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Impact of Thermal Non-equilibrium on the Stability of Natural Convection in an Oldroyd-B Fluid-Saturated Vertical Porous Layer with Internal Heat Sources

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Abstract

The stability of natural convection in a vertical layer of heat-generating Darcy porous medium saturated with an Oldroyd-B fluid using a local thermal non-equilibrium (LTNE) model has been investigated numerically. The impermeable vertical walls of the porous layer are maintained at different uniform temperatures. As a consequence of LTNE model, two temperature equations representing the fluid and solid phases separately are used for the heat transport equation. A uniform volumetric heating in both fluid and solid phases is considered and the transfer of heat between the phases is considered in the basic state. The internal heating introduced asymmetry in the basic flow which led to the existence of competing modes. The intricacies of internal heat source strength in the fluid and the solid phases are clearly discerned on the stability of the system. The stress relaxation parameter \(\Lambda_{1}\), fluid-heat generation parameter \(Q_{f}\), solid-heat generation parameter \(Q_{s}\) and the porosity-modified conductivities ratio \(\gamma\) were found to exhibit destabilizing effect on the system, while the strain retardation parameter \(\Lambda_{2}\) shows an opposite trend even in the presence of internal heating.

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Abbreviations

\(a\) :

Vertical wave number

\(c\) :

Wave speed

\(c_{i}\) :

Growth rate

\(c_{r}\) :

Phase velocity

2d :

Thickness of the porous layer

g :

Gravitational acceleration

h :

Inter-phase heat transfer coefficient

\(H\) :

Scaled inter-phase heat transfer coefficient

k :

Thermal conductivity

K :

Permeability

\(q^{\prime\prime\prime}_{{}}\) :

Rate of heat generation per unit volume

\(Q\) :

Heat generation parameter

\(R_{D}\) :

Darcy-Rayleigh number

\(t\) :

Time

T :

Temperature

\(T_{1}\) :

Temperature of the left boundary

\(T_{2}\) :

Temperature of the right boundary

x :

Coordinate across the channel

\(z\) :

Coordinate along the channel

\(\alpha\) :

Diffusivity ratio

β :

Thermal expansion coefficient

\(\gamma\) :

Porosity-modified conductivity ratio

\(\varepsilon\) :

Porosity of the medium

\(\Theta\) :

Disturbance fluid temperature

\(\kappa\) :

Thermal diffusivity

\(\lambda _{1}\) :

Stress relaxation time constant

\(\lambda_{2}\) :

Strain retardation time constant

\(\Lambda_{1}\) :

Relaxation parameter

\(\Lambda_{2}\) :

Retardation parameter

\(\mu\) :

Fluid viscosity

\(\rho _{0}\) :

Reference density at \(T_{0}\)

\(\Phi\) :

Disturbance solid temperature

\(\psi\) :

Stream function

\(\Psi\) :

Disturbance stream function

b :

Basic state

c :

Critical state

f :

Fluid phase

s :

Solid phase

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Acknowledgements

We are indebted to Professor D.A.S. Rees, and Professor Stetsyuk for providing the English translated version of the paper of Alishaev and Mirzadjanzade (1975). We thank the referees for their most valuable comments that helped us to modify the paper to the present form. The authors are grateful to their respective institutes of working for their encouragement.

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Authors and Affiliations

  1. Department of Mathematics, PES University, Bangalore, 560085, India

    B. M. Shankar

  2. Department of Mathematics, Bangalore University, Bangalore, 560056, India

    I. S. Shivakumara & S. B. Naveen

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  1. B. M. Shankar
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  2. I. S. Shivakumara
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  3. S. B. Naveen
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Correspondence to B. M. Shankar.

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Shankar, B.M., Shivakumara, I.S. & Naveen, S.B. Impact of Thermal Non-equilibrium on the Stability of Natural Convection in an Oldroyd-B Fluid-Saturated Vertical Porous Layer with Internal Heat Sources. Transp Porous Med 133, 437–458 (2020). https://doi.org/10.1007/s11242-020-01431-y

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