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Effect of Prescribed Heat Sources on Convective Unsteady MHD Flow of Williamson Nanofluid Through Porous Media: Darcy–Forchheimer Model

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Abstract

An unsteady two-dimensional mixed convection stagnation point flow induced past a surface stretching linearly embedded in a porous medium is examined numerically in this study, utilizing the Darcy–Forchheimer model. The medium is supposed to be jammed with an incompressible non-Newtonian Williamson nanofluid. The flow is examined for two different heat transfer mechanisms namely, PST and PHF, in the presence of a time-dependent magnetic field using the Buongiorno nanofluid model. The relevant governing PDEs such as continuity, momentum, energy, and concentration equations are first converted into non-linear ODEs, utilizing similarity transformations, and then solved using bvp4c solver in MATLAB. The results have been displayed in the form of figures and tables for both heat transfer mechanisms, PST and PHF. The purpose of this work is to examine the combined effects of unsteadiness, mixed convection, and magnetic field using the Darcy–Forchheimer model on the flow of non-Newtonian Williamson nanofluid near the stagnation-point region past a linearly stretching sheet for two different cases of heat transfer namely, PST and PHF. Our study concludes that escalating unsteadiness, thermal convection, and velocity ratio correspond to a rise in the local Nusselt number which corresponds to a better rate of heat transfer. Conductive heat transfer dominates for increasing permeability of the medium in both PST and PHF heat transfer mechanisms. These findings are useful for enhancing the rate of cooling of the sheet during extrusion of polymer sheets, emulsion coated sheets, plastic film drawing, glass fiber production, etc.

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Abbreviations

BVP:

Boundary value problem

HAM:

Homotopy analysis method

MHD:

Magnetohydrodynamics

ODE:

Ordinary differential equation

PDE:

Partial differential equation

PHF:

Prescribed heat flux

PST:

Prescribed surface temperature

\(\beta \) :

Unsteady parameter

\(\epsilon \) :

Velocity ratio parameter

\(\eta \) :

Dimensionless variable

\(\lambda \) :

Local porosity parameter

\(\phi \) :

Dimensionless concentration function

\(\theta \) :

Dimensionless temperature function

\(C_b\) :

Drag coefficient

\(C_{fx}\) :

Skin friction coefficient

f :

Dimensionless stream function

Fr :

Forcheimmer number

G :

Thermal convection parameter

\(Gr_x\) :

Thermal buoyancy parameter

M :

Magnetic parameter

N :

Buoyancy ratio parameter

\(N_c\) :

Specific heat ratio

\(N_{bt}\) :

Diffusivity ratio

\(Nu_x\) :

Nusselt number

Pr :

Prandtl’s number

Re :

Reynold’s number

Sc :

Schmidt number

\(Sh_x\) :

Sherwood number

We :

Williamson parameter

\(\alpha \) :

Thermal diffusivity (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\(\Gamma \) :

Time constant (\(\mathrm{s}^{-1}\))

\(\kappa \) :

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

\(\mu \) :

Dynamic viscosity (Pa s)

\(\nu \) :

Kinematic viscosity (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\(\rho \) :

Density (\(\mathrm{kg}\,\mathrm{m}^{-3})\)

\(\rho _p\) :

Density at constant pressure

\(\sigma \) :

Electric conductivity of the fluid (\(\mathrm{S}\,\mathrm{m}^{-1}\))

\(\tau _w\) :

Shear stress at the wall (\(\mathrm{N}\,\mathrm{m}^{-2}\))

a, b :

Positive constant for stretching rate (\(\mathrm{s}^{-1}\))

B :

Applied magnetic field intensity (\(\mathrm{A}\,\mathrm{m}^{-1}\))

C :

Concentration (\(\mathrm{k}\,\mathrm{m}^{-3})\)

\(C_p\) :

Specific heat at constant pressure (\(\mathrm{J}\,\mathrm{kg}^{-1}\,\mathrm{K}\))

\(D_B\) :

Brownian diffusion coefficient (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\(D_T\) :

Thermophoretic diffusion coefficient (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\(k^*\) :

Permeability (\(\mathrm{H}\,\mathrm{m}^{-1}\))

\(q_m\) :

Mass flux (\(\mathrm{kg}\,\mathrm{s}^{-1}\))

\(q_w\) :

Heat flux (\(\mathrm{W}\,\mathrm{m}^{-2}\))

T :

Temperature (K)

u, v :

Horizontal and vertical velocity components (\(\mathrm{m}\,\mathrm{s}^{-1}\))

\(U_\infty \) :

Free stream velocity (\(\mathrm{m}\,\mathrm{s}^{-1}\))

\(U_w\) :

Stretching velocity (\(\mathrm{m}\,\mathrm{s}^{-1}\))

x, y :

Cartesian coordinates (m)

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  1. Department of Mathematics and Statistics, Career Point University, Hamirpur, HP, India

    Archie Thakur & Shilpa Sood

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  1. Archie Thakur
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Thakur, A., Sood, S. Effect of Prescribed Heat Sources on Convective Unsteady MHD Flow of Williamson Nanofluid Through Porous Media: Darcy–Forchheimer Model. Int. J. Appl. Comput. Math 8, 74 (2022). https://doi.org/10.1007/s40819-022-01271-y

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