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Entropy generation in two-immiscible MHD flow of pulsating Casson fluid in a vertical porous space with Slip effects

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Abstract

The unsteady two immiscible MHD free convective flow of Casson liquid through a vertical channel, with a porous medium, is studied numerically in this investigation. Using a double perturbation approach, the governing flow equations are reduced to a system of connected partial differential equations, which are then solved using the 4th-order numerical Runge–Kutta method coupled with the shooting approach in Mathematica. The velocity and thermal slip conditions have been accommodated in this model. The interaction of permeability of the porous medium, energy dissipation, Joule heating, and thermal radiation are taken into consideration. The computational upshot is also described explicitly to examine the consequences of pertinent parameters. The computational results are analyzed to investigate the effects associated with pertinent parameters which include the Hartmann number of two regions, Darcy number, a ratio of Porous medium permeability, Grashof number, Radiation parameter, heat source, and Prandtl number. The characteristics of the essential regulating parameters on flow frameworks of velocity, temperature, Entropy generation, Bejan number, and heat transfer rate are analyzed correctly via plots, and skin friction and flow rate are given in tabular form. As the radiation parameter rises both velocity and temperature of the Casson fluid decreases. The rate of entropy generation falls with a rise in the magnetic field. The Bejan number escalates as it moves forward from the lower wall to the channel’s center. In the later part of the channel, the Bejan number starts to drastically fall and reaches a minimum at the upper wall.

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Abbreviations

\(x^*,y^*\) :

Cartesian coordinates (m )

x, y :

Dimensionaless cartesian coordinates

\(u_1^*,u_2^*\) :

Dimensional velocity in \(x^*-direction\) and \(y^*-direction\) (\({m}{\text {s}^{-1}}\) )

b :

Fluid width parameter (m)

\(B_0\) :

Uniform Magnetic field

\(T_1^*,T_2^*\) :

Temperature of the fluids of both regions (K )

\(t*\) :

Time(s)

P :

Dimensionless pressure

\(k*\) :

Rosseland mean absorption coefficient

g :

Gravational acceleration (\(\text {m}{\text {s}^{-2}}\) )

t :

Dimensionless time

p :

Fluid pressure (\(\text {N}{\text {m}^{-2}}\) )

Fs :

Forchheimer number

h :

Distance of the channel (m)

\(q_r\) :

Radiative Heat flux (\(\text {Kg}{\text {s}^{-3}}\) )

Re :

Reynolds number

Gr :

Grashoff number

Rd :

Thermal Radiation

Da :

Darcy number (\({K}{{{L}^{-2}}}\))

Ec :

Eckert number

Pr :

Prandtl number

Cp :

Specififc heat (\(Jk{{g}^{-1}}{{K}^{-1}}\))

\(k_1,k_2\) :

Ratio of thermal conductivity of two regions (\(W{{m}^{-1}}{{K}^{-1}}\))

\(k_{\text{p}}\) :

Ratio of permeabilty of two regions (\({{m}^{2}}\))

\(K_1,K_2\) :

Are the permeabilty of two regions (\({{m}^{2}}\))

H :

Frequency parameter

\(Q_1,Q_2\) :

Heat sources of both regions

Nu :

Nusselt number

\(Eg_1,Eg_2\) :

Entropy generations of region I and II

\(M_1,M_2\) :

Hartmann numbers of two regions I and II

I :

Non Darcy parameter

\(Be_1,Be_2\) :

Bejan numbers of region I and II

\(\beta _1\) :

Ratio of viscosity of two regions

\({{\alpha }_{1}}\) :

Is the slip coefficient at the porous walls

\({{h}_{\text{f}}}\) :

Is the slip coefficient at the surface of the porous walls

\(\alpha \) :

Ratio of densities of both regions

\(\beta \) :

Casson fluid parameter.

\({{\pi }_{\text{c}}}\) :

Critical value of this product based on the non-Newtonian fluid.

\({{\mu }_{\text{B}}}\) :

Plastic dynamic viscosity of the non-Newtonian fluid

\({{P}_{\text{y}}}\) :

Yield stress of the fluid

\(\sigma *\) :

Stefan-Boltzmann constant

\(\rho _1,\rho _2\) :

Densities of two regions(\(\text {Kg}{{m}^{-3}}\) )

\(\nu \) :

Kinemetic viscosity (\({{m}^{2}}{{s}^{-1}}\) )

\(\theta _1,\theta _2\) :

Dimensionless temperatures of two regioms

\(u_{\text{t}}\) :

Unsteady velocity

\(\theta _{\text{t}}\) :

Unsteady temperatures

\(\sigma _1,\sigma _2\) :

Ratio of electrical conductivities (\(\Omega ^{-1}{{m}^{-1}}\) )

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Author notes

  1. M. Padma Devi and K. Vajravelu are contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, VIT-AP University, Inavolu, Guntur, Andhra Pradesh, 522237, India

    M. Padma Devi & S. Srinivas

  2. Department of Mathematics, University of Central Florida, Orlanda, FL, 32816, USA

    K. Vajravelu

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Contributions

PThe writing up of the manuscript and completion of the work is by M. Padma Devi. Problem formulation and Analytical work by Prof. S. Srinivas, suggestion and verification of the results by Prof. K. Vajravelu.

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Correspondence to M. Padma Devi.

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Appendix

Appendix

$$\begin{aligned}{} & {} {{p}_{1}}=M_{1}^{2}+\dfrac{1}{{{K}_{1}}}, {{p}_{2}}={{p}_{1}}+i\omega R, {{p}_{3}}=M_{2}^{2}+\dfrac{1}{{{K}_{2}}}, {{p}_{4}}=\dfrac{\beta {{\beta }_{1}}Gr}{\alpha }, {{p}_{5}}={{p}_{3}}+\dfrac{1}{{{K}_{2}}}+R\beta i\omega , {{p}_{6}}=1+\dfrac{4}{3}Rd,\\ {{p}_{7}} & ={{\phi }_{1}}+R\Pr i\omega , {{p}_{8}}={{\phi }_{2}}K+\dfrac{R\Pr i\omega \beta }{\alpha }, {{p}_{9}}=\dfrac{{{\phi }_{1}}}{{{p}_{6}}}. \end{aligned}$$

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Padma Devi, M., Srinivas, S. & Vajravelu, K. Entropy generation in two-immiscible MHD flow of pulsating Casson fluid in a vertical porous space with Slip effects. J Therm Anal Calorim (2024). https://doi.org/10.1007/s10973-024-13337-8

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