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Boundary and inertia effects on the stability of natural convection in a vertical layer of an anisotropic Lapwood–Brinkman porous medium

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Abstract

The stability of natural convection in a fluid-saturated vertical anisotropic porous layer is investigated. The vertical rigid walls of the porous layer are maintained at different constant temperatures, and anisotropy in both permeability and thermal diffusivity is considered. The flow in the porous medium is described by the Lapwood–Brinkman model, and the stability of the basic flow is analysed numerically using Chebyshev collocation method. The presence of inertia is to inflict instability on the system and in the absence of which the system is always found to be stable. The mechanical and thermal anisotropies exhibit opposing contributions on the stability characteristics of the system. The mode of instability is interdependent on the values of Prandtl number and thermal anisotropy parameter, while it remains unaltered with the mechanical anisotropy parameter. The effect of increasing Prandtl and Darcy numbers shows a destabilizing effect on the system. Besides, simulations of secondary flow and energy spectrum have been analysed for various values of physical parameters at the critical state.

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Abbreviations

a :

Vertical wave number

c :

Wave speed

\(c_\mathrm{r}\) :

Phase velocity

\(c_i \) :

Growth rate

Da :

Darcy number

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

Disturbance kinetic energies

\(\vec {g}\) :

Acceleration due to gravity

G :

Grashof number

h :

Half-width of the porous layer

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

Unit vector in x-direction

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

Unit vector in z-direction

\(\underset{{\thicksim }}{K}\) :

Second-order permeability tensor

\({K}_{x}\) :

Transverse component of the permeability

\(K_z\) :

Longitudinal component of the permeability

\(K_1\) :

Mechanical anisotropy parameter

p :

Pressure

Pr :

Prandtl number

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

Velocity vector

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 co-ordinates

\(\alpha \) :

Volumetric thermal expansion coefficient

\(\beta \) :

Temperature gradient

\(\chi \) :

Ratio of heat capacities

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

Porosity of the porous medium

\(\underset{{\thicksim }}{\kappa }\) :

Thermal diffusivity tensor

\(\kappa _x \) :

Transverse component of the thermal diffusivity

\(\kappa _z \) :

Longitudinal component of the thermal diffusivity

\(\kappa _1 \) :

Thermal anisotropy parameter

\(\mu \) :

Fluid viscosity

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

Effective fluid viscosity

\(\nu \) :

Kinematic viscosity

\(\theta \) :

Amplitude of perturbed temperature

\(\rho \) :

Fluid density

\(\rho _0 \) :

Reference density at \(T_0 \)

\(\psi \) :

Stream function

\(\Psi \) :

Amplitude of vertical component of perturbed velocity

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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

Authors
  1. B. M. Shankar
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  2. Jai Kumar
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  3. I. S. Shivakumara
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Correspondence to B. M. Shankar.

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Shankar, B.M., Kumar, J. & Shivakumara, I.S. Boundary and inertia effects on the stability of natural convection in a vertical layer of an anisotropic Lapwood–Brinkman porous medium. Acta Mech 228, 2269–2282 (2017). https://doi.org/10.1007/s00707-017-1831-6

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