Hall current effect on unsteady rotational flow of carbon nanotubes with dust particles and nonlinear thermal radiation in Darcy–Forchheimer porous media

The present discussion is about the unsteady two-dimensional flow of mixed convection and nonlinear thermal radiation in the presence of water-based carbon nanotubes over the vertically convected stretched sheet embedded in a Darcy’ Forchheimer porous media. Saffman’s proposed model is used for the suspension of fine dust particles in the nanofluid. A strong magnetic field (MHD) is applied normal to the flow which governs the Hall current effects. Khanafer Vafai Lightstone model estimated the effect of thermal conductivity and viscosity of the carbon nanotubes. Boundary layer approximation is utilized to built the nonlinear partial differential equations (PDEs). Similarity transformation is applied to convert these PDEs into the system of ordinary differential equations. Problem is solved numerically by bvp4c, using MATLAB software. It is observed through the analysis that the thermal field of nanofluid and the temperature boundary layer are much more higher than that of the dust phase, and these are further enhanced for the higher radiation parameter.

\(b,\alpha \) :

A real constant

\(B_{0}\) :

Magnetic induction \((\hbox {kg}\,\hbox {s}^{-2}\,\hbox {A}^{-1})\)

Bi :

Biot number

C :

Specific heat \((\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1})\)

E :

Intensity vector of the electric field

\(F^{\prime }\) :

Dimensionless velocity of dust

F :

Inertia coefficient of porous medium

\(F_{\mathrm{p}}\) :

Force due to dust particles

\(F_{\mathrm{b}}\) :

Body forces

\(f^{\prime }\) :

Dimensionless velocity

\(F_{\mathrm{r}}\) :

Inertia coefficient

g :

Gravitational acceleration \((\hbox {m}\,\hbox {s}^{-2})\)

\(Gr_{\mathrm{x}}\) :

Grashof number

h :

Dimensionless transverse velocity

\(h_{\mathrm{c}}\) :

Heat flux coefficient \((\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1})\)

J :

Current density vector

\(K^{*}\) :

Permeability of porous medium

l :

Mass concentration of dust

\({m}_{\mathrm{p}}\) :

Mass of dust particles

Nu :

Local Nusselt number

\(n_{\mathrm{e}}\) :

Number density of electron

\(P_{\mathrm{e}}\) :

Electron pressure

P :

Pressure

Pr :

Prandtl number

\(Q_{\mathrm{p}}\) :

Thermal interaction between nanoparticles and dust phase

\(q_{\mathrm{w}}\) :

Surface heat flux

\(q_{\mathrm{r}}\) :

Radiation heat flux

Rd :

Thermal radiation parameter

\(r_{\mathrm{p}}\) :

Radius of dust particles

\(Re_{\mathrm{x}}\) :

Local Reynolds number

S :

Unsteadyness parameter

t :

Time

t :

Fluid temperature \((\hbox {K})\)

Uw :

Deformation velocity of the sheet

U(u, v, w):

Velocity vector \((\hbox {m}\,\hbox {s}^{-1})\)

V(u, v, w):

Velocity vector \((\hbox {m}\,\hbox {s}^{-1})\)

(x, y, z):

Axial and normal coordinates

\(\beta _{{\upnu }}\) :

Momentum dust parameter

\(\beta _{\mathrm{T}}\) :

Thermal expansion coefficient

\(\beta _{\mathrm{t}}\) :

Thermal dust parameter

\(\gamma \) :

Specific heat ratio

\(\delta \) :

Porosity parameter

\(\eta \) :

Dimensionless normal distance

\(\theta \) :

Dimensionless temperature

\(\theta _{\mathrm{w}}\) :

Temperature ratio parameter

\(\kappa \) :

Thermal conductivity \((\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1})\)

\(\lambda \) :

Mixed convection parameter

\(\mu \) :

Dynamic viscosity

\(\sigma \) :

Electric conductivity

\(\sigma ^{*}\) :

Stefan–Boltzmann constant \((\hbox {m}^{2}\,\hbox {s}^{-1})\)

\(\tau _{\mathrm{e}}\) :

Electron collision time

\(\tau _{\mathrm{T}}\) :

Thermal relaxation time of dust phase

\(\tau _{\upnu }\) :

Momentum relaxation time of dust phase

\(\tau _{\mathrm{wx}}\) :

Wall shear stress in x direction

\(\omega \) :

Frequency

\(\omega _{\mathrm{e}}\) :

Electron frequency

\(\varkappa \) :

Nanoparticle volume fraction

\(\psi \) :

Stream function \((\hbox {m}^{2}\,\hbox {s}^{-1}) \)

\(\varOmega \) :

Rotational parameter

\(\rightthreetimes \) :

Absorption coefficient

nf:

Nanofluid

f:

Base fluid

s:

Nanoparticles (CNTs)

p:

Dust phase/particles

w:

Wall

e:

Charge on electron

Authors and Affiliations

1. Department of Mathematics, University of Lahore, Chenab Campus, Gujrat, Pakistan

   M. Bilal

2. Bahria University, Islamabad, Pakistan

   M. Ramzan

Corresponding author

Correspondence to M. Bilal.

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