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Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

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Abstract

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

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Abbreviations

u, v :

components of velocity

v f :

kinematic fluid viscosity

Pr :

Prandtl number

x, y :

Cartesian coordinates

k f :

fluid thermal conductivity

τ w :

wall shear stress

c pf :

fluid specific heat

μ f :

fluid dynamic viscosity

U 0, U ∞ :

stretching and free stream velocities

f′, θ, φ :

dimensionless velocity, temperature, and concentration

α f :

thermal diffusivity of the fluid

α 11 :

viscoelastic parameter

α 22 :

cross viscous parameter

γ 11, γ 22, γ 33 :

fourth-grade fluid parameters

M :

melting parameter

Me :

magnetic parameter

σ f :

electrical conductivity

B 0 :

magnetic field strength

c b :

drag coefficient

φ1 :

porosity of the porous medium

λ f :

fluid latent heat

Fr :

Forchheimer number

k 1 :

permeability of the porous medium

λ :

porosity parameter

T ∞ :

ambient temperature

β :

third-grade fluid parameter

ρ f :

fluid density

τ*:

heat flux relaxation time

β 11 :

Biot number for temperature

Re x :

local Reynolds number

ħ f, ħ θ, ħ φ :

convergence control variables

β 22 :

Biot number for concentration

f 0, θ 0, φ 0 :

initial guesses

τ :

Cauchy stress tensor

γ :

thermal relaxation parameter

α i, β i, γ i :

material constants

C s :

surface heat capacity

T w :

wall temperature

ψ :

stream function

T 0 :

ambient temperature

U e(x):

free-stream velocity

ε f, ε θ :

average square residual errors at the mth-order approximation.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan

    T. Hayat & K. Muhammad

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 80203, Saudi Arabia

    A. Alsaedi

Authors
  1. T. Hayat
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Corresponding author

Correspondence to K. Muhammad.

Additional information

Citation: HAYAT, T., MUHAMMAD, K., and ALSAEDI, A. Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium. Applied Mathematics and Mechanics (English Edition), 42(12), 1787–1798 (2021) https://doi.org/10.1007/s10483-021-2798-6

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Hayat, T., Muhammad, K. & Alsaedi, A. Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium. Appl. Math. Mech.-Engl. Ed. 42, 1787–1798 (2021). https://doi.org/10.1007/s10483-021-2798-6

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  • DOI: https://doi.org/10.1007/s10483-021-2798-6

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