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Study of Heat Transport in Bénard-Darcy Convection with g-Jitter and Thermo-Mechanical Anisotropy in Variable Viscosity Liquids

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Abstract

The effects of temperature-dependent viscosity, gravity modulation and thermo-mechanical anisotropies on heat transport in a low-porosity medium are studied using the Ginzburg–Landau model. The effect of gravity modulation is to decrease the Nusselt number, Nu and variable viscosity leads to increase in Nu. The thermo-mechanical anisotropies have opposite effect on Nu with thermal anisotropy decreasing the heat transport.

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Abbreviations

A :

Amplitude of convection

C :

Specific heat capacity

g = (0, 0, −g):

Acceleration due to gravity

k :

Permeability tensor

k :

Wave number

K h, K v :

Permeability in horizontal and vertical direction

Nu :

Nusselt number

p :

Dynamic pressure

q = (u, 0, w):

Velocity vector

Ra :

Darcy-Rayleigh number

T :

Temperature

t :

Time

V :

Variable viscosity parameter

Va :

Darcy–Prandtl number or Vadasz number

(x, z):

Horizontal and vertical coordinates

α :

Coefficient of thermal expansion

γ :

Ratio of heat capacity

δ 0 :

Small parameter indicating variation of viscosity with temperature

δ 1 :

Amplitude of gravity modulation

\({\epsilon = \frac{K_{\rm h}}{K_{\rm v}}}\) :

Ratio of permeabilities in horizontal and vertical direction (mechanical anisotropy parameter)

\({\epsilon_{1}}\) :

Small quantity indicative of weak variation

\({\eta = \frac{\chi_{\rm h}}{\chi_{\rm v}}}\) :

Ratio of conductivities in horizontal and vertical direction (thermal anisotropy parameter)

μ :

Fluid viscosity

ν :

Kinematic viscosity

ρ, ρ 0 :

Fluid density and reference fluid density

χ h, χ v :

Thermometric conductivity in horizontal and vertical direction

ψ :

Stream function

Ω 0 :

Dimensional frequency of gravity modulation

b:

Basic state

f:

Property pertaining to fluid

m:

Property pertaining to porous media

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Authors and Affiliations

  1. Department of Mathematics, Central College Campus, Bangalore University, Bangalore, 560 001, India

    P. G. Siddheshwar

  2. Department of Mathematics, Maharani’s Science College for Women, Bangalore, 560 001, India

    R. K. Vanishree & Anthony Christy Melson

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  1. P. G. Siddheshwar
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  2. R. K. Vanishree
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  3. Anthony Christy Melson
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Correspondence to R. K. Vanishree.

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Siddheshwar, P.G., Vanishree, R.K. & Melson, A.C. Study of Heat Transport in Bénard-Darcy Convection with g-Jitter and Thermo-Mechanical Anisotropy in Variable Viscosity Liquids. Transp Porous Med 92, 277–288 (2012). https://doi.org/10.1007/s11242-011-9901-z

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