate the direction of the force.</p>
Full article ">Figure 2
<p>Complex interactions between mechanical forces and various cell types. Mechanical forces affect various cells in the intestine and alter the intestinal environment. These forces are orchestrated in the gut and contribute to intestinal homeostasis in a complex manner, including cell differentiation, proliferation, and immune regulation.</p>
Full article ">Figure 3
<p>Mechanical forces and intestinal disease. This schematic shows the connections between mechanical stress and intestinal diseases. “−” means a negative correlation, and “+” means a positive correlation between stress and intestinal diseases [<a href="#B85-biology-14-00110" class="html-bibr">85</a>,<a href="#B86-biology-14-00110" class="html-bibr">86</a>,<a href="#B109-biology-14-00110" class="html-bibr">109</a>,<a href="#B115-biology-14-00110" class="html-bibr">115</a>,<a href="#B116-biology-14-00110" class="html-bibr">116</a>,<a href="#B117-biology-14-00110" class="html-bibr">117</a>,<a href="#B118-biology-14-00110" class="html-bibr">118</a>,<a href="#B119-biology-14-00110" class="html-bibr">119</a>,<a href="#B120-biology-14-00110" class="html-bibr">120</a>,<a href="#B121-biology-14-00110" class="html-bibr">121</a>,<a href="#B122-biology-14-00110" class="html-bibr">122</a>,<a href="#B123-biology-14-00110" class="html-bibr">123</a>,<a href="#B124-biology-14-00110" class="html-bibr">124</a>,<a href="#B125-biology-14-00110" class="html-bibr">125</a>,<a href="#B126-biology-14-00110" class="html-bibr">126</a>,<a href="#B127-biology-14-00110" class="html-bibr">127</a>,<a href="#B129-biology-14-00110" class="html-bibr">129</a>,<a href="#B130-biology-14-00110" class="html-bibr">130</a>,<a href="#B131-biology-14-00110" class="html-bibr">131</a>,<a href="#B132-biology-14-00110" class="html-bibr">132</a>,<a href="#B133-biology-14-00110" class="html-bibr">133</a>,<a href="#B134-biology-14-00110" class="html-bibr">134</a>].</p>
Full article ">
74 pages, 7040 KiB  
Article
The Lattice Boltzmann Method with Deformable Boundary for Colonic Flow Due to Segmental Circular Contractions
by Irina Ginzburg
Fluids 2025, 10(2), 22; https://doi.org/10.3390/fluids10020022 - 21 Jan 2025
Viewed by 597
Abstract
We extend the 3D Lattice Boltzmann method with a deformable boundary (LBM-DB) for the computations of the full-volume colonic flow of the Newtonian fluid driven by the peristaltic segmented circular contractions which obey the three-step “intestinal law”: (i) deflation, (ii) inflation, and (iii) [...] Read more.
We extend the 3D Lattice Boltzmann method with a deformable boundary (LBM-DB) for the computations of the full-volume colonic flow of the Newtonian fluid driven by the peristaltic segmented circular contractions which obey the three-step “intestinal law”: (i) deflation, (ii) inflation, and (iii) elastic relaxation. The key point is that the LBM-DB accurately prescribes a curved deforming surface on the regular computational grid through precise and compact Dirichlet velocity schemes, without the need to recover for an adaptive boundary mesh or surface remesh, and without constraint of fluid volume conservation. The population “refill” of “fresh” fluid nodes, including sharp corners, is reformulated with the improved reconstruction algorithms by combining bulk and advanced boundary LBM steps with a local sub-iterative collision update. The efficient parallel LBM-DB simulations in silico then extend the physical experiments performed in vitro on the Dynamic Colon Model (DCM, 2020) to highly occlusive contractile waves. The motility scenarios are modeled both in a cylindrical tube and in a new geometry of “parabolic” transverse shape, which mimics the dynamics of realistic triangular lumen aperture. We examine the role of cross-sectional shape, motility pattern, occlusion scenario, peristaltic wave speed, elasticity effect, kinematic viscosity, inlet/outlet conditions and numerical compressibility on the temporal localization of pressure and velocity oscillations, and especially the ratio of retrograde vs antegrade velocity amplitudes, in relation to the major contractile events. The developed numerical approach could contribute to a better understanding of the intestinal physiology and pathology due to a possibility of its straightforward extension to the non-Newtonian chyme rheology and anatomical geometry. Full article
(This article belongs to the Special Issue Lattice Boltzmann Methods: Fundamentals and Applications)
Show Figures

Figure 1

Figure 1
<p>The large intestine. Reproduced from [<xref ref-type="bibr" rid="B54-fluids-10-00022">54</xref>].</p>
Full article ">Figure 2
<p>“The 3D model of the biomechanical Dynamic Colon Model of human proximal colon with focus on caecum—ascending region”, reproduced from Figure S2a [<xref ref-type="bibr" rid="B9-fluids-10-00022">9</xref>]; segments 1 and 10 are adjacent to the Caecum and the Hepatic flexures, respectively.</p>
Full article ">Figure 3
<p>“PC cine-MRI image of the DCM at the midpoint (cross-section) of segment 6”, reproduced from Figure 7C [<xref ref-type="bibr" rid="B64-fluids-10-00022">64</xref>].</p>
Full article ">Figure 4
<p>Relative occlusion variation <inline-formula><mml:math id="mm4274"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and the relative total volume variation <inline-formula><mml:math id="mm4275"><mml:semantics><mml:mrow><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="normal">o</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are displayed for 10 segments together; the motility cycle is followed by equilibration towards the steady state at the neutral occlusion degree. (<bold>left</bold>): <inline-formula><mml:math id="mm4276"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4277"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> from <xref ref-type="table" rid="fluids-10-00022-t001">Table 1</xref> with <inline-formula><mml:math id="mm4278"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.6</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>right</bold>): <inline-formula><mml:math id="mm4279"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4280"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> from <xref ref-type="table" rid="fluids-10-00022-t001">Table 1</xref> with <inline-formula><mml:math id="mm4281"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>0.6</mml:mn><mml:mo>,</mml:mo><mml:mn>0.4</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 5
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4282"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4284"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.293</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4285"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4286"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4287"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4288"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo>Δ</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4289"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4290"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4291"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4292"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4293"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A2">Table A2</xref>.</p>
Full article ">Figure 6
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4294"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4296"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.201</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4297"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4298"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4299"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4300"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.704</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4301"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4302"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4303"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4304"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4305"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A3">Table A3</xref>.</p>
Full article ">Figure 7
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4306"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4308"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.431</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4309"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4310"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4311"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4312"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.862</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4313"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4314"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4315"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4316"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4317"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A4">Table A4</xref>.</p>
Full article ">Figure 8
<p>(<bold>top-left</bold>) <inline-formula><mml:math id="mm4318"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in “P” pipe; (<bold>top-right</bold>) <inline-formula><mml:math id="mm4320"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.246</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4321"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>25</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-left</bold>) <inline-formula><mml:math id="mm4322"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mo>[</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4323"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>28</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>bottom-right</bold>) <inline-formula><mml:math id="mm4324"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.862</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4325"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mspace width="3.33333pt"/><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Velocity field <inline-formula><mml:math id="mm4326"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4327"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> <inline-formula><mml:math id="mm4328"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm4329"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t0A5">Table A5</xref>.</p>
Full article ">Figure 9
<p>“C” pipe with <inline-formula><mml:math id="mm4331"><mml:semantics><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mrow><mml:mi mathvariant="normal">f</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>3</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> [<xref ref-type="bibr" rid="B68-fluids-10-00022">68</xref>]. (<bold>left</bold>) <inline-formula><mml:math id="mm4332"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4333"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo>Δ</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>middle</bold>) <inline-formula><mml:math id="mm4334"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4335"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>; (<bold>right</bold>) <inline-formula><mml:math id="mm4336"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm4337"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>2</mml:mn><mml:mn>5</mml:mn></mml:mfrac></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mrow><mml:mi>π</mml:mi></mml:mfrac></mml:mstyle><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 10
<p>Case <inline-formula><mml:math id="mm4338"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4339"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4340"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4341"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4342"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 11
<p>Case <inline-formula><mml:math id="mm4343"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4344"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4345"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4346"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4347"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 12
<p>Case <inline-formula><mml:math id="mm4348"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4349"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4350"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4351"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4352"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 13
<p>Case <inline-formula><mml:math id="mm4353"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4354"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4355"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4356"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4357"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 14
<p>Case <inline-formula><mml:math id="mm4358"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4359"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4360"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4361"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4362"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 15
<p>Case <inline-formula><mml:math id="mm4363"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4364"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4365"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4366"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4367"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>right</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>left</bold>-<bold>middle</bold>).</p>
Full article ">Figure 16
<p>Case <inline-formula><mml:math id="mm4368"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4369"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4370"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4371"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4372"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 17
<p>Case <inline-formula><mml:math id="mm4373"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4374"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4375"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4376"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4377"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 18
<p>Case <inline-formula><mml:math id="mm4378"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4379"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4380"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4381"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4382"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>.Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 19
<p>Case <inline-formula><mml:math id="mm4383"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4384"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4385"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4386"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4387"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 20
<p>Case <inline-formula><mml:math id="mm4388"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4389"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4390"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4391"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4392"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure 21
<p>Case <inline-formula><mml:math id="mm4393"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4394"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4395"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4396"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4397"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A1
<p>The set of fluid grid points <inline-formula><mml:math id="mm1806"><mml:semantics><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mi>h</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>]</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> is displayed at <inline-formula><mml:math id="mm1807"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>300</mml:mn><mml:mo>,</mml:mo><mml:mn>500</mml:mn><mml:mo>,</mml:mo><mml:mn>700</mml:mn><mml:mo>}</mml:mo><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi mathvariant="normal">steps</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> (from <bold>left</bold> to <bold>right</bold> and <bold>top</bold> to <bold>bottom</bold>). The entry and exit are initially vertical and distanced by <inline-formula><mml:math id="mm1808"><mml:semantics><mml:mrow><mml:mi mathvariant="script">L</mml:mi><mml:mo>=</mml:mo><mml:mn>49</mml:mn><mml:mspace width="3.33333pt"/><mml:mo>[</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>.</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>; they are subjected to permanent stretching according to the parabolic velocity profile <inline-formula><mml:math id="mm1809"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>u</mml:mi><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mstyle><mml:msub><mml:mover accent="true"><mml:mn>1</mml:mn><mml:mo>→</mml:mo></mml:mover><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. Data [<inline-formula><mml:math id="mm1810"><mml:semantics><mml:mrow><mml:mi>l</mml:mi><mml:mo>.</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>]: <inline-formula><mml:math id="mm1811"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1812"><mml:semantics><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1813"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure A2
<p>Velocity and pressure deviations from the exact solution are output for all fluid grid nodes in a deformable channel when <inline-formula><mml:math id="mm1835"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mn>300</mml:mn><mml:mo>,</mml:mo><mml:mn>500</mml:mn><mml:mo>,</mml:mo><mml:mn>700</mml:mn><mml:mo>}</mml:mo><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi mathvariant="normal">steps</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>top</bold>) <inline-formula><mml:math id="mm1836"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>y</italic>; (<bold>second from top</bold>) <inline-formula><mml:math id="mm1837"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>x</italic>; (<bold>second from bottom</bold>) <inline-formula><mml:math id="mm1838"><mml:semantics><mml:mrow><mml:mrow><mml:mi>err</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> versus <italic>x</italic>; (<bold>bottom</bold>) <inline-formula><mml:math id="mm1839"><mml:semantics><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. This figure: all cut links apply <inline-formula><mml:math id="mm1841"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1842"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> (<bold>right</bold>).</p>
Full article ">Figure A3
<p>Similar as in <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. (<bold>left</bold>): <inline-formula><mml:math id="mm1862"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1863"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1864"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. (<bold>right</bold>): <inline-formula><mml:math id="mm1865"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1866"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1867"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>. (<bold>left</bold>,<bold>right</bold>): <inline-formula><mml:math id="mm1868"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">two</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1869"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>; <inline-formula><mml:math id="mm1870"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">one</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1871"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>; <inline-formula><mml:math id="mm1872"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1873"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1874"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> (<bold>right</bold>).</p>
Full article ">Figure A4
<p>Similar to <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. (<bold>left</bold>): <inline-formula><mml:math id="mm1894"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1895"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1896"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> where all cut links apply <inline-formula><mml:math id="mm1898"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>. (<bold>right</bold>): <inline-formula><mml:math id="mm1899"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1900"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1901"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, where the <inline-formula><mml:math id="mm1902"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> apply <inline-formula><mml:math id="mm1903"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, all others apply <inline-formula><mml:math id="mm1904"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure A5
<p>Similar to <xref ref-type="fig" rid="fluids-10-00022-f0A2">Figure A2</xref>. The reconstruction algorithm is <inline-formula><mml:math id="mm1936"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">r</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">equil</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>left</bold>) and <inline-formula><mml:math id="mm1937"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">r</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">bc</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">corner</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<bold>right</bold>). <inline-formula><mml:math id="mm1938"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1939"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>–<inline-formula><mml:math id="mm1940"><mml:semantics><mml:msub><mml:mrow><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> is applied in <inline-formula><mml:math id="mm1941"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">two</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1942"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">one</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1943"><mml:semantics><mml:mrow><mml:mi mathvariant="italic">no</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="italic">nb</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="italic">links</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, respectively.</p>
Full article ">Figure A6
<p>Case <inline-formula><mml:math id="mm3974"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3975"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3976"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3977"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3978"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A7
<p>Case <inline-formula><mml:math id="mm3979"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3980"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3981"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3982"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3983"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>right</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>left</bold>-<bold>middle</bold>).</p>
Full article ">Figure A8
<p><inline-formula><mml:math id="mm3984"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3985"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3986"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3987"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3988"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A9
<p>Case <inline-formula><mml:math id="mm3989"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3990"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3991"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3992"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3993"><mml:semantics><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow><mml:mrow><mml:mo> </mml:mo><mml:mo>★</mml:mo><mml:mo>★</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> (case <inline-formula><mml:math id="mm3994"><mml:semantics><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">L</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">V</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo> </mml:mo><mml:mo>★</mml:mo><mml:mo>★</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> in <xref ref-type="table" rid="fluids-10-00022-t005">Table 5</xref> with <inline-formula><mml:math id="mm3995"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac></mml:mstyle><mml:mo>×</mml:mo><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mstyle></mml:mrow></mml:semantics></mml:math></inline-formula>). Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A10
<p>Case <inline-formula><mml:math id="mm3996"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3997"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3998"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm3999"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4000"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A11
<p>Case <inline-formula><mml:math id="mm4001"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4002"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4003"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4004"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">H</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4005"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mi mathvariant="normal">X</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A12
<p>Case <inline-formula><mml:math id="mm4006"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4007"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4008"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4009"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4010"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>middle</bold>) to (<bold>left</bold>-<bold>top</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">Figure A13
<p>Case <inline-formula><mml:math id="mm4011"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mtext>-</mml:mtext><mml:mrow><mml:mi mathvariant="normal">P</mml:mi><mml:mi mathvariant="normal">E</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4012"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">T</mml:mi><mml:mtext>-</mml:mtext><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4013"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">L</mml:mi></mml:mrow><mml:mtext>-</mml:mtext><mml:mn>5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4014"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">V</mml:mi><mml:mi mathvariant="normal">I</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>:<inline-formula><mml:math id="mm4015"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">L</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mi mathvariant="normal">W</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. Legende (<bold>left</bold>-<bold>top</bold>) to (<bold>left</bold>-<bold>middle</bold>) and (<bold>right</bold>-<bold>middle</bold>).</p>
Full article ">
13 pages, 1311 KiB  
Article
Exploring Bone Morphogenetic Protein-2 and -4 mRNA Expression and Their Receptor Assessment in a Dynamic In Vitro Model of Vascular Calcification
by Manuela Cabiati, Federico Vozzi, Elisa Ceccherini, Letizia Guiducci, Elisa Persiani, Ilaria Gisone, Agnese Sgalippa, Antonella Cecchettini and Silvia Del Ry
Cells 2024, 13(24), 2091; https://doi.org/10.3390/cells13242091 - 18 Dec 2024
Viewed by 635
Abstract
Background: Vascular calcification (VC) is a dynamic, tightly regulated process driven by cellular activity and resembling the mechanisms of bone formation, with specific molecules playing pivotal roles in its progression. We aimed to investigate the involvement of the bone morphogenic proteins (BMP-2 [...] Read more.
Background: Vascular calcification (VC) is a dynamic, tightly regulated process driven by cellular activity and resembling the mechanisms of bone formation, with specific molecules playing pivotal roles in its progression. We aimed to investigate the involvement of the bone morphogenic proteins (BMP-2, BMP-4, BMPR-1a/1b, and BMPR-2) system in this process. Our study used an advanced in vitro model that simulates the biological environment of the vascular wall, assessing the ability of a phosphate mixture to induce the osteoblastic switch in human coronary artery smooth muscle cells (HCASMCs). Methods: HCASMCs were grown in mono- and co-culture with human coronary artery endothelial cells (HCAECs) in a double-flow bioreactor (LiveBox2 and IVTech), allowing static and dynamic conditions through a peristaltic pump. The VC was stimulated by incubation in a calcifying medium for 7 days. A BMP system Real-Time PCR was performed at the end of each experiment. Results: In monocultures, BMP-2 expression increased in calcified HCASMCs in static (p = 0.01) and dynamic conditions. BMP-4 and the biological receptors were expressed in all the experimental settings, increasing mainly in dynamic flow conditions. In co-cultures, we observed a marked increase in BMP-2 and BMP-4, BMPR-1a (p = 0.04 and p = 0.01, respectively), and BMPR-2 (p = 0.001) in the calcifying setting mostly in dynamic conditions. Conclusions: The increase in BMP-2/4 in co-culture suggests that these genes might promote the switch towards an osteogenic-like phenotype, data also supported by the rise of both BMPR-1a and BMPR-2. Thus, our findings provide insights into the mechanisms by which dynamic co-culture modulates the BMP system activation in an environment mimicking in vivo VC’s cellular and mechanical characteristics. Full article
(This article belongs to the Special Issue Role of Vascular Smooth Muscle Cells in Cardiovascular Disease)
Show Figures

Figure 1

Figure 1
<p>Expression of <span class="html-italic">RUNX-2</span> in calcifying environment: (<b>A</b>) <span class="html-italic">RUNX-2</span> fold increase in HCASMC subjected to static (dark grey bar) and dynamic conditions (black bar) compared with reference; and (<b>B</b>) <span class="html-italic">RUNX-2</span> fold increase in the HCASMC co-culture subjected to static (dark grey bar) and dynamic conditions (black bar) compared with reference.</p>
Full article ">Figure 2
<p>Relative mRNA expression levels of (<b>A</b>) <span class="html-italic">BMP-2</span>, (<b>B</b>) <span class="html-italic">BMP-4</span>, (<b>C</b>) <span class="html-italic">BMPR-1a</span>, (<b>D</b>) <span class="html-italic">BMPR-1b</span>, and (<b>E</b>) <span class="html-italic">BMPR-2</span> in HCASMC in the absence/presence of calcifying environment and subjected to static (white and dark grey bars) and dynamic conditions (light grey and black bars).</p>
Full article ">Figure 3
<p>Transcript levels: (<b>A</b>) relative mRNA expression levels of <span class="html-italic">TGF-β1</span> in the absence/presence of calcifying environment and subjected to static (white and dark grey bars) and dynamic conditions (light grey and black bars) [Fisher’ test: <span class="html-italic">p</span> = 0.03; Bonferroni/Dunn’s post hoc test: <span class="html-italic">p</span> = ns calcifying medium static vs. dynamic]; and (<b>B</b>) mRNA expression ratio of <span class="html-italic">TGF-β1/BMP-2</span> and (<b>C</b>) <span class="html-italic">TGF-β1/BMP-4</span> in HCASMC expressed as box plots in the absence/presence of calcifying environment and subjected to static (white and dark grey plots) and dynamic conditions (light grey and black plots). Each box consists of five horizontal lines displaying the 10th, 25th, 50th, 75th and 90th percentiles of the variable. All values above the 90th percentile and below the 10th percentile, are plotted separately and represents the outliers.</p>
Full article ">Figure 4
<p>Relative mRNA expression levels of (<b>A</b><span class="html-italic">) BMP-2</span>, (<b>B</b>) <span class="html-italic">BMP-4</span>, (<b>C</b>) <span class="html-italic">BMPR-1a</span>, (<b>D</b>) <span class="html-italic">BMPR-1b</span>, and (<b>E</b>) <span class="html-italic">BMPR-2</span> in the HCASMC co-culture system in the absence/presence of a calcifying environment and subjected to static (white and dark grey bars) and dynamic conditions (light grey and black bars).</p>
Full article ">
13 pages, 6810 KiB  
Article
Open-Source Equipment Design for Cost-Effective Redox Flow Battery Research
by Trinh V. Dung, Nguyen T. T. Huyen, Nguyen L. T. Huynh, Nguyen T. Binh, Nguyen T. Dat, Nguyen T. T. Nga, Nguyen T. Lan, Hoang V. Tran, Nguyen T. T. Mai and Chinh D. Huynh
ChemEngineering 2024, 8(6), 120; https://doi.org/10.3390/chemengineering8060120 - 28 Nov 2024
Viewed by 890
Abstract
Redox flow batteries (RFBs), with distinct characteristics that are suited for grid-scale applications, stand at the forefront of potential energy solutions. However, progress in RFB technology is often impeded by their prohibitive cost and the limited availability of essential research and development test [...] Read more.
Redox flow batteries (RFBs), with distinct characteristics that are suited for grid-scale applications, stand at the forefront of potential energy solutions. However, progress in RFB technology is often impeded by their prohibitive cost and the limited availability of essential research and development test cells. Addressing this bottleneck, we present herein an open-source device tailored for RFB laboratory research. Our proposed device significantly lowers the financial barriers to research and enhances the accessibility of vital equipment for RFB studies. Employing innovative fabrication methods such as laser cutting, 3D printing, and CNC machining, a versatile and efficient flow cell has been designed and fabricated. Furthermore, our open laboratory research equipment comprises the Opensens potentiostat, charge/discharge testing devices, peristaltic pumps, and inexpensive rotating electrodes. Every individual element contributes significantly to the establishment of an all-encompassing experimental configuration that is both economical and efficient, thereby facilitating expedited progress in RFB research and development. Full article
Show Figures

Figure 1

Figure 1
<p>The PCB layout of the power module.</p>
Full article ">Figure 2
<p>The Nodered dashboard used to control and monitor the charge/discharge module.</p>
Full article ">Figure 3
<p>(<b>a</b>) The setup used for the CV experiments, (<b>b</b>) the experimental setup of the RDE and Opensens used in the experiments, (<b>c</b>) the original design with a one-brush electrode, and (<b>d</b>) the dual-brush electrode.</p>
Full article ">Figure 4
<p>Cyclic voltammograms of a Pt electrode in: (<b>a</b>) 1 M of vanadium electrolyte with different concentrations of H<sub>2</sub>SO<sub>4</sub>, (<b>b</b>) 0.6M Ce(CH<sub>3</sub>SO<sub>3</sub>)<sub>3</sub> + 4M MSA + x M H<sub>2</sub>SO<sub>4</sub>.</p>
Full article ">Figure 5
<p>Cyclic voltammograms of a Pt electrode in 0.6 M Ce(CH<sub>3</sub>SO<sub>3</sub>)<sub>3</sub> + x M MSA (x = 1, 2, 3, 4, 5).</p>
Full article ">Figure 6
<p>(<b>a</b>) Linear sweep voltammetry of 0.6 M Ce(CH<sub>3</sub>SO<sub>3</sub>)<sub>3</sub> + 4M MSA + 0.5 M H<sub>2</sub>SO<sub>4</sub> + 2% mol L-Lysine electrolyte with different rotation speeds at a scan rate of 5 mV/s. (<b>b</b>) Levich plot of limiting current versus the square root of the rotation rate (ω<sup>1/2</sup>). (<b>c</b>) Koutecky–Levich plot at different overpotentials. (<b>d</b>) Tafel plot of the logarithm of kinetically limited current versus overpotential.</p>
Full article ">Figure 7
<p>Charge/discharge curves of the V-Ce RFB (current I = 0.05 A, with voltage limits at 2.0–0.6 V).</p>
Full article ">Figure 8
<p>Coulombic, voltage, and energy efficiency as a function of the cycle number.</p>
Full article ">
14 pages, 6553 KiB  
Article
An Arteriovenous Bioreactor Perfusion System for Physiological In Vitro Culture of Complex Vascularized Tissue Constructs
by Florian Helms, Delia Käding, Thomas Aper, Arjang Ruhparwar and Mathias Wilhelmi
Bioengineering 2024, 11(11), 1147; https://doi.org/10.3390/bioengineering11111147 - 14 Nov 2024
Viewed by 758
Abstract
Background: The generation and perfusion of complex vascularized tissues in vitro requires sophisticated perfusion techniques. For multiscale arteriovenous networks, not only the arterial, but also the venous, biomechanical and biochemical conditions that physiologically exist in the human body must be accurately emulated. For [...] Read more.
Background: The generation and perfusion of complex vascularized tissues in vitro requires sophisticated perfusion techniques. For multiscale arteriovenous networks, not only the arterial, but also the venous, biomechanical and biochemical conditions that physiologically exist in the human body must be accurately emulated. For this, we here present a modular arteriovenous perfusion system for the in vitro culture of a multi-scale bioartificial vascular network. Methods: The custom-built perfusion system consisted of two circuits: in the arterial circuit, physiological arterial biomechanical and biochemical conditions were simulated using a modular set-up with a pulsatile peristaltic pump, compliance chambers, and resistors. In the venous circuit, venous conditions were emulated accordingly. In the center of the system, a bioartificial multi-scale vascularized fibrin-based tissue was perfused by both circuits simultaneously under biomimetic arteriovenous conditions. Culture conditions were monitored continuously using a multi-sensor monitoring system. Results: The physiological arterial and venous pressure- and flow-curves, as well as the microvascular arteriovenous oxygen partial pressure gradient, were accurately emulated in the perfusion system. The multi-sensor monitoring system facilitated live monitoring of the respective parameters and data-logging. In a proof-of-concept experiment, vascularized three-dimensional fibrin tissues showed sustained cell viability and homogenous microvessel formation after culture in the perfusion system. Conclusions: The arteriovenous perfusion system facilitated the in vitro culture of a multiscale vascularized tissue under physiological pressure-, flow-, and oxygen-gradient conditions. With that, it presents a promising technique for the in vitro generation and culture of complex large-scale vascularized tissues. Full article
Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Generation of the vascularized fibrin-based matrix. (<b>A</b>) Schematic cross-section of the targeted multi-scale vasculature. Venous and arterial fibrin-based macrovessels (1 + 2) were placed in parallel to each other and interconnected via four microchannels (3). Vascular sprouts arising from the microchannels (4) were intended to interconnect the microchannels to a capillary network built-up by the co-culture of human umbilical vein derived endothelial cells (HUVECs) and adipogenous stem cells (5) seeded throughout a low-density fibrin matrix (6). Both macrovessels and microchannels were enothelialized by a HUVEC monolayer (7). Black arrows indicate the media flow direction during perfusion. (<b>B</b>) Perfusion chamber with the integrated fibrin-based tissue construct. Two hose nozzles on each side facilitated connection of the integrated macrovessels to the respective arterial and venous perfusion circuit, and the perforated sheath on the bottom allowed for insertion of needles during the molding process for the generation of the microchannels. (<b>C</b>) Macroscopic morphology of the explanted fibrin-based tissue matrix after 48 h of culture in the arteriovenous perfusion system. Scale bar = 1 cm.</p>
Full article ">Figure 2
<p>(<b>A</b>) Schematic representation of the arteriovenous perfusion system setup and desired pressure and flow curves. 1: Pulsatile peristaltic pump; 2: upstream compliance chamber; 3: pressure sensor; 4: flow sensor; 5: perfusion chamber with the integrated fibrin-based matrix and vessels; 6: variable resistor; 7: reservoir; 8: dissolved oxygen sensor; 9: oxygen inflow canula; 10: downstream arterial compliance chamber; 11: backflow line. (<b>B</b>) Photographic top-view of the assembled system.</p>
Full article ">Figure 3
<p>Pressure curve analysis. (<b>A</b>) Pressure curve monitored in the arterial circuit; (<b>B</b>) systolic (black) and diastolic (grey) pressures observed in the arterial circuit over 48 h. (<b>C</b>) Pressure curve monitored in the venous circuit; (<b>D</b>) systolic (black) and diastolic (grey) pressures observed in the venous circuit over 48 h.</p>
Full article ">Figure 4
<p>Flow curve analysis. (<b>A</b>) Flow curve monitored in the arterial circuit. (<b>B</b>) Flow curve monitored in the venous circuit.</p>
Full article ">Figure 5
<p>Arterial (black) and venous (grey) oxygen partial pressure monitored in the system over 48 h.</p>
Full article ">Figure 6
<p>(<b>A</b>) Fluorescence microscopic view of the fibrin-based tissue matrix. Capillary tubes were visualized based on red fluorescent protein expression of human umbilical vein endothelial cells. (<b>B</b>) Angiotool analysis of the capillary network depicted in (<b>A</b>). Crossing points were marked by blue dots, capillary tubes were depicted in red, and outlines were marked in yellow. Scale bar = 100 µm.</p>
Full article ">
29 pages, 5055 KiB  
Review
Comparative Analysis of In Vitro Pumps Used in Cardiovascular Investigations: Focus on Flow Generation Principles and Characteristics of Generated Flows
by Noaman Mazhar, Munshi Sajidul Islam, Muhammad Zohaib Raza, SM. Khaled Hossain Mahin, Mohammed Riazul Islam, Muhammad E. H. Chowdhury, Abdulla Al-Ali, Abdelali Agouni and Huseyin C. Yalcin
Bioengineering 2024, 11(11), 1116; https://doi.org/10.3390/bioengineering11111116 - 5 Nov 2024
Viewed by 1164
Abstract
A comprehensive analysis of in vitro pumps used in cardiovascular research is provided in this review, with a focus on the characteristics of generated flows and principles of flow generations. The cardiovascular system, vital for nutrient circulation and waste removal, generates complex hemodynamics [...] Read more.
A comprehensive analysis of in vitro pumps used in cardiovascular research is provided in this review, with a focus on the characteristics of generated flows and principles of flow generations. The cardiovascular system, vital for nutrient circulation and waste removal, generates complex hemodynamics critical for endothelial cell function. Cardiovascular diseases (CVDs) could be caused by the disturbances in these flows, including aneurysms, atherosclerosis, and heart defects. In vitro systems simulate hemodynamic conditions on cultured cells in the laboratory to study and evaluate these diseases to advance therapies. Pumps used in these systems can be classified into contact and non-contact types. Contact pumps, such as piston and gear pumps, can generate higher flow rates, but they have a higher risk of contamination due to the direct interaction of pump with the fluid. Non-contact pumps, such as peristaltic and lab-on-disk centrifugal pumps, minimize contamination risks, but they are limited to lower flow rates. Advanced pumps including piezoelectric and I-Cor diagonal pumps are focused on improving the accuracy of flow replication and long-term stability. The operational principles, advantages, and some disadvantages of these pump categories are evaluated in this review, while providing insights for optimizing in vitro cardiovascular models and advancing therapeutic strategies against CVDs. The outcomes of the review elaborate the importance of selecting an appropriate pump system, to accurately replicate cardiovascular flow patterns. Full article
(This article belongs to the Section Biomedical Engineering and Biomaterials)
Show Figures

Figure 1

Figure 1
<p>Working principle of peristaltic pump and examples of modifications. (<b>a</b>) Generic peristaltic pump construction schematic, (<b>b</b>) schematic of a pulsatile flow mimicking setup using a peristaltic pump and one syringe pump [<a href="#B2-bioengineering-11-01116" class="html-bibr">2</a>], (<b>c</b>) schematic diagram of the left ventricle chamber using peristaltic pump and piston pump [<a href="#B3-bioengineering-11-01116" class="html-bibr">3</a>], (<b>d</b>) pulsatile bioreactor system, (<b>e</b>) simple bioreactor setup using peristaltic pump [<a href="#B4-bioengineering-11-01116" class="html-bibr">4</a>], (<b>f</b>) schematic of a novel roller pump for physiological flow mimicking setup [<a href="#B7-bioengineering-11-01116" class="html-bibr">7</a>], (<b>g</b>) schematic diagram illustrates a Braille display pump [<a href="#B6-bioengineering-11-01116" class="html-bibr">6</a>].</p>
Full article ">Figure 1 Cont.
<p>Working principle of peristaltic pump and examples of modifications. (<b>a</b>) Generic peristaltic pump construction schematic, (<b>b</b>) schematic of a pulsatile flow mimicking setup using a peristaltic pump and one syringe pump [<a href="#B2-bioengineering-11-01116" class="html-bibr">2</a>], (<b>c</b>) schematic diagram of the left ventricle chamber using peristaltic pump and piston pump [<a href="#B3-bioengineering-11-01116" class="html-bibr">3</a>], (<b>d</b>) pulsatile bioreactor system, (<b>e</b>) simple bioreactor setup using peristaltic pump [<a href="#B4-bioengineering-11-01116" class="html-bibr">4</a>], (<b>f</b>) schematic of a novel roller pump for physiological flow mimicking setup [<a href="#B7-bioengineering-11-01116" class="html-bibr">7</a>], (<b>g</b>) schematic diagram illustrates a Braille display pump [<a href="#B6-bioengineering-11-01116" class="html-bibr">6</a>].</p>
Full article ">Figure 2
<p>Working principle of osmosis pumps and example modifications. (<b>a</b>) Generic osmosis pump construction schematic, (<b>b</b>) schematic diagrams of electrofusion chip driven by surface tension [<a href="#B10-bioengineering-11-01116" class="html-bibr">10</a>,<a href="#B11-bioengineering-11-01116" class="html-bibr">11</a>], (<b>c</b>) schematic of gradient generation by an osmotic pump setup [<a href="#B12-bioengineering-11-01116" class="html-bibr">12</a>,<a href="#B13-bioengineering-11-01116" class="html-bibr">13</a>].</p>
Full article ">Figure 3
<p>Working principle of non-contact centrifugal pumps and example modifications. (<b>a</b>) Generic centrifugal pump or LAB on disk construction schematic, (<b>b</b>) schematic of cell positioning inside microfluidic device experiment [<a href="#B14-bioengineering-11-01116" class="html-bibr">14</a>], (<b>c</b>) schematic of cell culture system in LAB in disk setup [<a href="#B15-bioengineering-11-01116" class="html-bibr">15</a>].</p>
Full article ">Figure 4
<p>Working principle of centrifugal pump and examples of modifications. (<b>a</b>) Generic centrifugal pump or LAB on disk construction schematic, (<b>b</b>) schematic of cell positioning inside microfluidic devices experiment [<a href="#B28-bioengineering-11-01116" class="html-bibr">28</a>], (<b>c</b>) schematic of cell culture system in LAB in disk setup [<a href="#B29-bioengineering-11-01116" class="html-bibr">29</a>].</p>
Full article ">Figure 4 Cont.
<p>Working principle of centrifugal pump and examples of modifications. (<b>a</b>) Generic centrifugal pump or LAB on disk construction schematic, (<b>b</b>) schematic of cell positioning inside microfluidic devices experiment [<a href="#B28-bioengineering-11-01116" class="html-bibr">28</a>], (<b>c</b>) schematic of cell culture system in LAB in disk setup [<a href="#B29-bioengineering-11-01116" class="html-bibr">29</a>].</p>
Full article ">Figure 5
<p>Working principle of piston pump and examples of modifications. (<b>a</b>) Generic piston pump configuration, (<b>b</b>) schematic of a new pulsatile hydrostatic pressure bioreactor using piston pump [<a href="#B30-bioengineering-11-01116" class="html-bibr">30</a>], (<b>c</b>) schematic of control system and generated flow profiles for in vitro cardiovascular emulation [<a href="#B33-bioengineering-11-01116" class="html-bibr">33</a>], (<b>d</b>) schematic diagram of pulsatile hydrostatic pressure bioreactor components used for vascular tissue-engineered construction [<a href="#B31-bioengineering-11-01116" class="html-bibr">31</a>], (<b>e</b>) schematic of the perfusion-based bioreactor system [<a href="#B32-bioengineering-11-01116" class="html-bibr">32</a>].</p>
Full article ">Figure 5 Cont.
<p>Working principle of piston pump and examples of modifications. (<b>a</b>) Generic piston pump configuration, (<b>b</b>) schematic of a new pulsatile hydrostatic pressure bioreactor using piston pump [<a href="#B30-bioengineering-11-01116" class="html-bibr">30</a>], (<b>c</b>) schematic of control system and generated flow profiles for in vitro cardiovascular emulation [<a href="#B33-bioengineering-11-01116" class="html-bibr">33</a>], (<b>d</b>) schematic diagram of pulsatile hydrostatic pressure bioreactor components used for vascular tissue-engineered construction [<a href="#B31-bioengineering-11-01116" class="html-bibr">31</a>], (<b>e</b>) schematic of the perfusion-based bioreactor system [<a href="#B32-bioengineering-11-01116" class="html-bibr">32</a>].</p>
Full article ">Figure 6
<p>Working principle of diaphragm pump and examples of modifications. (<b>a</b>) Generic construction of diaphragm pump, (<b>b</b>) schematic of the perfusion-based bioreactor system: shows the components of the integrally designed pulsatile perfusion-based bioreactor system developed for the successful creation of small diameter tissue-engineered vascular vessels [<a href="#B37-bioengineering-11-01116" class="html-bibr">37</a>]; (<b>c</b>) schematic of the circulatory mock loop for biventricular device testing: figure presents the schematic of a circulatory mock loop designed for testing biventricular devices under various heart conditions [<a href="#B38-bioengineering-11-01116" class="html-bibr">38</a>].</p>
Full article ">Figure 7
<p>Working principle of syringe pump and examples of modifications. (<b>a</b>) Generic construction of syringe pump, (<b>b</b>) schematic diagram of the experimental system for applying hydrostatic pressure to cells, showing the experimental setup designed to apply hydrostatic pressure (HP) to endothelial cells, mimicking pressure therapy conditions [<a href="#B41-bioengineering-11-01116" class="html-bibr">41</a>].</p>
Full article ">Figure 8
<p>Working Principle and Generic Construction of Vacuum Pump.</p>
Full article ">Figure 9
<p>Working principle of gear pump and examples of modifications. (<b>a</b>) Generic construction of gear pump, (<b>b</b>) schematic diagram of the experimental setup for pulse duplication system, presenting the schematic diagram of the experimental setup for an affordable pulse duplication system designed for in vitro cardiovascular experiments [<a href="#B46-bioengineering-11-01116" class="html-bibr">46</a>], (<b>c</b>) schematic diagram of the flow pumping system for physiological waveforms, showing the schematic diagram of the flow pumping system designed to generate physiological waveforms using conjugated operation of gear pump and piston pump [<a href="#B45-bioengineering-11-01116" class="html-bibr">45</a>].</p>
Full article ">Figure 10
<p>Working Principle and Generic Construction and Operation of Piezo Pump [<a href="#B52-bioengineering-11-01116" class="html-bibr">52</a>].</p>
Full article ">Figure 11
<p>Working Principle and Generic Construction of I-Cor Diagonal Pump [<a href="#B55-bioengineering-11-01116" class="html-bibr">55</a>].</p>
Full article ">Figure 12
<p>Working principle and schematic of the fluidic device for SIS-based vascular graft evaluation, presenting the schematic of the fluidic device and the necessary accessories for cell seeding on the SIS-based vascular graft [<a href="#B57-bioengineering-11-01116" class="html-bibr">57</a>].</p>
Full article ">
11 pages, 11641 KiB  
Article
Comparative Analysis of Blood Transfusion Accuracy and Hemolysis Rate of Transfusion Cartridge Set Between Conventional Infusion Pumps and Cylinder-Type Infusion Pumps
by Hee-Young Lee, Sun-Ju Kim, Kang-Hyun Lee, Il-Hwan Park, Hyeok-Jin Jeon and Hyun Youk
Biomedicines 2024, 12(11), 2421; https://doi.org/10.3390/biomedicines12112421 - 22 Oct 2024
Viewed by 1082
Abstract
Background: Infusion pumps are critical in delivering fluids, including medications and blood products, in controlled amounts. However, conventional pumps can cause hemolysis and other issues such as flow variations and infection risks, especially during blood transfusions. To address these limitations, a novel cylinder-type [...] Read more.
Background: Infusion pumps are critical in delivering fluids, including medications and blood products, in controlled amounts. However, conventional pumps can cause hemolysis and other issues such as flow variations and infection risks, especially during blood transfusions. To address these limitations, a novel cylinder-type infusion pump, the Anyfusion H-100, was developed, which includes a specialized blood transfusion cartridge set that combines syringe and peristaltic infusion methods. This study evaluates the accuracy and hemolysis rates of the Anyfusion H-100 compared to conventional pumps, aiming to confirm its viability as a safe and effective medical device for blood transfusions. Methods: This study evaluated six different infusion rates (10–180 cc/hr) and conducted 57 transfusion trials, 20 of which used a 1:1 blood–saline dilution. Blood transfusion accuracy was measured using research-grade packed red blood cells, and hemolysis rates were assessed before and after transfusion by chi-square tests and independent sample t-tests. Results: Anyfusion demonstrated an average transfusion error rate of 3.77%, compared to 4.00% for the Terufusion, with no statistically significant difference in hemolysis rates (p = 0.697). Bland–Altman plots confirmed their equivalent performance, with hemolysis rates of 0.566 ± 0.095% for Anyfusion and 0.518 ± 0.126% for Terufusion. Conclusions: Anyfusion provides an accurate and reliable blood transfusion performance comparable to that of Terufusion, with no significant difference in hemolysis rates; its integration of syringe and infusion methods shows a potential for safer and more efficient transfusion practices, especially in pediatric and emergency settings. Full article
(This article belongs to the Section Biomedical Engineering and Materials)
Show Figures

Figure 1

Figure 1
<p>Photo of a cylinder-type infusion pump and transfusion cartridge set.</p>
Full article ">Figure 2
<p>Photo of a cylinder cartridge fixed to the infusion pump body with an automatic locking system and mechanism of cylinder-type pump.</p>
Full article ">Figure 3
<p>Infusion pump medical devices. (<b>a</b>) Test device (Anyfusion H-100); (<b>b</b>) comparator (Terufusion TE-LM700).</p>
Full article ">Figure 4
<p>Research-grade blood products supplied by Korean Red Cross Blood Services.</p>
Full article ">Figure 5
<p>Flowchart of this study using the research-grade blood products for comparative analysis of blood transfusion accuracy and hemolysis rate.</p>
Full article ">Figure 6
<p>Agreement of hemolysis rates between two medical devices using Bland–Altman plot.</p>
Full article ">
17 pages, 6040 KiB  
Article
3D Printed Microfluidic Separators for Solid/Liquid Suspensions
by Marijan-Pere Marković, Krunoslav Žižek, Ksenija Soldo, Vjeran Sunko, Julijan Zrno and Domagoj Vrsaljko
Appl. Sci. 2024, 14(17), 7856; https://doi.org/10.3390/app14177856 - 4 Sep 2024
Viewed by 1060
Abstract
This study investigates the fabrication of 3D-printed microfluidic devices for solid/liquid separation, focusing on additive manufacturing technologies. Stereolithography (SLA) and fused filament fabrication (FFF) were used to create microseparators with intricate designs optimized for separation efficiency. Model suspensions containing quartz sand, nano-calcium carbonate, [...] Read more.
This study investigates the fabrication of 3D-printed microfluidic devices for solid/liquid separation, focusing on additive manufacturing technologies. Stereolithography (SLA) and fused filament fabrication (FFF) were used to create microseparators with intricate designs optimized for separation efficiency. Model suspensions containing quartz sand, nano-calcium carbonate, and talc-based baby powder in water were prepared using an electric magnetic stirrer and conveyed into the microseparator via a peristaltic pump. Different flow rates were tested to evaluate their influence on the separation efficiency. The highest separation efficiency for the model systems was observed at a flow rate of 200 mL min−1. This was due to the increased turbulence at higher flow rates, which hindered the secondary flow perpendicular to the primary flow direction. The particle size distribution before and after separation was analyzed using sieve and laser diffraction, and particle morphology was inspected by scanning electron microscopy. The laser diffraction analysis revealed post-separation particle size distributions, indicating that Outlet 1 (external stream) consistently captured larger particles more effectively than Outlet 2 (internal stream). This work highlights the potential of additive manufacturing to produce customized microfluidic devices, enabling rapid prototyping and fine-tuning of complex geometries, thus enhancing separation efficiency across various industrial applications. Full article
Show Figures

Figure 1

Figure 1
<p>Separator model with dimensions in millimeters.</p>
Full article ">Figure 2
<p>3D-printed separators: SLA (<b>left</b>) and FFF with added blue color for easier channel visualization (<b>right</b>).</p>
Full article ">Figure 3
<p>PSD of pre-separated baby powder; sieve analysis (<b>left</b>) and laser diffraction analysis (<b>right</b>).</p>
Full article ">Figure 4
<p>SEM micrographs of the baby powder sample at 200× (<b>left</b>) and 2000× (<b>right</b>) magnification.</p>
Full article ">Figure 5
<p>PSD of pre-separated calcium carbonate; sieve analysis (<b>left</b>) and laser diffraction analysis (<b>right</b>).</p>
Full article ">Figure 6
<p>SEM micrographs of the calcium carbonate sample at 1000× (<b>left</b>) and 5000× (<b>right</b>) magnification.</p>
Full article ">Figure 7
<p>PSD of pre-separated quartz sand; sieve analysis (<b>left</b>) and laser diffraction analysis (<b>right</b>).</p>
Full article ">Figure 8
<p>SEM micrographs of the quartz sand sample at 30× (<b>left</b>) and 100× (<b>right</b>) magnification.</p>
Full article ">Figure 9
<p>PSDs of baby powder before and after separation at 200 mL min<sup>−1</sup>; laser diffraction analysis.</p>
Full article ">Figure 10
<p>PSDs of calcium carbonate before and after separation at 200 mL min<sup>−1</sup>; laser diffraction analysis.</p>
Full article ">Figure 11
<p>PSDs of quartz sand before and after separation at 200 mL min<sup>−1</sup>; laser diffraction analysis.</p>
Full article ">
12 pages, 950 KiB  
Article
Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel
by Mary G. Thoubaan, Dheia G. Salih Al-Khafajy, Abbas Kareem Wanas, Daniel Breaz and Luminiţa-Ioana Cotîrlă
Symmetry 2024, 16(9), 1144; https://doi.org/10.3390/sym16091144 - 3 Sep 2024
Viewed by 890
Abstract
This study aims to analyze how the parameter flow rate and amplitude of walling waves affect the peristaltic flow of Jeffrey’s fluid through an irregular channel. The movement of the fluid is described by a set of non-linear partial differential equations that consider [...] Read more.
This study aims to analyze how the parameter flow rate and amplitude of walling waves affect the peristaltic flow of Jeffrey’s fluid through an irregular channel. The movement of the fluid is described by a set of non-linear partial differential equations that consider the influential parameters. These equations are transformed into non-dimensional forms with appropriate boundary conditions. The study also utilizes dynamic systems theory to analyze the effects of the parameters on the streamline and to investigate the position of critical points and their local and global bifurcation of flow. The research presents numerical and analytical methods to illustrate the impact of flow rate and amplitude changes on fluid transport. It identifies three types of streamline patterns that occur: backwards, trapping, and augmented flow resulting from changes in the value of flow rate parameters. Full article
Show Figures

Figure 1

Figure 1
<p>The graphical representation of the geometry of a wall surface.</p>
Full article ">Figure 2
<p>A streamline pattern is shown for system (<a href="#FD18-symmetry-16-01144" class="html-disp-formula">18</a>) with different value flow rates <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> with constant amplitude <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. Panel (<b>A<sub>1</sub></b>) shows the backward flow of fluid with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> and there are just saddle points. The Panels (<b>A<sub>2</sub></b>,<b>A<sub>3</sub></b>) display two significant changes physically and dynamically with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.089</mn> <mo>,</mo> <mo> </mo> <mn>1.3</mn> </mrow> </semantics></math> birth red/black critical points refer to center and saddle points, respectively. Also, the appearance of a trapping zone is inside the heteroclinic connection that is created between two different saddle points. The last panel, (<b>A<sub>4</sub></b>), illustrates augmented flow where <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> </mrow> </semantics></math> and changes the number of saddle points with a new formation heteroclinic connection.</p>
Full article ">Figure 3
<p>(The description is as for <a href="#symmetry-16-01144-f002" class="html-fig">Figure 2</a>, except for the change in the value of the parameter <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>). Streamline patterns for different value flow rates <math display="inline"><semantics> <msub> <mi>Q</mi> <mn>1</mn> </msub> </semantics></math> with constant amplitude <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. Panel (<b>B<sub>1</sub></b>) shows the backward flow of fluid with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>, and there are just saddle points. Panels (<b>B<sub>2</sub></b>,<b>B<sub>3</sub></b>) display two significant changes physically and dynamically with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>1.3</mn> </mrow> </semantics></math> birth red/black critical points refer to center and saddle points, respectively. In addition, there is an appearance of a trapping zone inside the heteroclinic connection that is created between two different saddle points. The last panel, (<b>B<sub>4</sub></b>), illustrates augmented flow where <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> and changes the number of saddle points with a new formation heteroclinic connection.</p>
Full article ">Figure 4
<p>This figure shows the bifurcation diagram with <math display="inline"><semantics> <msub> <mi>Q</mi> <mn>1</mn> </msub> </semantics></math> against <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> for Equation (<a href="#FD18-symmetry-16-01144" class="html-disp-formula">18</a>) with the various values of parameters <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mo> </mo> <mn>0.7</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> </mrow> </semantics></math> in the panels (<b>A</b>, <b>B</b> and <b>C</b>), respectively. Red/black lines refer to stable/unstable points, green indicates periodic orbit, and a single saddle node is represented by a blue line. At (<b>E</b>), there are two branch lines of saddle-node bifurcation (dash–circle lines) and three regions have a light color (blue, yellow, green) that indicate the existence of three complicated behaviors (backward, trapping, augment), respectively. The symbols I, III, and IV are the number of critical points in every zone.</p>
Full article ">
24 pages, 19259 KiB  
Article
Synergistic Exploration of Heat Transfer for Integration Magnetohydrodynamics of Nanofluids Peristaltic Transport within Annular Tubes
by Muhammad Magdy, Ramzy Abumandour, Islam Eldesoky and Hammad Alotaibi
Mathematics 2024, 12(13), 2024; https://doi.org/10.3390/math12132024 - 29 Jun 2024
Viewed by 1097
Abstract
The problem of treating cancer is considered one of the most important daily challenges that affect the lives of people with cancer. This research deals with solving this problem theoretically. Through previous studies, it has been proven that gold nanoparticles are able to [...] Read more.
The problem of treating cancer is considered one of the most important daily challenges that affect the lives of people with cancer. This research deals with solving this problem theoretically. Through previous studies, it has been proven that gold nanoparticles are able to remove these cancer cells. The idea of this research is theoretically based on injecting a cancer patient with gold nanoparticles that are exposed to a magnetic field. When these particles penetrate cancerous cells and are exposed to a magnetic field, this causes their temperature to rise. The high temperature of the nanometer gold particles that penetrate the cells of the affected body leads to the explosion of the cancer cells. In this research, the various external forces that affect the flow movement of the nanofluid are studied and how its physical and thermal properties are affected by those external forces. The MHD peristaltic flow of a nanofluid in an annulus pipe as a result of the effect of the wall properties has been investigated. This has been achieved through slip and thermal conditions. Wave velocity u0 leads to flow development. The inner annulus wall is rigid, while the outer wall of the artery moves under the influence of wave peristaltic movement. The nonlinear equations that describe the flow are solved under long-wavelength assumptions. The results were compared with other numerical methods, such as finite volume and finite element and the long wavelength method and proved to be accurate and effective. The expressions of pressure difference, velocity, stream function, wall shear stress, and temperature are analyzed. It is noted that the flow velocity increases with the Knudsen number, and the increased source heat suggests an increased temperature. The increasing amplitude ratio at most of the interface points between the artery wall and the catheter results in increased velocity. The streamlines are affected by the magnetic field, as increasing the influencing magnetic field leads to a decrease in flow lines. It is observed that this stress decreases when nanoparticles increase, in contrast to the effect of the magnetic field and also the occurrence of slipping. It was found that the mass of the wall cells relative to their area works to decrease the pressure difference, in contrast to the tension between those cells, which works to increase the pressure difference. Without slipping Kn=0 and with slipping Kn=0.1, the temperature decreases with increasing in nanoparticle concentration φ. The temperature also increases with the amplitude ratio δ. This strongly affects the generated drag on the catheter wall, which is mainly responsible for the enhanced temperature on this wall. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 2nd Edition)
Show Figures

Figure 1

Figure 1
<p>Mathematical formulation of the catheterized artery.</p>
Full article ">Figure 2
<p>Velocity with artery wall damping (<span class="html-italic">A</span><sub>3</sub>) at (<span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">K</span>n = 0, <span class="html-italic">A</span><sub>2</sub> = 0.01, φ = 0, Ha = 1.5, δ = 0.2, 1/k = 0, J = 0).</p>
Full article ">Figure 3
<p>Velocity with different artery locations at (<span class="html-italic">K</span>n = 0, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, φ = 0.2, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1).</p>
Full article ">Figure 4
<p>Velocity distribution with wall amplitude ratio (δ) at (φ = 0.2, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1 <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05). (<b>a</b>) <span class="html-italic">K</span>n = 0.0; (<b>b</b>) <span class="html-italic">K</span>n = 0.15.</p>
Full article ">Figure 5
<p>Velocity with catheter size (J) at (<span class="html-italic">K</span>n = 0.15, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, φ = 0.2, <span class="html-italic">Ha</span> = 3, k = 0.1, δ = 0.1).</p>
Full article ">Figure 6
<p>Velocity with magnetic field (<span class="html-italic">Ha</span>) at (<span class="html-italic">K</span>n = 0.15, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, k = 0.1, φ = 0.2, J = 0.1, δ = 0.1).</p>
Full article ">Figure 7
<p>Velocity with Knudsen number at (<span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, φ = 0.2, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1, δ = 0.1).</p>
Full article ">Figure 8
<p>Velocity with artery wall damping (<span class="html-italic">A</span><sub>3</sub>) at (<span class="html-italic">K</span>n = 0.15, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, φ = 0.2, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1, δ = 0.1).</p>
Full article ">Figure 9
<p>Velocity with artery wall tension (<span class="html-italic">A</span><sub>1</sub>) at (<span class="html-italic">K</span>n = 0.15, φ = 0.2, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">Ha</span> = 3, k = 0.1, <span class="html-italic">A</span><sub>2</sub> = 0.05, J = 0.1, δ = 0.1).</p>
Full article ">Figure 10
<p>Velocity with mass per unit area of the artery (<span class="html-italic">A</span><sub>2</sub>) at (<span class="html-italic">K</span>n = 0.15, k = 0.1, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, J = 0.1, φ = 0.2, <span class="html-italic">Ha</span> = 3, δ = 0.1).</p>
Full article ">Figure 11
<p>Velocity with nanoparticle concentration (φ)at (<span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1, δ = 0.1, <span class="html-italic">K</span>n = 0.15).</p>
Full article ">Figure 12
<p>Velocity with permeability (1/k) at (<span class="html-italic">K</span>n = 0, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 0.1, <span class="html-italic">φ</span> = 0.2, <span class="html-italic">Ha</span> = 3, J = 0.1, δ = 0.1).</p>
Full article ">Figure 13
<p>Wall wave and catheter motions at (<span class="html-italic">K</span>n = 0, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.01, <span class="html-italic">A</span><sub>3</sub> = 0.1, <span class="html-italic">φ</span> = 0.02, k = 0.1, J = 0.2, δ = 0.2).</p>
Full article ">Figure 14
<p>Streamlines with Knudsen number at (δ = 0.1, <span class="html-italic">φ</span> = 0.2, <span class="html-italic">Ha</span> = 3, <span class="html-italic">A</span><sub>1</sub> = 1, k = 0.1, <span class="html-italic">A</span><sub>2</sub> = 0.02, J = 0.1, <span class="html-italic">A</span><sub>3</sub> = 1). (<b>a</b>) <span class="html-italic">K</span>n = 0; (<b>b</b>) <span class="html-italic">K</span>n = 0.15.</p>
Full article ">Figure 15
<p>Streamlines with Hartman number at (δ = 0.1, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">K</span>n = 0.15, <span class="html-italic">A</span><sub>2</sub> = 0.02, <span class="html-italic">φ</span> = 0.2, k = 0.1, <span class="html-italic">A</span><sub>3</sub> = 1, J = 0.1). (<b>a</b>) <span class="html-italic">Ha</span> = 1; (<b>b</b>) <span class="html-italic">Ha</span> = 3.</p>
Full article ">Figure 16
<p>Streamlines with wall damping at (<span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.02, <span class="html-italic">Ha</span> = 3, δ = 0.1, <span class="html-italic">K</span>n = 0.15, <span class="html-italic">φ</span> = 0.2, k = 0.1, J = 0.1). (<b>a</b>) <span class="html-italic">A</span><sub>3</sub> = 0; (<b>b</b>) <span class="html-italic">A</span><sub>3</sub> = 2.</p>
Full article ">Figure 17
<p>Streamlines with artery wall tension (<span class="html-italic">A</span><sub>1</sub>) at (<span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.02, <span class="html-italic">Ha</span> = 3, δ = 0.1, <span class="html-italic">K</span>n = 0.15, <span class="html-italic">φ</span> = 0.2, k = 0.1, J = 0.1). (<b>a</b>) <span class="html-italic">A</span><sub>1</sub> = 0; (<b>b</b>) <span class="html-italic">A</span><sub>1</sub> = 1.</p>
Full article ">Figure 18
<p>Streamlines with mass per unit area of the artery (<span class="html-italic">A</span><sub>2</sub>) at (<span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">Ha</span> = 3, δ = 0.1, <span class="html-italic">K</span>n = 0.15, <span class="html-italic">φ</span> = 0.2, k = 0.1, J = 0.1). (<b>a</b>) <span class="html-italic">A</span><sub>2</sub> = 0; (<b>b</b>) <span class="html-italic">A</span><sub>2</sub> = 0.05.</p>
Full article ">Figure 19
<p>Streamlines with nanofluid concentration (<span class="html-italic">φ)</span> at (<span class="html-italic">Ha</span> = 3, δ = 0.1, <span class="html-italic">A</span><sub>2</sub> = 0.02, <span class="html-italic">K</span>n = 0.15, <span class="html-italic">A</span><sub>3</sub> = 1, k = 0.1, <span class="html-italic">A</span><sub>1</sub> = 1, J = 0.1). (<b>a</b>) <span class="html-italic">φ</span> = 0; (<b>b</b>) <span class="html-italic">φ</span> = 0.3.</p>
Full article ">Figure 20
<p>Shear stress at the artery wall with (<span class="html-italic">φ</span>) at (<span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, <span class="html-italic">Ha</span> = 3, δ = 0.1, <span class="html-italic">K</span>n = 0, k = 0.1, J = 0.1).</p>
Full article ">Figure 21
<p>Shear stress at the artery wall with <span class="html-italic">Ha</span> at (<span class="html-italic">φ</span> = 0.2, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, δ = 0.1, <span class="html-italic">K</span>n = 0, k = 0.1, J = 0.1).</p>
Full article ">Figure 22
<p>Shear stress at the artery wall with <span class="html-italic">K</span>n at (<span class="html-italic">φ</span> = 0.2, <span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, δ = 0.1, <span class="html-italic">Ha</span> = 3, k = 0.1, J = 0.1).</p>
Full article ">Figure 23
<p>Pressure difference with artery wall damping <span class="html-italic">(A</span><sub>3</sub>) at (<span class="html-italic">A</span><sub>2</sub> = 0.05, <span class="html-italic">A</span><sub>1</sub> = 1, δ = 0.1, J = 0.1).</p>
Full article ">Figure 24
<p>Pressure difference with mass per unit area of the artery <span class="html-italic">(A</span><sub>2</sub>) at (<span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>1</sub> = 1, δ = 0.1, J = 0.1).</p>
Full article ">Figure 25
<p>Pressure difference with artery wall tension <span class="html-italic">(A</span><sub>1</sub>) at (<span class="html-italic">A</span><sub>3</sub> = 1, <span class="html-italic">A</span><sub>2</sub> = 0.05, δ = 0.1, J = 0.1).</p>
Full article ">Figure 26
<p>Temperature distribution with the axial coordinate (z) at (δ = 0.05, <span class="html-italic">φ</span> = 0.02, J = 0.2, S = 0.1).</p>
Full article ">Figure 27
<p>Temperature distribution with concentration of nanoparticles (<span class="html-italic">φ</span>) at (δ = 0.05, J = 0.2, S = 0.1).</p>
Full article ">Figure 28
<p>Temperature distribution with the amplitude ratio (δ) at (φ = 0.02, S = 0.1, J = 0.2).</p>
Full article ">Figure 29
<p>Temperature distribution with catheter size (J) at (δ = 0.05, S = 0.1, φ = 0.02).</p>
Full article ">Figure 30
<p>Temperature distribution with heat source at (<span class="html-italic">φ</span> = 0.02, J = 0.2, δ = 0.05).</p>
Full article ">
25 pages, 18531 KiB  
Article
The Impact of Heat Transfer and a Magnetic Field on Peristaltic Transport with Slipping through an Asymmetrically Inclined Channel
by Muhammad Magdy, Ahmed G. Nasr, Ramzy M. Abumandour and Mohammed A. El-Shorbagy
Mathematics 2024, 12(12), 1827; https://doi.org/10.3390/math12121827 - 12 Jun 2024
Cited by 1 | Viewed by 854
Abstract
This theoretical investigation explores the intricate interplay of slip, heat transfer, and magneto-hydrodynamics (MHD) on peristaltic flow within an asymmetrically inclined channel. The channel walls exhibit sinusoidal undulations to simulate flexibility. The governing equations for continuity, momentum, and energy are utilized to mathematically [...] Read more.
This theoretical investigation explores the intricate interplay of slip, heat transfer, and magneto-hydrodynamics (MHD) on peristaltic flow within an asymmetrically inclined channel. The channel walls exhibit sinusoidal undulations to simulate flexibility. The governing equations for continuity, momentum, and energy are utilized to mathematically represent the flow dynamics. Employing the perturbation method, these nonlinear equations are systematically solved, yielding analytical expressions for key parameters such as stream function, temperature distribution, and pressure gradient. This study meticulously examines the influence of various physical parameters on flow characteristics, presenting comprehensive visualizations of flow streamlines, fluid axial velocity profiles, and pressure gradient distributions. Noteworthy findings include the observation that the axial velocity of the fluid increases by 55% when the slip parameter is increased from 0 to 0.1, indicative of enhanced fluid transport. Furthermore, the analysis reveals that the pressure gradient amplifies by 80% with increased magnetic field strength from 0.5 to 4, underscoring the significant role of MHD effects on overall flow behavior. In essence, this investigation elucidates the complex dynamics of peristaltic flow in an asymmetrically inclined channel under the combined influence of slip, heat transfer, and magnetohydrodynamics. It sheds light on fundamental mechanisms that govern fluid dynamics in complex geometries and under diverse physical conditions. Full article
(This article belongs to the Special Issue Mathematical Modeling for Fluid Mechanics)
Show Figures

Figure 1

Figure 1
<p>Geometry of the problem.</p>
Full article ">Figure 2
<p>The comparison of axial velocity (<span class="html-italic">u</span>) with the results of Kothandapani and Srinivas for <span class="html-italic">M</span> = 0, <span class="html-italic">Gr</span> = 0, and <span class="html-italic">Kn</span> = 0.</p>
Full article ">Figure 3
<p>The variation in axial velocity (u) for different cross-sections: <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 4
<p>The effect of magnetic field (<span class="html-italic">M</span>) on the flow velocities at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 5
<p>The effect of magnetic field (M) on the flow pressure gradient at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 6
<p>The effect of magnetic field (<span class="html-italic">M</span>) on the flow pressure difference at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 7
<p>The effect of magnetic field (<span class="html-italic">M</span>) on the streamlines at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 8
<p>The effect of the walls’ amplitude (<span class="html-italic">a</span>) on the flow behavior at <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 9
<p>The effect of the walls’ amplitude (<span class="html-italic">a</span>) on the flow pressure gradient at <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 10
<p>The effect of the walls’ amplitude (<span class="html-italic">a</span>) on the flow pressure difference at <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 11
<p>The effect of the walls’ amplitude (<span class="html-italic">a</span>) on the streamlines at <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">S</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 12
<p>The effect of a heat source/sink (<math display="inline"><semantics> <mi>S</mi> </semantics></math>) on the flow velocity on the flow behavior at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">M</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 13
<p>The effect of a heat source /sink (S) on the flow pressure difference on the flow behavior at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">M</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 14
<p>The effect of a eat source /sink (<math display="inline"><semantics> <mi>S</mi> </semantics></math>) on the temperature distribution on the flow behavior at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">M</span> = 1, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01.</p>
Full article ">Figure 15
<p>The effect of geometric parameters (δ) on the flow pressure gradient at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">S</span> = 1, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 16
<p>The effect of gravity (Gr) on the flow pressure difference at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">S</span> = 1, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 17
<p>The influence of thermal radiation (Nr) on the flow pressure gradient at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Gr</span> = 2, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">S</span> = 1, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 18
<p>The influence of thermal radiation (<span class="html-italic">Nr</span>) on the temperature distribution at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <span class="html-italic">π</span>/4, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Gr</span> = 2, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">S</span> = 1, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 19
<p>The variation of pressure difference (Δp) with inclines of the pipe (<span class="html-italic">ϕ</span>) at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Q</span> = 2.4, <span class="html-italic">Gr</span> = 2, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">Gr</span> = 2, <span class="html-italic">S</span> = 1, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 20
<p>The effect of flow rate (<span class="html-italic">Q</span>) on the flow velocity at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 21
<p>The effect of flow rate (<span class="html-italic">Q</span>) on the flow pressure gradient at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 22
<p>The effect of flow rate (<span class="html-italic">Q</span>) on the flow pressure difference at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 23
<p>The effect of flow rate (<span class="html-italic">Q</span>) on the streamlines at <span class="html-italic">a</span> = 0.1, <span class="html-italic">d</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 24
<p>The effect of slipping (<span class="html-italic">Kn</span>) on the flow velocities at <span class="html-italic">a</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 25
<p>The effect of slipping (<span class="html-italic">Kn</span>) on the flow pressure gradient at <span class="html-italic">a</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">Figure 26
<p>The effect of slipping (<span class="html-italic">Kn</span>) on the flow pressure difference gradient at <span class="html-italic">a</span> = 1, <span class="html-italic">ϕ</span> = <math display="inline"><semantics> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </semantics></math>, <span class="html-italic">S</span> = 1, <span class="html-italic">Nr</span> = 0.8, <span class="html-italic">Gr</span> = 2, <span class="html-italic">Re</span> = 0.005, <span class="html-italic">Pr</span> = 0.7, <span class="html-italic">δ</span> = 0.01, <span class="html-italic">M</span> = 1.</p>
Full article ">
18 pages, 7532 KiB  
Article
A Novel and Self-Calibrating Weighing Sensor with Intelligent Peristaltic Pump Control for Real-Time Closed-Loop Infusion Monitoring in IoT-Enabled Sustainable Medical Devices
by Chiang Liang Kok, Chee Kit Ho, Yuwei Dai, Teck Kheng Lee, Yit Yan Koh and Jian Ping Chai
Electronics 2024, 13(9), 1724; https://doi.org/10.3390/electronics13091724 - 30 Apr 2024
Cited by 4 | Viewed by 1631
Abstract
Technological advancements are propelling medical technology towards automation through the application and widespread use of automatic control, sensing, and Internet of Things (IoT) technologies. Currently, IoT technology has been extensively applied in medical devices, aiming to ensure patient safety through more real-time detection [...] Read more.
Technological advancements are propelling medical technology towards automation through the application and widespread use of automatic control, sensing, and Internet of Things (IoT) technologies. Currently, IoT technology has been extensively applied in medical devices, aiming to ensure patient safety through more real-time detection and more effective management. In the monitoring of intravenous infusion, accurately sensing the infusion conditions in real time is particularly important. This article introduces a low-cost smart infusion device based on IoT technology, which controls the infusion rate with a peristaltic pump and monitors the volume of fluid delivered. It uses an improved, self-calibrating weighing sensor to achieve the real-time closed-loop control of the flow rate, ensuring patient safety. Additionally, the Blynk dashboard can be used for monitoring and controlling the flow rate and infusion volume. Full article
(This article belongs to the Section Circuit and Signal Processing)
Show Figures

Figure 1

Figure 1
<p>Overall block diagram of the project [<a href="#B26-electronics-13-01724" class="html-bibr">26</a>].</p>
Full article ">Figure 2
<p>Weight scale calibration condition.</p>
Full article ">Figure 3
<p>Reading drift vs. time graph.</p>
Full article ">Figure 4
<p>Variance vs. time range.</p>
Full article ">Figure 5
<p>Offset calibration setup.</p>
Full article ">Figure 6
<p>Drift vs. time with offset calibration.</p>
Full article ">Figure 7
<p>Reading drift vs. time with scale factor calibration.</p>
Full article ">Figure 8
<p>Variance vs. time range.</p>
Full article ">Figure 9
<p>CAD model for the peristaltic pump.</p>
Full article ">Figure 10
<p>Fluid displacement per revolution fitting result.</p>
Full article ">Figure 11
<p>Block diagram of calibration process and integration with IoT.</p>
Full article ">Figure 12
<p>Mobile Blynk IoT user interface.</p>
Full article ">Figure 13
<p>Unfiltered (<b>Left</b>) and filtered (<b>Right</b>) reading drift with no load.</p>
Full article ">Figure 14
<p>Reading deviation after bootup vs. days.</p>
Full article ">Figure 15
<p>0.1 mL/min actual flow rate reading.</p>
Full article ">Figure 16
<p>0.5 mL/min actual flow rate reading.</p>
Full article ">Figure 17
<p>1 mL/min actual flow rate reading.</p>
Full article ">Figure 18
<p>Overall measurement setup for weight sensor feedback.</p>
Full article ">Figure 19
<p>Overall measurement setup for opposed laser photoelectric sensor feedback.</p>
Full article ">
17 pages, 2039 KiB  
Article
Multivariate Peristalsis in a Straight Rectangular Duct for Carreau Fluids
by Iosif C. Moulinos, Christos Manopoulos and Sokrates Tsangaris
Computation 2024, 12(3), 62; https://doi.org/10.3390/computation12030062 - 20 Mar 2024
Viewed by 1460
Abstract
Peristaltic flow in a straight rectangular duct is examined imposed by contraction pulses implemented by pairs of horizontal cylindrical segments with their axes perpendicular to the flow direction. The wave propagation speed is considered in such a range that triggers a laminar fluid [...] Read more.
Peristaltic flow in a straight rectangular duct is examined imposed by contraction pulses implemented by pairs of horizontal cylindrical segments with their axes perpendicular to the flow direction. The wave propagation speed is considered in such a range that triggers a laminar fluid motion. The setting is analyzed over a set of variables which includes the propagation speed, the relative occlusion, the modality of the squeezing pulse profile and the Carreau power index. The numerical solution of the equations of motion on Cartesian meshes is grounded in the immersed boundary method. An increase in the peristaltic pulse modality leads to the reduction in the shear rate levels on the central tube axis and to the movement of the peristaltic characteristics to higher pressure values. The effect of the no slip side walls (NSSWs) is elucidated by the collation with relevant results for the flow field produced under the same assumptions though with slip side walls (SSWs). Shear thinning behavior exhibits a significantly larger effect on transport efficiency for the NSSWs duct than on the SSWs duct. Full article
(This article belongs to the Section Computational Engineering)
Show Figures

Figure 1

Figure 1
<p>Rectangular duct and the contraction wave in a trimetric view. The upper symmetric half with reference to the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> plane is depicted. The squeezing pulse profile is a bimodal function of <span class="html-italic">x</span> with respect to the <span class="html-italic">y</span> direction. Yellow: inlet, green: wall, orange: excitation area of peristaltic wave.</p>
Full article ">Figure 2
<p>Shape of the developed velocity profile, <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> in the rectangular duct under consideration, with <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> mm and <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> mm and for <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>800</mn> </mrow> </semantics></math><math display="inline"><semantics> <mfrac> <mrow> <mi>m</mi> <msup> <mi>m</mi> <mn>3</mn> </msup> </mrow> <mi>s</mi> </mfrac> </semantics></math>.</p>
Full article ">Figure 3
<p>The convergence rate with respect to the number (#) of iterations. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, the modality is <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>90</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 4
<p>Streamwise velocity component magnitude for multiple squeezing cylindrical segments pulse at the middle of the width and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> mm for the case with no slip side walls (NSSWs) and for the case with slip side walls (SSWs) [<a href="#B35-computation-12-00062" class="html-bibr">35</a>]. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 5
<p>The streamwise velocity profile at <span class="html-italic">x</span> direction for a set of streamwise positions in the middle plane <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm. The unit for <span class="html-italic">x</span> is also mm.</p>
Full article ">Figure 6
<p>The vertical velocity profile at <span class="html-italic">y</span> direction for a set of streamwise positions in the middle plane <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm. The unit for <span class="html-italic">x</span> is also mm.</p>
Full article ">Figure 7
<p>Pressure for multiple squeezing cylindrical segments pulse at the middle of the width and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> mm for the case with no slip side walls (NSSWs) and for the case with slip side walls (SSWs) [<a href="#B35-computation-12-00062" class="html-bibr">35</a>]. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 8
<p>Shear rate for multiple squeezing cylindrical segments pulse at the middle of the width and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> mm for the case with no slip side walls (NSSWs) and for the case with slip side walls (SSWs) [<a href="#B35-computation-12-00062" class="html-bibr">35</a>]. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 9
<p>Shear rate as a function of height for various <span class="html-italic">x</span> positions in the middle plane <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the power index is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>180</mn> </mrow> </semantics></math> mm. The unit for <span class="html-italic">x</span> is also mm.</p>
Full article ">Figure 10
<p>Streamwise velocity component magnitude for Carreau–Yasuda fluids, at the middle of the width and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> mm for the case with no slip side walls (NSSWs) and for the case with slip side walls (SSWs) [<a href="#B35-computation-12-00062" class="html-bibr">35</a>]. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the modality is <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>90</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 11
<p>Pressure for Carreau–Yasuda fluids, at the middle of the width and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> mm for the case with no slip side walls (NSSWs) and for the case with slip side walls (SSWs) [<a href="#B35-computation-12-00062" class="html-bibr">35</a>]. The relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, the peristaltic wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> <mfrac> <mi>mm</mi> <mi mathvariant="normal">s</mi> </mfrac> </mrow> </semantics></math>, the modality is <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and the length is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>90</mn> </mrow> </semantics></math> mm.</p>
Full article ">Figure 12
<p>Peristaltic characteristics for various combinations of the values of the parameters. Results for duct of the same geometry with slip side walls are presented [<a href="#B35-computation-12-00062" class="html-bibr">35</a>].</p>
Full article ">Figure 13
<p>The streamwise velocity field at selected crosswise sections for Newtonian fluids. One cylindrical segment centered at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>45</mn> </mrow> </semantics></math> mm is placed at an <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>90</mn> </mrow> </semantics></math> mm duct. The wave speed is <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> mm/s, and the relative occlusion is <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>O</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>. A quarter of the cross-section is presented.</p>
Full article ">
14 pages, 539 KiB  
Article
Viscous Dissipation and Mixed Convection Effects on the Induced Magnetic Field for Peristaltic Flow of a Jeffrey Nanofluid
by Borhen Halouani and Khalid Nowar
Symmetry 2024, 16(3), 329; https://doi.org/10.3390/sym16030329 - 8 Mar 2024
Viewed by 1142
Abstract
The issue of Jeffrey nanofluid peristaltic flow in an asymmetric channel being affected by an induced magnetic field was studied. In addition, mixed convection and viscous dissipation were considered. Under the supposition of a long wave length and a low Reynolds number, the [...] Read more.
The issue of Jeffrey nanofluid peristaltic flow in an asymmetric channel being affected by an induced magnetic field was studied. In addition, mixed convection and viscous dissipation were considered. Under the supposition of a long wave length and a low Reynolds number, the problem was made simpler. The system and corresponding boundary conditions were solved numerically by using the built-in package NDSolve in Mathematica software. This software ensures that the boundary value problem solution is accurate when the step size is set appropriately. It computes internally using the shooting method. Axial velocity, temperature distribution, nanoparticle concentration, axial induced magnetic field, and density distribution were all calculated numerically. An analysis was conducted using graphics to show how different factors affect the flow quantities of interest. The results showed that when the Jeffrey fluid parameter is increased, the magnitude of axial velocity increases at the upper wall of the channel, while it decreases close to the lower walls. Increasing the Hartmann number lads to increases in the axial velocity near the channel walls and in the concentration of nanoparticles. Additionally, as the Brownian motion parameter is increased, both temperature and nanoparticle concentration grow. Full article
(This article belongs to the Special Issue Symmetry in Micro/Nanofluid and Fluid Flow)
Show Figures

Figure 1

Figure 1
<p>The problem’s geometry.</p>
Full article ">Figure 2
<p>Variations in velocity profile <math display="inline"><semantics> <mi mathvariant="normal">u</mi> </semantics></math> with <math display="inline"><semantics> <mi mathvariant="normal">y</mi> </semantics></math> for various values of Hartmann number <span class="html-italic">M</span> (<b>a</b>), local Grashof number <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics></math> (<b>b</b>), nanoparticles’ Grashof number <math display="inline"><semantics> <mrow> <mi>Q</mi> <mi>r</mi> </mrow> </semantics></math> (<b>c</b>), and Jeffrey fluid parameter <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> (<b>d</b>). The other parameters chosen are <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>a</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>b</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>c</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>1.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (<b>d</b>).</p>
Full article ">Figure 3
<p>Variations in a fluid’s local temperature <math display="inline"><semantics> <mi>θ</mi> </semantics></math> with respect to <math display="inline"><semantics> <mi mathvariant="normal">y</mi> </semantics></math> for different values of the Hartmann number <span class="html-italic">M</span>, thermophoresis parameter <math display="inline"><semantics> <msub> <mi>N</mi> <mi>t</mi> </msub> </semantics></math>, Brownian motion parameter <math display="inline"><semantics> <msub> <mi>N</mi> <mi>b</mi> </msub> </semantics></math>, and local temperature Grashof number <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics></math> are shown in panels a through d. The other factors considered are <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>a</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>b</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>c</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (<b>d</b>).</p>
Full article ">Figure 4
<p>Variations in nanoparticle concentration <math display="inline"><semantics> <mo>Ω</mo> </semantics></math> with <math display="inline"><semantics> <mi mathvariant="normal">y</mi> </semantics></math> for various values of Hartmann number <span class="html-italic">M</span> (<b>a</b>), Brownian motion parameter <math display="inline"><semantics> <msub> <mi>N</mi> <mi>b</mi> </msub> </semantics></math> (<b>b</b>), thermophoresis parameter <math display="inline"><semantics> <msub> <mi>N</mi> <mi>t</mi> </msub> </semantics></math> (<b>c</b>), and local temperature Grashof number <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics></math> (<b>d</b>). The other parameters chosen are <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>a</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>b</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>c</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (panel <b>d</b>).</p>
Full article ">Figure 5
<p>Variations in axial induced magnetic field <math display="inline"><semantics> <msub> <mi>h</mi> <mi>x</mi> </msub> </semantics></math> against space variable <span class="html-italic">y</span> for different values of Hartmann number <span class="html-italic">M</span> (panel <b>a</b>), Jeffrey fluid parameter <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> (panel <b>b</b>), magnetic Reynolds number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>m</mi> </msub> </semantics></math> (panel <b>c</b>), and electric field intensity <span class="html-italic">E</span> (panel <b>d</b>). The other parameters chosen are <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>a</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>b</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (panel <b>c</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>d</b>).</p>
Full article ">Figure 6
<p>Variations in the current density distribution <math display="inline"><semantics> <msub> <mi>J</mi> <mi>z</mi> </msub> </semantics></math> within <span class="html-italic">y</span> for different values of Hartmann number <span class="html-italic">M</span> (panel <b>a</b>), Jeffrey fluid parameter <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> (panel <b>b</b>), magnetic Reynolds number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>m</mi> </msub> </semantics></math> (panel <b>c</b>), and electric field intensity <span class="html-italic">E</span> (panel <b>d</b>). The other parameters chosen are <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mo> </mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>a</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>b</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (panel <b>c</b>); <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>Q</mi> <mi>r</mi> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (panel <b>d</b>).</p>
Full article ">
14 pages, 3225 KiB  
Communication
Metabolic Rates of Rainbow Trout Eggs in Reconstructed Salmonid Egg Pockets
by Rudy Benetti, Tobia Politi, Marco Bartoli and Nerijus Nika
Water 2024, 16(4), 612; https://doi.org/10.3390/w16040612 - 19 Feb 2024
Viewed by 1395
Abstract
In situ evaluations of the metabolic rates (i.e., respiration and excretion) of salmonid eggs are mostly indirect, focusing on the sampling of hyporheic water from wild or artificial nests. Comparatively, experimental studies carried out under controlled, laboratory conditions are less abundant due to [...] Read more.
In situ evaluations of the metabolic rates (i.e., respiration and excretion) of salmonid eggs are mostly indirect, focusing on the sampling of hyporheic water from wild or artificial nests. Comparatively, experimental studies carried out under controlled, laboratory conditions are less abundant due to methodological difficulties. This study presents a novel experimental setup aimed to address this issue and enable the measurement of oxygen and dissolved inorganic nitrogen fluxes in simulated rainbow trout (O. mykiss) egg pockets. The experimental setup consists of reconstructed egg pockets in cylindrical cores under flow-through conditions. Live and dead eyed-stage eggs were incubated in a natural, sterilised gravel substrate. Hyporheic water circulation was ensured using peristaltic pumps, with the possibility to collect and analyse inflowing and outflowing water for chemical analyses. Microcosm incubations, with closed respirometry of eggs in water alone, were also carried out in order to infer the importance of microbial respiration in the simulated egg pockets. The results show an increasing trend in oxygen demand, due to the development of biofilm in the reconstructed egg pockets and increased egg respiration rates. Moreover, egg pockets showed positive ammonium net fluxes connected with the advancing developmental egg stage, while nitrate removal peaked during the last phase of the experiment, mainly due to the formation of oxic-hypoxic interfaces, leading to couple nitrification–denitrification processes. The suggested approach enables to test a number of in situ situations, including the effects of extreme hydrological conditions, sediment clogging and sudden changes in water chemistry or temperature on the survival and metabolic performances of nests, at different egg development stages. Full article
(This article belongs to the Section Biodiversity and Functionality of Aquatic Ecosystems)
Show Figures

Figure 1

Figure 1
<p>The photos summarise the sequence of actions for the mesocosm apparatus and the measurements: (<b>a</b>) preparation of the cores with spawning gravel, (<b>b</b>) egg pocket construction and laying of eggs, (<b>c</b>) falcon tube sampled with the needle oxygen logger, and (<b>d</b>) chamber used to incubate eggs for the microcosm assay.</p>
Full article ">Figure 2
<p>Mesocosm setup in which reconstructed trout nests were incubated. The aquarium was laid in one pool of the RAS. Arrows indicate the water circulation direction.</p>
Full article ">Figure 3
<p>Boxplot portraying the (<b>a</b>) DO demand (mg O<sub>2</sub> egg<sup>−1</sup> h<sup>−1</sup>) and (<b>b</b>) N-NH<sub>4</sub><sup>+</sup> excretion rates (μg N-NH<sub>4</sub><sup>+</sup> egg<sup>−1</sup> h<sup>−1</sup>) of rainbow trout eggs at the three main developmental stages: eyed egg (294–310 dd), nearly hatched alevin with yolk sack (350–358 dd) and prior to the emergence time (swim up, 454–462 dd). The three eggs stages were incubated in closed glass chambers (see the text for more details).</p>
Full article ">Figure 4
<p>Line plots depicting the (<b>a</b>) oxygen (mg O<sub>2</sub> mesocosm<sup>−1</sup> h<sup>−1</sup>) demand, (<b>b</b>) ammonium (mg N-NH<sub>4</sub><sup>+</sup> mesocosm<sup>−1</sup> h<sup>−1</sup>) and (<b>c</b>) nitrates fluxes (mg N-NO<sub>3</sub><sup>−</sup> mesocosm<sup>−1</sup> h<sup>−1</sup>) for the simulated salmonid egg pockets (cores), for the whole incubation period until the complete alevin yolk sack absorption. Different colours represent the two main stages “eyed egg” and “alevin”, while different geometries stand for the live (L) and live and dead (L + D) setups and the control (bare sediments). Boxplots of 100-egg respiration and excretion data from the microcosm incubation are inserted for comparison. Please note that the boxplot position along the degree day axis is representative of the degree day intervals in which fluxes were measured from the microcosm experiments.</p>
Full article ">
Back to TopTop