Open AccessArticle

Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect

Muhammad Zubair

¹,

Zahir Shah

Abdullah Dawar

Saeed Islam

Poom Kumam

^4,5,6,*

and

Aurangzeb Khan

⁷

Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber, Pakhtunkhwa 23200, Pakistan

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Department of Mathematics, Qurtuba University of Science and Information Technology, Peshawar 25000, Pakistan

⁴

KMUTT-Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

⁵

KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

⁶

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

⁷

Department of Physics, Abdul Wali Khan University, Mardan 23200, Pakistan

Author to whom correspondence should be addressed.

Entropy 2019, 21(8), 747; https://doi.org/10.3390/e21080747

Submission received: 20 June 2019 / Revised: 20 July 2019 / Accepted: 24 July 2019 / Published: 30 July 2019

(This article belongs to the Special Issue Challenges and Progresses in the Modelling of Entropy Generation in Fluid Mechanics, Heat Transfer and Porous Media)

Download

Browse Figures

Versions Notes

Abstract

In this research article, the investigation of the three-dimensional Casson nanofluid flow in two rotating parallel plates has been presented. The nanofluid has been considered in steady state. The rotating plates have been considered porous. The heat equation is considered to study the magnetic field, joule heating, and viscous dissipation impacts. The nonlinear ordinary system of equations has been solved analytically and numerically. For skin friction and Nusslt number, numerical results are tabulated. It is found that velocity declines for higher values of magnetic and porosity parameter while it is heightened through squeezing parameter. Temperature is an enhancing function for Eckert number and nanoparticles volume fraction. Entropy generation is augmented with radiation parameter, Prandtl, and Eckert numbers. The Casson, porosity, magnetic field, and rotation parameters were reduced while the squeezing and suction parameters increased the velocity profile along x-direction. The porosity parameter increased the Bejan number while the Eckert and Prandtl numbers decreased the Bejan number. Skin friction was enhanced with increasing the Casson, porosity, and magnetic parameters while it decreased with enhancing rotation and squeezing parameters. All these impacts have been shown via graphs. The influences by fluid flow parameters over skin friction and Nusselt number are accessible through tables.

Keywords:

Casson fluid; magnetohydrodynamic (MHD); heat transfer; porosity; viscous dissipation; entropy; homotopy analysis method (HAM)

1. Introduction

In the modern world of science and technology, we need more development in the direction of the exhaustion of energy in engineering and industrial fields. Thus, for the exhaustion and transfer of heat, the study of nanofluids discloses extraordinary thermal conductivity and heat transfer coefficients compared to conventional fluids. These specific aspects of nanofluids make them appropriate for the succeeding generation of flow and heat transfer fluids. Therefore, the revolutionary nanofluids investigation has encouraged researchers and engineers all over the world. Problems related to the heat and mass transfer via nanofluid flow between two permeable walls/plates has remained under discussion since last few years. This is because heat and mass transfer have numerous applications in the industries of petroleum supplies, oil conveyance, and separation processes in chemical companies, etc. The advancement of power energy in technology is the essential objective of scientists and engineers. This is because of the extreme interest in cooling/heating in modern systems. Very high thermal conductivity of the nanofluid separates them from the other ordinary fluids as an ideal fluid. Since nanofluids have ideal thermal conductivity, they are used as cooling agents in computers, nuclear reactors, cancer therapy, to lower the level of cholesterol in blood, make surgeries safer, electronics, micro channels in defence sector, printing and press systems, food and drinks industries, as well as in oil, gas, and chemicals industries.

A key test for analysts is to upgrade the thermal conductivity applications of usual coolants corresponding to oil, ethylene glycol, and water, which have low thermal conductivity. Because of such inspiration, the first endeavor made by Choi [1] on this path was to encourage thermal conductivity of customary fluids including metallic nano-sized particles into a base fluid. Yu et al. [2] inspected the heat transmission and thermal conductivity through these nanofluid flows. Tyler et al. [3] investigated diamond-based nanofluid. Liu et al. [4] studied the carbon nanotubes nanofluid with thermal radiation. Ellahi et al. [5] fined the power series results of the nanofluids analytically. Nadeem et al. [6,7] examined the magnetohydrodynamic (MHD) nanofluids flow convection conditions. Shah et al. [8,9,10] inspected the nanofluids flows with Brownian, thermophoresis, and Hall impacts. Ramzan et al. [11] explored radiative flow of nanofluids by applying the impression of magnetic field. Sheikholeslami et al. [12] studied the flow of nanofluids via porous enclosure making an allowance for the magnetic effect. Besthapu et al. [13] explored the combined convection in nanofluid flow under the impressions of viscous dissipation and magnetic field influences. Dawar et al. [14] analyzed the nanofluid flow over a porous extending surface. In a rotating frame, Shah et al. [15] probed the Darcy–Forchheimer flow of nanofluid. Khan et al. [16] examined [15] considering the thermal radiation and magnetic field impacts. Khan et al. [17] evaluated two-dimensional flow of nanofluid over extending surface. Sheikholeslami [18] witnessed the convective nanofluid flow via porous medium by implementing effects of electric field. Sheikholeslami [19] studied the flow of nanofluid based on water with properties of Brownian motion. Dawar et al. [20] analyzed Darcy flow by using nanofluids past extending sheet. Ramzan et al. [21] observed the 3D flow of couple stress nanofluid. Sajid et al. [22] studied the fluid flow over a radially extending sheet. Attia et al. [23] examined the nanofluid flow through a porous median over radially extending sheet.

The measure of disorder in a system or unavailable energy in a closed thermodynamically system is known as entropy. The entropy generation is derived from second law of thermodynamics. In 1980, the entropy production rate was introduced by Bejan [24]. Hayat et al. [25] investigated the nanomaterial fluid flow in a revolving system. Nouri et al. [26] studded the nanofluids flow through entropy generation considering the spherical heat source. Dalir et al. [27] scrutinized the Jeffrey nanofluid flow. Rashidi et al. [28] studied the steady MHD nanofluid flow in a rotating system. Das et al. [29] examined the MHD flow of nanofluids via porous medium. Seth et al. [30] calculated the Hall effect on Couette flow with entropy generation. Mkwizu et al. [31] numerically studied the Couette flow. Adesanya and Makinde [32] investigated the flow of couple stress nanofluid with entropy generation. Sheremet et al. [33] evaluated the nanofluid natural convection in a square cavity. In another article, Sheremet et al. [34] investigated the same fluid in wavy cavity. Alharbi et al. [35] evaluated the flow of nanofluid over porous extending sheet. Dawar et al. [36] examined the MHD carbon nanotubes nanofluid flow passing through rotating channels under the effects of viscous dissipation. Kumam et al. [37] studied the MHD nanofluid flow in rotating channels with entropy generation. Alain Portavoce et al. [38] explored transmission of atom in crystalline thin film. Oudina [39] explored heat transfer in Titania nanofluids in cylindrical annulus with discontinuous heat source. Oudina [40] studded MHD flow with natural convection between vertical coaxial cylinders. Reza et al. [41] explored nanofluids flow with MHD in a narrow channel with stretching walls. Jawad et al. [42] explored the MHD flow between two plates with different angle. Alkasassbeh et al. [43] investigated heat transfer by the fin attached to hybrid generator. Salim et al. [44] analyzed MHD jaffery suction flow through homotopy analysis method. Batti et al. [45] explored entropy production with MHD in nanofluid flow via porous stretching plate. Rashidi et al. [46] explored the entropy production in a single-slope solar still. Batti et al. [47] explored entropy production in entropy production with MHD during thermal radiation past a shrinking surface. Esfahani et al. [48] explored entropy production in nanofluid flow streaming past a wavy wall. For more study on entropy analysis, see [49,50,51,52,53,54,55,56,57].

2. Problem Formulation

Here, we assumed 3D unsteady flow of Casson nanofluid to have viscous dissipation, joule heating, and entropy generation properties in two rotating parallel plates. Both plates are permeable. The suction velocity of the fluid via upper porous plate is

V_{0}

. The magnetic field’s strength

B = \frac{B_{0}}{\sqrt{1 - s t}}

is also considered in the nanofluid flow. Darcy’s relation is implemented in order to study the nanofluid flow in a porous medium. The lower sheet is stretched with velocity

U_{w} = \frac{a x}{1 - s t}

and is positioned at

y = 0

. The upper sheet is squeezed towards the lower sheet with velocity

V_{h} = \frac{d h (t)}{d t}

and is positioned at

h (t) = \sqrt{\frac{υ_{n f} (1 - s t)}{a}}

where

a

and

s

are the stretching and time parameters. The whole system rotates with angular velocity

Ω_{0}

. The lower plate temperature is denoted by

T_{w}

and the upper plate temperature is denoted by

T_{h}

. It is also supposed that

T_{w} > T_{h}

. Figure 1 shows the symmetric flow sketch of the nanofluids.

Nanofluid flow governing equations are [53,54,55,56]:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,

(1)

\begin{array}{l} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} + 2 \frac{Ω_{0}}{1 - s t} w + \frac{1}{ρ_{n f}} \frac{\partial P}{\partial x} \\ - υ_{n f} (1 + \frac{1}{β_{0}}) (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) + (\frac{σ_{n f} B_{0}^{2}}{ρ_{n f}} + \frac{υ_{n f}}{κ^{*}}) \frac{u}{1 - s t} = 0, \end{array}

(2)

\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + \frac{1}{ρ_{n f}} \frac{\partial P}{\partial y} - υ_{n f} (1 + \frac{1}{β_{0}}) (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}) + \frac{υ_{n f}}{κ^{*}} \frac{v}{1 - s t} = 0,

(3)

\begin{array}{l} \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} - 2 \frac{Ω_{0}}{1 - b t} u \\ - υ_{n f} (1 + \frac{1}{β_{0}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}) + (\frac{σ_{n f} B_{0}^{2}}{ρ_{n f}} + \frac{υ_{n f}}{κ^{*}}) \frac{w}{1 - s t} = 0, \end{array}

(4)

\begin{array}{l} \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} - \frac{κ_{n f}}{{(ρ C_{p})}_{n f}} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}}) \\ - \frac{μ_{n f}}{{(ρ C_{p})}_{n f}} [\begin{array}{l} 2 {(\frac{\partial u}{\partial x})}^{2} + 2 {(\frac{\partial v}{\partial y})}^{2} + 2 {(\frac{\partial w}{\partial z})}^{2} + {(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y})}^{2} \\ + {(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x})}^{2} + {(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y})}^{2} \end{array}] \\ - \frac{σ_{n f} B_{o}^{2}}{{(ρ C_{p})}_{n f}} \frac{(u^{2} + w^{2})}{(1 - s t)} = 0 . \end{array}

(5)

Problem constrains at the boundary is of the form:

\begin{array}{l} u = U_{w} = \frac{a x}{1 - s t}, v = \frac{- V_{o}}{1 - s t}, w = 0, T = T_{w}, a t y = 0, \\ u = 0, v = - \frac{s}{2} \sqrt{\frac{υ_{n f}}{a (1 - s t)}}, w = 0, T = T_{h}, a t y = h (t) . \end{array}

(6)

The applied two forces are equal in measurement but reverse in direction, keeping the plates stretched. The lower plate is extended because of these equivalent forces in size and inverted direction, while the lower plate is squeezed down with velocity

V_{h} (t)

u = U_{w} = \frac{a x}{1 - s t}

Illustrates the stretching velocity at

y = 0

while

u = 0

illustrates the free stream velocity at

y = h (t)

. At the surface of the lower plate, there is a suction represented by

v = - \frac{V_{o}}{1 - s t}

while at

y = h (t)

there is a squeezing velocity represented by

v = - \frac{s}{2} \sqrt{\frac{υ_{n f}}{a (1 - s t)}}

T = T_{w}

and

T = T_{h}

characterize the constant temperature at the lower plate and rise of mercury at upper plate at

y = 0

and

y = h (t)

, respectively. In the above equations

u, v and w

are the velocity components in their respective directions.

κ^{*}

demonstrates the absorption coefficient or porosity of the medium and

Ω_{0}

define the angular velocity. Furthermore,

μ_{n f}

describes the nanofluid viscosity,

ρ_{n f}

describes the nanofluid density,

κ_{n f}

defines the nanofluid thermal conductivity, and

{(C_{p})}_{n f}

illustrates the nanofluid specific heat capacity, which are defined as

\begin{array}{l} \frac{μ_{n f}}{μ_{f}} = \frac{1}{{(1 - Φ)}^{2.5}}, \frac{ρ_{n f}}{ρ_{f}} = (1 - Φ) + \frac{ρ_{s}}{ρ_{f}} Φ, \frac{{(ρ C_{p})}_{n f}}{{(ρ C_{p})}_{f}} = (1 - Φ) + \frac{{(ρ C_{p})}_{s}}{{(ρ C_{p})}_{f}} Φ, \\ \frac{κ_{n f}}{κ_{f}} = \frac{κ_{s} + 2 κ_{f} - 2 Φ (κ_{f} - κ_{s})}{κ_{s} + 2 κ_{f} + 2 Φ (κ_{f} - κ_{s})}, \frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{s}}{σ_{f}} - 1) Φ}{(\frac{σ_{s}}{σ_{f}} + 2) - (\frac{σ_{s}}{σ_{f}} - 1) Φ} . \end{array}

(7)

where

Φ, μ_{f}, ρ_{f}, κ_{f}, σ_{f}, {(C_{p})}_{f}, s, f

, and

n f

indicate the nanoparticle volume fraction, dynamic viscosity, density, thermal conductivity, electric conductivity, specific heat capacity, solid nanoparticles, base fluid, and nano-fluid, respectively.

The transformation variables are demarcated:

\begin{array}{l} u = U_{w} f^{'} (ξ), v = - \sqrt{\frac{a υ_{n f}}{1 - s t}} f (ξ), w = U_{w} g (ξ), \\ θ (ξ) = \frac{T - T_{h}}{T_{w} - T_{h}}, h (t) = \sqrt{\frac{υ_{n f} (1 - s t)}{a}}, ξ = \frac{y}{h (t)} . \end{array}

(8)

Obviously, Equation (1) satisfies, and Equations (2)–(6) becomes:

(1 + \frac{1}{β_{0}}) f^{i v} - f^{'} f^{″} + f f^{‴} - \frac{β}{2} (3 f^{″} + ξ f^{‴}) - 2 α g^{'} - \frac{1}{N_{1}} (N_{2} M + N_{3} λ) f^{″} = 0,

(9)

(1 + \frac{1}{β_{0}}) g^{″} + f g^{'} - f^{'} g - β (g + \frac{ξ}{2} g^{'}) + 2 α f^{'} - \frac{1}{N_{1}} (N_{2} M + N_{3} λ) g = 0,

(10)

\frac{N_{1} N_{4}}{N_{3} N_{5}} \frac{1}{\Pr} θ^{″} + (f - \frac{β}{2} ξ) θ^{'} + \frac{1}{N_{5}} [N_{3} E_{c} (4 {f^{'}}^{2} + g^{2}) + E_{d} {\begin{cases} N_{1} ({g^{'}}^{2} + 2 {f^{″}}^{2}) \\ + N_{2} M ({f^{'}}^{2} + g^{2}) \end{cases}}] = 0 .

(11)

Satisfying the succeeding boundary conditions:

\begin{array}{l} f^{'} = 1, f = δ, g = 0, θ = 1, at ξ = 0, \\ f^{'} = 0, f = \frac{β}{2}, g = 0, θ = 0 at ξ = 1 . \end{array}

(12)

Here the following ratios are defined between the nanofluid and base fluid.

N_{1} = \frac{ρ_{n f}}{ρ_{f}}

N_{2} = \frac{σ_{n f}}{σ_{f}}

N_{3} = \frac{μ_{n f}}{μ_{f}}

N_{4} = \frac{κ_{n f}}{κ_{f}}

, and

N_{5} = \frac{{(ρ C_{p})}_{n f}}{{(ρ C_{p})}_{f}}

define the density, electric conductivity, dynamic viscosity, thermal conductivity, and specific heat capacitance.

Also,

β_{o}

clarifies the Casson parameter,

M = \frac{σ_{f} B_{o}^{2}}{a ρ_{f}}

indicates the magnetic field,

α = \frac{Ω_{o}}{a}

demonstrates the rotation parameter,

β = \frac{s}{a}

displays the squeezing parameter,

δ = \frac{V_{o}}{a h}

defines the suction parameter,

λ = \frac{υ_{f}}{a κ^{*}}

determines the porosity parameter,

\Pr = \frac{μ_{f} {(c_{p})}_{f}}{κ_{f}}

shows the Prandtl number, and

E c = \frac{υ_{f}^{2}}{{(c_{p})}_{f} (T_{w} - T_{h}) h^{2}}

and

E d = \frac{U_{w}^{2}}{{(c_{p})}_{f} (T_{w} - T_{h})}

are local Eckert numbers.

The skin friction and Nusselt number of the nanofluid can be characterized as:

C_{f x} = \frac{- 2 τ_{w}}{ρ_{n f} U_{w}^{2}}, N u_{x} = \frac{x q_{w}}{κ_{n f} (T_{w} - T_{h})},

(13)

where

τ_{w}

defines wall share stress and

q_{w}

defines heat flux. Mathematically, we have:

τ_{w} = {μ_{n f} (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}) |}_{y = 0}, {q_{w} = - κ_{n f} \frac{\partial T}{\partial y} |}_{y = 0},

(14)

Substituting Equation (14) in Equation (13), in a simple way, is

C_{f x} \sqrt{({Re}_{x})} = - 2 \sqrt{\frac{N_{3}}{N_{1}}} f^{″} (0), \frac{N u_{x}}{\sqrt{({Re}_{x})}} = - N_{4} \sqrt{\frac{N_{3}}{N_{1}}} θ^{'} (0),

(15)

where

{Re}_{x} (= \frac{x U_{w}}{υ_{f}})

defines the Reynolds number.

3. Entropy Analysis

The rate of entropy production (

E_{G}

) is obtained as [47,48]:

\begin{array}{l} E_{G} = \frac{k_{n f}}{T_{h}^{2}} [{(\frac{\partial T}{\partial x})}^{2} + {(\frac{\partial T}{\partial y})}^{2} + {(\frac{\partial T}{\partial z})}^{2}] + \frac{σ_{n f} B_{o}^{2}}{{(ρ C_{p})}_{n f}} (u^{2} + w^{2}) \\ + \frac{μ_{n f}}{T_{h}} [\begin{array}{l} 2 {(\frac{\partial u}{\partial x})}^{2} + 2 {(\frac{\partial v}{\partial y})}^{2} + 2 {(\frac{\partial w}{\partial z})}^{2} \\ + {(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y})}^{2} + {(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x})}^{2} + {(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y})}^{2} \end{array}] + \frac{μ_{n f}}{T_{h} κ^{*} (1 - s t)} (u^{2} + v^{2} + w^{2}) . \end{array}

(16)

Using Equations (8) and (16) reduced as:

E_{G} = E_{G_{_{O}}} [N_{4} {θ^{'}}^{2} + \frac{T_{c} \Pr N_{3}^{2}}{N_{1} N_{4}} {\begin{cases} E_{c} (4 {f^{'}}^{2} + g^{2}) + \frac{N_{1} E_{d}}{N_{3}} ({f^{″}}^{2} + {g^{'}}^{2}) + \\ \frac{N_{2} M E_{d}}{N_{3}} ({f^{'}}^{2} + g^{2}) + λ E_{d} ({f^{'}}^{2} + \frac{N_{3}}{N_{1} {Re}_{x}} f^{2} + g^{2}) \end{cases}}],

(17)

where

E_{G}_{_{o}} (= \frac{κ_{f} (T_{w} - T_{h})}{T_{h}^{2} h^{2}})

describes the characteristics entropy analysis quotient while

T_{c} = \frac{T_{h}}{T_{w} - T_{h}}

defines the temperature ratio.

The entropy generation rate is reduced as:

N s = \frac{E_{G}}{E_{G_{o}}} = [N_{4} {θ^{'}}^{2} + \frac{T_{c} \Pr N_{3}^{2}}{N_{1} N_{4}} {\begin{cases} E_{c} (4 {f^{'}}^{2} + g^{2}) + \frac{N_{1} E_{d}}{N_{3}} ({f^{″}}^{2} + {g^{'}}^{2}) + \\ \frac{N_{2} M E_{d}}{N_{3}} ({f^{'}}^{2} + g^{2}) + λ E_{d} ({f^{'}}^{2} + \frac{N_{3}}{N_{1} {Re}_{x}} f^{2} + g^{2}) \end{cases}}] .

(18)

In components form, Equation (18) is expressed as:

N s = N h + N f + N j + N p .

(19)

The complete volumetric entropy production is demarcated as:

E_{G_{t o t a l}} = \int_{0}^{h} E_{G} d y .

(20)

The non-dimensional total entropy generation is reduced as:

N s_{t o t a l} = \int_{0}^{h} N s d y = \int_{0}^{1} [N_{4} {θ^{'}}^{2} + \frac{T_{c} \Pr N_{3}^{2}}{N_{1} N_{4}} {\begin{cases} E_{c} (4 {f^{'}}^{2} + g^{2}) + \frac{N_{1} E_{d}}{N_{3}} ({f^{″}}^{2} + {g^{'}}^{2}) \\ + \frac{N_{2} M E_{d}}{N_{3}} ({f^{'}}^{2} + g^{2}) \\ + λ E_{d} ({f^{'}}^{2} + \frac{N_{3}}{N_{1} {Re}_{x}} f^{2} + g^{2}) \end{cases}}] d ξ .

(21)

3.1. Bejan Number

The Bejan number shows the quotient of entropy production rate for heat transfer to entire entropy production. It is characterized as:

B e = \frac{N h}{N s} .

(22)

The Bejan number permanently lies in the range of 0 and 1. The

N h

falls down in the region

0 \leq B e < \frac{1}{2}

. The

N f + N j + N p

dominates in the region

0.5 < B e \leq 1.0

. For

B e = 0.5

in cooperation, the properties are balanced.

3.2. Solution by HAM

To solve the proposed model by analytical method called HAM [49,50,51,52], the initial supposition for Equations (9)–(11) are supposed as:

f_{0} (ξ) = ξ - 2 ξ^{2} + ξ^{3} + 3 δ ξ^{2} - 2 δ ξ^{3}, g_{0} (ξ) = 0, θ_{0} (ξ) = 1 - ξ .

(23)

The linear operators are defined as:

L_{f} (f) = f^{i v}, L_{g} (g) = g^{″}, L_{θ} (θ) = θ^{″} .

(24)

The initial solutions are:

L_{f} (k_{1} + k_{2} ξ + k_{3} ξ^{2} + k_{4} ξ^{3}) = 0, L_{g} (k_{5} + k_{6} ξ) = 0, L_{θ} (k_{7} + k_{8} ξ) = 0 .

(25)

Here,

\sum_{n = 1}^{8} k_{n},

where

n = 1, 2, 3, \dots

are random constants.

3.3. HAM Convergence

When we compute the series solutions of the velocity and temperature functions in order to use HAM, the assisting parameters

ℏ_{f, g, θ}

appear. These assisting parameters are responsible for adjusting the convergence of these solutions. The combined

ℏ -

curve of

f^{″} (0), g^{'} (0)

and

θ (0)

at 10th-order approximations are plotted in Figure 2 for different values of the embedding parameter. The combined

ℏ -

curve consecutively displays the valid region.

4. Results and Discussion

This segment treaties with the physical impacts of concerned factors of the nanofluid flow on velocity profiles

(f^{'} (ξ), f (ξ))

, temperature profile

θ (ξ)

, entropy production

N s (ξ)

, Bejan number

B e (ξ)

, skin friction

C_{f x}

, and Nusselt number

N u_{x}

. These parameters include Casson-

β_{o}

, Magnetic field-

M

, squeezing-

β

, porosity-

λ

, suction-

δ

, rotation-

α

, Prandtl number-

\Pr

, and Eckert numbers-

E c, E d

The impact of Casson parameter

β_{o}

is depicted in Figure 3a,b. The growing Casson parameter declines the velocity profiles

(f^{'} (ξ), f (ξ))

of the nanofluid flow. Actually, the upsurge in Casson parameter

β_{o}

escalates the plastics dynamic viscosity of the nanofluid, which creates resistance during the flow of nanofluid and decline in the velocity profile occurs. The same impact can be seen in Figure 3b at

0.0 \leq ξ < 0.45

. In this figure, the velocity profile increases at

0.45 < ξ \leq 1.0

because of the extending of the lower plate. The impact of porosity parameter

λ

is depicted in Figure 4a,b. Physically, the porous media acts on the boundary layer flow, which produces the opposition to the fluid’s flow and thus fluid’s velocity declines. This impact is shown in Figure 4a. Similarly, the porosity parameter increases the velocity profile at

0.0 \leq ξ \leq 0.40

while it decreases at

0.4 \leq ξ \leq 1.0

as shown in Figure 4b. This impact is because of the extending of the lower plate. The impact of squeezing parameter

β

is portrayed in Figure 5a,b. The escalation in

β

upsurges the velocity profiles. Physically, the higher values of

β

move the upper surface descending and additional pressure utilized over fluid’s particles. Therefore, the velocity profiles increased. Figure 6a,b displays the result of

M

on velocity profiles. Actually, the Lorentz force declares that the induced

M

resists the fluid motion on the liquid boundary, which as a result, diminishes the velocity of the liquid. The impression of

δ

on velocity profiles is illustrated in Figure 7a,b. The higher values of

δ

upturns the velocity along x-direction while the greater values of

δ

reduce the velocity in y-direction. The impression of rotation

α

is depicted in Figure 8a,b. Clearly, from Figure 8a, the increase in

α

reduces the velocity profile in x-direction. From Figure 8b, the velocity profile reduces in y-direction at

0.0 \leq ξ < 0.40

while it increases at

0.4 \leq ξ \leq 1.0

. The impression of

E c

θ (ξ)

is offered in Figure 9. The increasing

E c

upsurges the fluid flow temperature. Actually, the Eckert number produces viscous resistance due to the occurrence of dissipation term, which increases the nanofluid thermal conductivity to upsurge the temperature field. A similar impact can be seen in Figure 10. The influence of squeezing

β

θ (ξ)

is depicted in Figure 11. Unmistakably, we saw that the temperature increases rapidly with the enlargement in

β

. Bigger

β

implies that the upper plate moves in a descending fashion in consequence with the fluid interatomic collision increments because of the little space accessible concerning the plates to the nanofluid particles. At the point when the impacts among the atoms of the nanofluid improve, the nanofluid temperature upsurges. Figure 12 exhibits the impression of

\Pr

θ (ξ)

. Here, we analyzed that temperature field diminishes by means of escalation in

\Pr

. Bigger estimations of

\Pr

connects to lesser heat conductivity, hence the temperature profile is diminished. Figure 13a illustrates the impression of

\Pr

N s (ξ)

. The value of

\Pr

escalates the entropy generation. The magnetic field boosts the form of the plate surface to its maximum values in the area of the plate. The opposite impact of Prandtl number

\Pr

B e (ξ)

is demonstrated in Figure 1. Figure 14a,b denotes the effect of porosity parameter

λ

over

N s (ξ)

and

B e (ξ)

. Both the entropy and Bejan number boost with the escalation in porous media. It is perceived that the entropy upsurges more speedily compared to Bejan number. Figure 15a,b demonstrates the impact of Eckert number

E c

over

N s (ξ)

and

B e (ξ)

. The escalation in Eckert number upsurges the entropy generation

N s (ξ)

while it diminishes the

B e (ξ)

. Similar impacts of

E d

N s (ξ)

and

B e (ξ)

are observed in Figure 16a,b.

The impressions of Casson parameter

β_{o}

, magnetic field

M

, rotation parameter

α

, squeezing parameter

β

, suction parameter

δ

, porosity parameter

λ

, Prandtl number

\Pr

, and local Eckert numbers

{E c, E d}

over skin friction

C_{f x}

and Nusselt number

N u_{x}

are displayed in Table 1 and Table 2, respectively. The escalating approximations of squeezing and rotation parameters reduced the

C_{f x}

while the rise in Casson, porosity, and magnetic field parameters augmented the

C_{f x}

. The rising approximations of squeezing parameter showed dual behavior in

N u_{x}

. The escalating approximations of Prandtl number, Eckert numbers, and magnetic field parameter declined the

N u_{x}

5. Comparison of HAM with Numerical Result

We solve the system the model Equations (9)–(11) with boundary conditions (12) by ND Solve in Mathematica 10 package. Comparison between analytical and numerical techniques is shown in Table 3 and Table 4. An excellent agreement is fund.

6. Conclusions

The squeezing Casson nanofluid flow with viscous dissipation and entropy generation between two parallel stretching plates has been presented in this paper. The stretching plates are considered porous. The impact of magnetic field is also reflected in the proposed model. The impacts of embedded parameters are shown through figures and tables.

The significant facts of the presented model are as follows:

The Casson parameter, porosity parameter, magnetic field parameter, and rotation parameter reduced the velocity profile $f (ξ)$ while the squeezing and suction parameters increased the velocity profile $f (ξ)$ .
The Casson parameter, porosity parameter, magnetic field parameter, and rotation parameter showed dual behavior in the velocity profile $f^{'} (ξ)$ .
The squeezing parameter \ increased the velocity profile $f^{'} (ξ)$ while the suction parameter reduced the velocity profile $f^{'} (ξ)$ .
The Eckert numbers and squeezing parameter increased the temperature profile $θ (ξ)$ while the Prandtl number increased the temperature profile $θ (ξ)$ .
The Prandtl number, porosity parameter, and Eckert numbers increased the Entropy generation rate $N s (ξ)$ .
The porosity parameter increased the Bejan number $B e (ξ)$ while the Eckert and Prandtl numbers increased the Bejan number.

Author Contributions

M.Z. and Z.S.: Conceptualization; Methodology; Software; Validation; Writing-Original Draft Preparation; Writing-Review & Editing. A.D.: Conceptualization; Methodology; Software; Visualization; Writing-Review & Editing. P.K.: Writing-Review & Editing; Visualization; Project Administration; Funding Acquisition; Investigation; Resources, S.I. and A.K.: Writing-Review & Editing; Resources; Visualization; Software.

Funding

This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

Acknowledgments

This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.

Conflicts of Interest

The author declares that they have no competing interests.

Nominiclature

Symboles	Description
$u, v, w$	Velocity Components
$x, y, z$	Coordinate Axes
$P$	Pressure
$B_{o}$	Magnetic Field
$s$	Time Parameter
$t$	Time
$C_{p}$	Specific Heat Capacity
$a$	Stretching Parameter
$h$	Gap between walls/ Plates
$T$	Temperature
$M$	Magnetic Parameter
$q_{w}$	Heat Flux
$U_{w}$	Stretching velocity
$Ω_{o}$	Angular Velocity
$V_{o}$	Suction Velocity
Greek Latters
$α$	Rotation Parameter
$β$	Squeezing Parameter
$β_{o}$	Casson Fluid Parameter
$δ$	Suction Parameter
$σ$	Electrical Conductivity
$τ$	Shear Stress
$λ$	Porosity Parameter
$μ$	Dynamic viscosity
$ν$	Kinematic viscosity
$ρ$	Density
$θ$	Temperature Profile
$Φ$	Nanoparticles fraction by Volume
$κ^{*}$	Thermal Conductivity
Dimensionless Numbers
$\Pr$	Prandtl Number
$B e$	Bejan Number
$E c, E d$	Eckert Numbers
$N u_{x}$	Nusselt number
${Re}_{x}$	Reynolds Number
$N_{i}, i = 1, 2, 3, 4, 5$	Ratios of Physical Quantities
Subscripts
$n f$	Nanofluid
$f$	Base Fluid
$s$	Solid Nanoparticles
$w$	At lower wall/ Plate
$h$	At Upper wall/ Plate

References

Choi, S.U.S. Enhancing thermal conductivity of fluids with nanoparticles. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, 12–17 November 1995; ASME: New York, NY, USA; pp. 99–105. [Google Scholar]
Yu, W.; France, D.M.; Routbort, J.L.; Choi, S.U.S. Review and comparison of nanofluid thermal conductivity and heat transfer enhancements. Heat Transf. Eng. 2008, 29, 432–460. [Google Scholar] [CrossRef]
Tyler, T.; Shenderova, O.; Cunningham, G.; Walsh, J.; Drobnik, J.; McGuire, G. Thermal transport properties of diamond based nanofluids and nano composites. Diam. Relat. Mater. 2006, 15, 2078–2081. [Google Scholar] [CrossRef]
Liu, M.S.; Lin, M.C.C.; Huang, I.T.; Wang, C.C. Enhancement of thermal conductivity with carbon nanotube for nanofluids. Int. Commun. Heat Mass Transf. 2005, 32, 1202–1210. [Google Scholar] [CrossRef]
Ellahi, R.; Raza, M.; Vafai, K. Series solutions of non-Newtonian nanofluids with Reynolds’ model and Vogel’s model by means of the homotopy analysis method. Math. Comput. Model. 2012, 55, 1876–1891. [Google Scholar] [CrossRef]
Nadeem, S.; Haq, R.U.; Khan, Z.H. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanopar-ticles. J. Taiwan Ins. Chem. Eng. 2014, 45, 121–126. [Google Scholar] [CrossRef]
Nadeem, S.; Haq, R.U. Effect of thermal radiation for megnetohydrody- namic boundary layer flow of a nanofluid past a stretching sheet with convective boundary conditions. J. Comput. Theor. Nanosci. 2013, 11, 32–40. [Google Scholar] [CrossRef]
Shah, Z.; Gul, T.; Khan, A.M.; Ali, I.; Islam, S. Effects of hall current on steady three dimensional non-Newtonian nanofluid in a rotating frame with Brownian motion and thermophoresis effects. J. Eng. Technol. 2017, 6, 280–296. [Google Scholar]
Shah, Z.; Islam, S.; Gul, T.; Bonyah, E.; Khan, M.A. The electrical MHD and Hall current impact on micropolar nanofluid flow between rotating parallel plates. Results Phys. 2018, 9, 1201–1214. [Google Scholar] [CrossRef]
Shah, Z.; Gul, T.; Islam, S.; Khan, M.A.; Bonyah, E.; Hussain, F.; Mukhtar, S.; Ullah, M. Three dimensional third grade nanofluid flow in a rotating system between parallel plates with Brownian motion and thermophoresis effects. Results Phys. 2018, 10, 36–45. [Google Scholar] [CrossRef]
Ramzan, M.; Chung, J.D.; Ullah, N. Radiative magnetohydrodynamic nanofluid flow due to gyrotactic microorganisms with chemical reaction and non-linear thermal radiation. Int. J. Mech. Sci. 2017, 130, 31–40. [Google Scholar] [CrossRef]
Sheikholeslami, M.; Shehzad, S. Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM. Int. J. Heat Mass Transf. 2017, 113, 796–805. [Google Scholar] [CrossRef]
Besthapu, P.; Haq, R.U.; Bandari, S.; Al-Mdallal, Q.M. Mixed convection flow of thermally stratified MHD nanofluid over an exponentially stretching surface with viscous dissipation effect. J. Taiwan Inst. Chem. Eng. 2017, 71, 307–314. [Google Scholar] [CrossRef]
Dawar, A.; Shah, Z.; Idrees, M.; Khan, W.; Islam, S.; Gul, T. Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface. Math. Comput. Appl. 2018, 23, 20. [Google Scholar] [CrossRef]
Shah, Z.; Dawar, A.; Islam, S.; Ching, D.L.C.; Khan, I. Darcy-Forchheimer flow of radiative carbon nanotubes with microstructure and inertial characteristics in the rotating frame. Case Stud. Therm. Eng. 2018, 12, 823–832. [Google Scholar] [CrossRef]
Khan, A.; Shah, Z.; Islam, S.; Dawar, A.; Bonyah, E.; Ullah, H.; Khan, A. Darcy-Forchheimer flow of MHD CNTs nanofluid radiative thermal behaviour and convective non uniform heat source/sink in the rotating frame with microstructure and inertial characteristics. AIP Adv. 2018, 8, 125024. [Google Scholar] [CrossRef]
Khan, A.S.; Nie, Y.; Shah, Z.; Dawar, A.; Khan, W.; Islam, S. Three-Dimensional Nanofluid Flow with Heat and Mass Transfer Analysis over a Linear Stretching Surface with Convective Boundary Conditions. Appl. Sci. 2018, 8, 2244. [Google Scholar] [CrossRef]
Sheikholeslami, M. Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure. J. Mol. Liq. 2018, 249, 1212–1221. [Google Scholar] [CrossRef]
Sheikholeslami, M. CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion. J. Mol. Liq. 2018, 249, 921–929. [Google Scholar] [CrossRef]
Dawar, A.; Shah, Z.; Khan, W.; Islam, S.; Idrees, M. An optimal analysis for Darcy-Forchheimer 3-D Williamson Nanofluid Flow over a stretching surface with convective conditions. Adv. Mech. Eng. 2019, 11. [Google Scholar] [CrossRef]
Ramzan, M.; Sheikholeslami, M.; Saeed, M.; Chung, J.D. On the convective heat and zero nanoparticle mass flux conditions in the flow of 3D MHD Couple Stress nanofluid over an exponentially stretched surface. Sci. Rep. 2019, 9, 562. [Google Scholar] [CrossRef]
Sajid, M.; Hayat, T.; Asghar, S. Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet. Acta Mech. 2007, 189, 193–205. [Google Scholar] [CrossRef]
Ewis, K.M.; Attia, H.A.; Abdeen, M.A.M. Stagnation Point Flow through a Porous Medium towards a Radially Stretching Sheet in the Presence of Uniform Suction or Injection and Heat Generation. J. Fluids Eng. 2012, 134, 081202. [Google Scholar]
Bejan, A. Second law analysis in heat transfer. Energy 1980, 5, 720–732. [Google Scholar] [CrossRef]
Hayat, T.; Rafiq, M.; Ahmad, B.; Asghar, S. Entropy generation analysis for peristaltic flow of nanoparticles in a rotating frame. Int. J. Heat Mass Transf. 2017, 108, 1775–1786. [Google Scholar] [CrossRef]
Nouri, D.; Pasandideh-Fard, M.; Oboodi, M.J.; Mahian, O.; Sahin, A.Z. Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden expansion and contraction. Int. J. Heat Mass Transf. 2018, 116, 1036–1043. [Google Scholar] [CrossRef]
Dalir, N.; Dehsara, M.; Nourazar, S.S. Entropy analysis for magnetohydrodynamic flow and heat transfer of a Jeffrey nanofluid over a stretching sheet. Energy 2015, 79, 351–362. [Google Scholar] [CrossRef]
Rashidi, M.M.; Abelman, S.; Mehr, N.F. Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Int. J. Heat Mass Transf. 2013, 62, 515–525. [Google Scholar] [CrossRef]
Das, S.; Banu, A.S.; Jana, R.N.; Makinde, O.D. Entropy analysis on MHD pseudo-plastic nanofluid flow through a vertical porous channel with convective heating. Alexand. Eng. J. 2015, 54, 325–337. [Google Scholar] [CrossRef] [Green Version]
Seth, G.S.; Sarkar, S.; Makinde, O.D. Combined Free and Forced Convection Couette-Hartmann Flow in a Rotating Channel with Arbitrary Conducting Walls and Hall Effects. J. Mech. 2016, 32, 613–629. [Google Scholar] [CrossRef]
Mkwizu, M.H.; Makinde, O.D.; Nkansah-Gyekye, Y. Numerical investigation into entropy generation in a transient generalized Couette flow of nanofluids with convective cooling. Sadhana 2015, 40, 2073–2093. [Google Scholar] [CrossRef] [Green Version]
Adesanya, S.O.; Makinde, O.D. Entropy generation in couple stress fluid flow through porous channel with fluid slippage. Int. J. Exergy 2014, 15, 344–362. [Google Scholar] [CrossRef]
Sheremet, M.A.; Oztop, H.F.; Pop, I.; Abu-Hamdeh, N. Analysis of entropy generation in natural convection of nanofluid inside a square cavity having hot solid block: Tiwari and das’ model. Entropy 2016, 18, 6. [Google Scholar] [CrossRef]
Sheremet, M.; Pop, I.; Oztop, H.F.; Abu-Hamdeh, N. Natural convection of nanofluid inside a wavy cavity with a non-uniform heating: Entropy generation analysis. Int. J. Numer. Meth. Heat Fluid Flow 2017, 27, 958–980. [Google Scholar] [CrossRef]
Alharbi, S.O.; Dawar, A.; Shah, Z.; Khan, W.; Idrees, M.; Islam, S.; Khan, I. Entropy Generation in MHD Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface under the Impact of Thermal Radiation and Heat Source/Sink. Appl. Sci. 2018, 8, 2588. [Google Scholar] [CrossRef]
Dawar, A.; Shah, Z.; Khan, W.; Idrees, M.; Islam, S. Unsteady squeezing flow of magnetohydrodynamic carbon nanotube nanofluid in rotating channels with entropy generation and viscous dissipation. Adv. Mech. Eng. 2019, 11, 1–18. [Google Scholar] [CrossRef]
Kumam, P.; Shah, Z.; Dawar, A.; Rasheed, H.U.; Islam, S. Entropy Generation in MHD Radiative Flow of CNTs Casson Nanofluid in Rotating Channels with Heat Source/Sink. Math. Probl. Eng. 2019, 2019, 1–14. [Google Scholar] [CrossRef] [Green Version]
Portavoce, A.; Hoummada, K.; Chow, L. Atomic Transport in Nano-crystalline Thin Films. Defect Diffus. Forum 2016, 367, 140–148. [Google Scholar] [CrossRef]
Mebarek-Oudina, F. Convective heat transfer of Titania nanofluids of different base fluids in cylindrical annulus with discrete heat source. Heat Trans. Asian Res. 2019, 48, 135–147. [Google Scholar] [CrossRef]
Mebarek-Oudina, F.; Bessaïh, R. Oscillatory Magnetohydrodynamic Natural Convection of Liquid Metal between Vertical Coaxial Cylinders. J. Appl. Fluid Mech. 2016, 9, 1655–1665. [Google Scholar] [CrossRef]
Reza, J.; Mebarek Oudina, F.; Makinde, O.D. MHD Slip Flow of Cu-Kerosene Nanofluid in a Channel with Stretching Walls Using 3-Stage Lobatto IIIA formula. Defect Diffus. Forum 2018, 387, 51–62. [Google Scholar] [CrossRef]
Raza, J.; Mebarek-Oudina, F.; Chamkha, A.J. Magnetohydrodynamic flow of molybdenum disulfide nanofluid in a channel with shape effects. Multidisc. Model. Mater. Struct. 2019, 15, 737–757. [Google Scholar] [CrossRef]
Alkasassbeh, M.; Omar, Z.; Mebarek-Oudina, F.; Raza, J.; Chamka, A. Heat transfer study of convective fin with temperature-dependent internal heat generation by hybrid block method. Heat Transf. Asian Res. 2019, 48, 1225–1244. [Google Scholar] [CrossRef]
Hamrelaine, S.; Mebarek-Oudina, F.; Rafik Sari, M. Analysis of MHD Jeffery Hamel Flow with Suction/Injection by Homotopy Analysis Method. J. Adv. Res. Fluid Mech. Therm. Sci. 2019, 58, 173–186. [Google Scholar]
Bhatti, M.M.; Abbas, T.; Rashidi, M.M.; Ali, M.E.-S.; Yang, Z. Entropy Generation on MHD Eyring–Powell Nanofluid through a Permeable Stretching Surface. Entropy 2016, 18, 224. [Google Scholar] [CrossRef]
Rashidi, S.; Akar, S.; Bovand, M.; Ellahi, R. Volume of fluid model to simulate the nanofluid flow and entropy generation in a single slope solar still. Renew. Energy 2018, 115, 400–410. [Google Scholar] [CrossRef]
Bhatti, M.M.; Abbas, T.; Rashidi, M.M.; Ali, M.E.-S. Numerical Simulation of Entropy Generation with Thermal Radiation on MHD Carreau Nanofluid towards a Shrinking Sheet. Entropy 2016, 18, 200. [Google Scholar] [CrossRef]
Esfahani, J.A.; Akbarzadeh, M.; Rashidi, S.; Rosen, M.A.; Ellahi, R. Influences of wavy wall and nanoparticles on entropy generation over heat exchanger plat. Int. J. Heat Mass Trans. 2017, 109, 1162–1171. [Google Scholar] [CrossRef]
Darbari, B.; Rashidi, S.; Esfahani, J.A. Sensitivity Analysis of Entropy Generation in Nanofluid Flow inside a Channel by Response Surface Methodology. Entropy 2016, 18, 52. [Google Scholar] [CrossRef]
Ellahi, R.; Alamri, S.Z.; Basit, A.; Majeed, A. Effects of MHD and slip on heat transfer boundary layer flow over a moving plate based on specific entropy generation. J. Taibah Univ. Sci. 2018, 12, 476–482. [Google Scholar] [CrossRef] [Green Version]
Abbas, T.; Ayub, M.; Bhatti, M.M.; Rashidi, M.M.; Ali, M.E.-S. Entropy Generation on Nanofluid Flow through a Horizontal Riga Plate. Entropy 2016, 18, 223. [Google Scholar] [CrossRef]
Qasim, M.; Khan, Z.H.; Khan, I.; Al-Mdallal, Q.M. Analysis of Entropy Generation in Flow of Methanol-Based Nanofluid in a Sinusoidal Wavy Channel. Entropy 2017, 19, 490. [Google Scholar] [CrossRef]
Mamourian, M.; Shirvan, K.M.; Ellahi, R.; Rahimi, A. Optimization of mixed convection heat transfer with entropy generation in a wavy surface square lid-driven cavity by means of Taguchi approach. Int. J. Heat Mass Transf. 2016, 102, 544–554. [Google Scholar] [CrossRef]
Sufian, M.; Ahmer, M.; Asif, A. Three diamensional squeezing flow in arotating chanel of lower stretching porous wall. Comput. Math. Appl. 2012, 64, 1575–1586. [Google Scholar]
Khan, A.S.; Nie, Y.; Shah, Z. Impact of Thermal Radiation on Magnetohydrodynamic Unsteady Thin Film Flow of Sisko Fluid over a Stretching Surface. Processes 2019, 7, 369. [Google Scholar] [CrossRef]
Saeed, A.; Shah, Z.; Islam, S.; Jawad, M.; Ullah, A.; Gul, T.; Kumam, P. Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation. Coatings 2019, 9, 248. [Google Scholar] [CrossRef]
Feroz, N.; Shah, Z.; Islam, S.; Alzahrani, E.O.; Khan, W. Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model. Entropy 2019, 21, 52. [Google Scholar] [CrossRef]

Figure 1. Physical sketch of the flow.

Figure 2. The combined

ℏ

curves for velocities

f^{'} (ξ) and g (ξ)

and temperature

θ (ξ)

profiles.

Figure 2. The combined

ℏ

curves for velocities

f^{'} (ξ) and g (ξ)

and temperature

θ (ξ)

profiles.

Figure 3. (a,b): Impression of

β_{o}

f (ξ)

and

f^{'} (ξ)

Figure 3. (a,b): Impression of

β_{o}

f (ξ)

and

f^{'} (ξ)

Figure 4. (a,b): Impression of

λ

f (ξ)

and

f^{'} (ξ)

Figure 4. (a,b): Impression of

λ

f (ξ)

and

f^{'} (ξ)

Figure 5. (a,b): Impression of

β

over

f (ξ)

and

f^{'} (ξ)

Figure 5. (a,b): Impression of

β

over

f (ξ)

and

f^{'} (ξ)

Figure 6. (a,b): Impact of

M

f (ξ)

and

f^{'} (ξ)

Figure 6. (a,b): Impact of

M

f (ξ)

and

f^{'} (ξ)

Figure 7. (a,b): Impression of

δ

f (ξ)

and

f^{'} (ξ)

Figure 7. (a,b): Impression of

δ

f (ξ)

and

f^{'} (ξ)

Figure 8. (a,b): Impression of

α

f (ξ)

and

f^{'} (ξ)

Figure 8. (a,b): Impression of

α

f (ξ)

and

f^{'} (ξ)

Figure 9. Impression of

E c

over

θ (ξ)

Figure 9. Impression of

E c

over

θ (ξ)

Figure 10. Impression of

E d

over

θ (ξ)

Figure 10. Impression of

E d

over

θ (ξ)

Figure 11. Impression of

β

over

θ (ξ)

Figure 11. Impression of

β

over

θ (ξ)

Figure 12. Impact of

\Pr

over

θ (ξ)

Figure 12. Impact of

\Pr

over

θ (ξ)

Figure 13. (a,b): Impression

\Pr

N s (ξ)

and

B e (ξ)

Figure 13. (a,b): Impression

\Pr

N s (ξ)

and

B e (ξ)

Figure 14. (a,b): Impression of

λ

N s (ξ)

and

B e (ξ)

Figure 14. (a,b): Impression of

λ

N s (ξ)

and

B e (ξ)

Figure 15. (a,b): Impression of

E c

N s (ξ)

and

B e (ξ)

Figure 15. (a,b): Impression of

E c

N s (ξ)

and

B e (ξ)

Figure 16. (a,b): Impression of

E d

N s (ξ)

and

B e (ξ)

Figure 16. (a,b): Impression of

E d

N s (ξ)

and

B e (ξ)

Table 1. Impacts of

β, β_{o}, α

, and

M

C_{f x}

Table 1. Impacts of

β, β_{o}, α

, and

M

C_{f x}

$β$	$β_{o}$	$λ$	$α$	$M$	$C_{f x}$
1.2	0.1	0.1	0.1	0.2	0.955529
1.3					0.344419
1.4					−0.247015
1.2	0.2				0.984942
	0.3				1.009420
	0.4				1.029430
1.2	0.1	0.2			0.956860
		0.4			0.959551
		0.5			0.962230
1.2	0.1	0.1	0.2		0.955522
			0.3		0.955510
			0.4		0.955493
1.2	0.1	0.1	0.1	0.3	0.956887
				0.4	0.958211
				0.5	0.959551

Table 2. The impacts of

β, \Pr, E c, E d

, and

M

N u_{x}

Table 2. The impacts of

β, \Pr, E c, E d

, and

M

N u_{x}

$β$	$\Pr$	$E c$	$E d$	$M$	$N u_{x}$
1.2	0.1	0.5	0.6	0.2	0.323922
1.3					0.534419
1.4					0.169332
1.2	0.2				0.256999
	0.3				0.190056
	0.4				0.123293
1.2	0.1				0.056644
		0.6			0.200577
		0.7			0.077723
		0.8			0.046114
1.2	0.1	0.5	0.7		0.313305
			0.8		0.302689
			0.9		0.292072
1.2	0.1	0.5	0.6	0.3	0.292126
				0.4	0.200330
				0.5	0.128534

Table 3. Comparison of HAM and numerical results for velocity profile.

$ξ$	HAM	Numerical	Difference
0.0	$0.1$	$- 6.9368 \times 10^{- 9}$	1
0.2	$0.269142$	$0.170298$	0.098844
0.4	$0.383833$	$0.288010$	0.095823
0.6	$0.454293$	$0.362311$	0.091982
0.8	$0.490022$	$0.401375$	0.088647
1.0	$0.5$	$0.412773$	0.087227

Table 4. Comparison of HAM and numerical results for temperature profile.

$ξ$	HAM	Numerical	Difference
0.0	$1$	$1$	0.0
0.2	$0.799983$	$0.800563$	$5.8 \times 10^{- 4}$
0.4	$0.599511$	$0.599659$	$1.48 \times 10^{- 4}$
0.6	$0.399278$	$0.398728$	$5.5 \times 10^{- 4}$
0.8	$0.199484$	$0.198671$	$8.13 \times 10^{- 4}$
1.0	$1.1102 \times 10^{- 16}$	$1.5240 \times 10^{- 9}$	$1.52 \times 10^{- 9}$

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Zubair, M.; Shah, Z.; Dawar, A.; Islam, S.; Kumam, P.; Khan, A. Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect. Entropy 2019, 21, 747. https://doi.org/10.3390/e21080747

AMA Style

Zubair M, Shah Z, Dawar A, Islam S, Kumam P, Khan A. Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect. Entropy. 2019; 21(8):747. https://doi.org/10.3390/e21080747

Chicago/Turabian Style

Zubair, Muhammad, Zahir Shah, Abdullah Dawar, Saeed Islam, Poom Kumam, and Aurangzeb Khan. 2019. "Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect" Entropy 21, no. 8: 747. https://doi.org/10.3390/e21080747

APA Style

Zubair, M., Shah, Z., Dawar, A., Islam, S., Kumam, P., & Khan, A. (2019). Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect. Entropy, 21(8), 747. https://doi.org/10.3390/e21080747

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Entropy Generation Optimization in Squeezing Magnetohydrodynamics Flow of Casson Nanofluid with Viscous Dissipation and Joule Heating Effect

Abstract

1. Introduction

2. Problem Formulation

3. Entropy Analysis

3.1. Bejan Number

3.2. Solution by HAM

3.3. HAM Convergence

4. Results and Discussion

5. Comparison of HAM with Numerical Result

6. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

Nominiclature

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI