Transport in Porous Media

, Volume 99, Issue 2, pp 359–376 | Cite as

Heat Transport in an Anisotropic Porous Medium Saturated with Variable Viscosity Liquid Under G-jitter and Internal Heating Effects

  • Alok Srivastava
  • B. S. Bhadauria
  • P. G. Siddheshwar
  • I. Hashim
Article
First Online:
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Accepted:

Abstract

Thermo-rheological effect of temperature-dependent viscous fluid saturating a porous medium has been studied in the presence of imposed time periodic gravity field and internal heat source. Weak nonlinear stability analysis has been performed by using the power series expansion in terms of the amplitude of gravity modulation, which is considered to be small. Nusselt number is calculated numerically using Ginzburg–Landau equation. The nonlinear effects of thermo-mechanical anisotropies, internal heat source parameter, Vadász number, thermo-rheological parameter and amplitude of gravity modulation have been obtained and depicted graphically. Streamlines and isotherms have been drawn for different times. Comparisons have been made between various physical systems.

Keywords

Ginzburg–Landau model Gravity modulation Porous media Anisotropy Temperature-dependent viscosity Internal heat source Nonlinear stability 

List of Symbols

Latin Symbols

\(A\)

Amplitude of convection

\(a_1\)

Amplitude of gravity modulation

\(d\)

Depth of the fluid layer

\(\mathbf g \)

Acceleration due to gravity

\(k_{\mathrm{c}}\)

Critical wave number

\(K_{x}\)

Permeability in x-direction

\(K_{z}\)

Permeability in z-direction

\(Nu\)

Nusselt number

\(p\)

Reduced pressure

\(R_{\mathrm{i}}\)

Internal heat source parameter \(R_{\mathrm{i}}=\frac{Q d^2}{\kappa _{Tz}}\)

\(Ra_{\mathrm{T}}\)

Thermal Rayleigh number, \(Ra_{\mathrm{T}}=\frac{\beta _{\mathrm{T}} g _0 \Delta T d K_{z}}{\nu \kappa _{Tz}}\)

\(R_{\mathrm{0c}}\)

Critical Rayleigh number

\(T\)

Temperature

\({Va}\)

Vadász number \({Va}=\frac{\nu d^2}{K_z\kappa _{Tz}}\)

\(V\)

Thermo-rheological parameter \(V=\delta _0 {\Delta T} \)

\(\Delta T\)

Temperature difference across the porous layer

\(t\)

Time

\((x,z)\)

horizontal and vertical coordinates

Greek Symbols

\(\beta _{\mathrm{T}}\)

Coefficient of thermal expansion

\(\delta _0\)

Small parameter indicating variation of viscosity with temperature

\(\delta ^2\)

Horizontal wave number \(\delta ^2=k_{\mathrm{c}}^2+\pi ^2\)

\(\epsilon \)

Perturbation parameter

\(\gamma \)

Heat capacity ratio \(\gamma =\frac{(\rho c)_{\mathrm{m}}}{(\rho c)_{\mathrm{f}}}\)

\(\eta \)

Thermal anisotropy parameter \(\kappa _{Tx} /\kappa _{Tz}\)

\(\xi \)

Mechanical anisotropy parameter \( K_{x} /K_{z}\)

\(\varOmega \)

Frequency of modulation

\(\kappa _{T}\)

\(\kappa _{Tx} (ii + jj) + \kappa _{Tz} (kk)\)

\(\kappa _{Tx}\)

Effective thermal diffusivity in \(x\)-direction

\(\kappa _{Tz}\)

Effective thermal diffusivity in \(z\)-direction

\(\mu \)

Dynamic viscosity of the fluid

\(\phi \)

Porosity

\(\nu \)

Kinematic viscosity, \(\left( {\frac{\mu }{\rho _{0}}} \right) \)

\(\rho \)

Fluid density

\(\psi \)

Stream function

\(\tau \)

Slow time \(\tau =\epsilon ^2 t\)

\(\varTheta \)

Perturbed temperature

Other Symbols

\(\nabla _{1}^2\)

\(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}\)

\(\nabla ^{2}\)

\(\nabla _{1}^2+\frac{\partial ^{2}}{\partial z^{2}}\)

Subscripts

b

Basic state

c

Critical

0

Reference value

Superscripts

\(^{\prime }\)

Perturbed quantity

\(*\)

Dimensionless quantity

st

Stationary

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Notes

Acknowledgments

This work was done during the lien sanctioned to the author by Banaras Hindu University, Varanasi to work as professor of Mathematics at Department of Applied Mathematics, School for Physical Sciences, BB Ambedkar University, Lucknow, India. The author BSB gratefully acknowledges Banaras Hindu University, Varanasi for the same. Further, author Alok Srivastava gratefully acknowledges the financial assistance from Banaras Hindu University as a research fellowship.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Alok Srivastava
    • 1
  • B. S. Bhadauria
    • 2
    • 1
  • P. G. Siddheshwar
    • 3
  • I. Hashim
    • 4
  1. 1.Department of Mathematics, Faculty of ScienceBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of Applied Mathematics, School for Physical SciencesBabasaheb Bhimrao Ambedkar UniversityLucknowIndia
  3. 3.Department of Mathematics, Central College CampusBangalore UniversityBangaloreIndia
  4. 4.School of Mathematical Sciences, Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia

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