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Transport in Porous Media

, Volume 98, Issue 2, pp 239–257 | Cite as

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

  • S. N. GaikwadEmail author
  • Irfana Begum
Article

Abstract

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.

Keywords

Double-diffusive convection Rotation Chemical reaction Porous layer Anisotropy Heat mass transfer 

List of Symbols

Variables

\(a\)

Wavenumber

\(d\)

Height of the porous layer [m]

\(\mathbf{g}\)

Gravitational acceleration, (0, 0,-g) [m s\(^{-2}\)]

K

Inverse anisotropic permeability tensor, \(K_x^{-1} \mathbf{ii}+K_y^{-1} \mathbf{jj}+K_z^{-1} \mathbf{kk}\)

\(k\)

Lumped effective reaction rate

Le

Lewis number, \({\kappa _{\mathrm{T}z} }/{\kappa _\mathrm{S} }\)

\(l, m\)

Horizontal wavenumbers

Nu

Nusselt number

\(p\)

Pressure [kg m\(^{-1}\) s\(^{-2}\)]

\(\mathbf{q}\)

Velocity vector (u,v,w) [m s\(^{-1}\)]

\(Ra_T \)

Darcy–Rayleigh number, \({\rho _{0} \beta _\mathrm{T} g\Delta TdK_z }/{ \mu \varepsilon \kappa _{\mathrm{T}z} }\)

\(Ra_\mathrm{S} \)

Solute Rayleigh number, \({\rho _{0} \beta _\mathrm{S}g\Delta SdK_z }/{\varepsilon \mu \kappa _{\mathrm{T}z} }\)

\(S\)

Solute concentration

\(S_\mathrm{eq} (T)\)

Equilibrium concentration of the solute at a given temperature

Sh

Sherwood number

\(\Delta S\)

Salinity difference between the walls

\(t\)

Time [s]

\(T\)

Temperature [K]

\(Ta\)

Darcy Taylor number \(({{2\Omega K_z }/\mu })^{2}\)

\(\Delta T\)

Temperature difference between the walls [K]

\(V_{z}\)

\(z\) component of vorticity vector \(\mathbf{V}=\nabla \times \mathbf{q} \; [{\text{ m} \text{ s}}^{-1}]\)

\(x, y, z\)

Space coordinates [d]

Greek Symbols

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

Thermal expansion coefficient

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

Solute expansion coefficient

\(\varepsilon \)

Porosity

\(\varvec{\Omega }\)

Angular velocity [radians s\(^{-1}\)] (\({0,0,\Omega }\))

\(\eta \)

Thermal anisotropy parameter, \({\kappa _{\mathrm{T}x} }/{\kappa _{\mathrm{T}z} }\)

\(\varvec{\kappa }_\mathrm{T}\)

Anisotropic thermal diffusion tensor, \(\kappa _{\mathrm{T}x} \mathbf{ii}+\kappa _{\mathrm{T}y} \mathbf{jj}+\kappa _{\mathrm{T}z} \mathbf{kk}\)

\(\kappa _\mathrm{S} \)

Solute diffusivity

\(\lambda \)

Normalized porosity parameter,\(\varepsilon \frac{({\rho c})_\mathrm{f} }{({\rho c})_\mathrm{m }}\)

\(\chi \)

Damkohler number, \({kd^{2}}/{\varepsilon \kappa _{\mathrm{T}z} }\)

\(\mu \)

Dynamic viscosity [N s m\(^{-2}\)]

\(\nu \)

Kinematic viscosity [m\(^{2}\) s\(^{-1}\)]

\(\rho \)

Density [kg m\(^{-3}\)]

\(\rho c\)

Volumetric heat capacity

\(\sigma \)

Growth rate

\(\xi \)

Mechanical anisotropy parameter, \({K_x }/{K_z }\)

\(\psi \)

Stream function

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

f

Fluid

m

Porous medium

0

Reference value

s

Solid

i

Imaginary part

Superscripts

*

Dimensionless quantity

Perturbed quantity

F

Finite amplitude

Osc

Oscillatory state

St

Stationary state

Notes

Acknowledgments

This study was supported by University Grants Commission, New Delhi, under Maulana Azad National Scholarship to Minority Students and partially under Major Research Project F. No. 37-174/2009 (SR) dated 12-01-2010. One of the authors (Irfana Begum) thanks UGC for awarding the Scholarship. The authors thank the reviewers for their useful suggestions.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsGulbarga UniversityGulbargaIndia

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