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Stability of natural convection in a vertical layer of Brinkman porous medium

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Abstract

A classical linear stability theory is applied to emphasize the effect of inertia on the stability of buoyancy-driven parallel shear flow in a vertical layer of porous medium. The Lapwood–Brinkman model with fluid viscosity different from effective viscosity is used to describe the flow in a porous medium. The resulting eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Darcy–Rayleigh number \(R_\mathrm{Dc} \), the critical wave number \(a_\mathrm{c}\) and the critical wave speed \(c_\mathrm{c}\) are computed over a wide range of values of the Darcy–Prandtl number \(Pr_\mathrm{D}\) and the Darcy number \({\tilde{D}}a\). Depending on the choice of physical parameters, instability occurs due to the presence of inertia. The value of \(Pr_\mathrm{D}\) at which the transition from stationary to traveling-wave mode instability takes place increases with decreasing \({\tilde{D}}a\). Besides, the effect of decreasing \({\tilde{D}}a\) shows destabilizing effect if the instability is via stationary mode, and on the contrary, it exhibits a dual behavior if the instability is through traveling-wave mode. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state. In the energy spectrum, transition of instability from one type to another is found to take place as a function of \(Pr_\mathrm{D}\). The disturbance kinetic energy due to surface drag and viscous force plays no significant role in the stability of flow throughout the domain of \(Pr_\mathrm{D}\) considered.

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Abbreviations

a :

Vertical wave number

c :

Wave speed

\(c_r\) :

Phase velocity

\(c_i\) :

Growth rate

\(D=\mathrm{d}/\mathrm{d}x\) :

Differential operator

\({\tilde{D}}a\) :

Darcy number

\(E_\mathrm{b}, E_\mathrm{d}, E_\mathrm{D}, E_\mathrm{s}\) :

Disturbance kinetic energies

\({\vec {g}}\) :

Acceleration due to gravity

h :

Half-width of the porous layer

\({\hat{i}}\) :

Unit vector in the x-direction

\({\hat{k}}\) :

Unit vector in the z-direction

k :

Permeability

p :

Pressure

\(Pr_\mathrm{D}\) :

Darcy–Prandtl number

\({\vec {q}}=(u,v,w)\) :

Velocity vector

\(R_\mathrm{D}\) :

Darcy–Rayleigh number

t :

Time

T :

Temperature

\(T_\mathrm{c}, T_\mathrm{d}\) :

Disturbance thermal energies

\(T_1\) :

Temperature of the left boundary

\(T_2\) :

Temperature of the right boundary

\(W_\mathrm{b}\) :

Basic velocity

\(\left( {x,y,z} \right) \) :

Cartesian coordinates

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Volumetric expansion coefficient

\(\theta \) :

Amplitude of the perturbed temperature

\(\mu \) :

Fluid viscosity

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

Effective fluid viscosity

\(\nu \) :

Kinematic viscosity

\(\rho \) :

Fluid density

\(\rho _0\) :

Reference density at \(T_0\)

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

Porosity of the porous medium

\(\chi \) :

Ratio of heat capacities

\(\psi \left( {x,z,t} \right) \) :

Stream function

\(\Psi \) :

Amplitude of the perturbed stream function

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Authors and Affiliations

  1. Department of Mathematics, PES University, Bangalore, 560 085, India

    B. M. Shankar

  2. ISRO Satellite Centre, Bangalore, 560 017, India

    Jai Kumar

  3. Department of Mathematics, Bangalore University, Bangalore, 560 056, India

    I. S. Shivakumara

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Shankar, B.M., Kumar, J. & Shivakumara, I.S. Stability of natural convection in a vertical layer of Brinkman porous medium. Acta Mech 228, 1–19 (2017). https://doi.org/10.1007/s00707-016-1690-6

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  • DOI: https://doi.org/10.1007/s00707-016-1690-6

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