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Cross Diffusion Convection in a Newtonian Fluid-Saturated Rotating Porous Medium

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Abstract

Effect of rotation on linear and nonlinear instability of cross-diffusive convection in an anisotropic porous medium saturated with Newtonian fluid has been investigated. Normal mode technique has been used for linear stability analysis, however nonlinear analysis is done using spectral method, involving only two terms. The Darcy model with Coriolis terms, has been employed in the momentum equation. Nonlinear analysis is used to find the thermal and concentration Nusselt numbers. The effects of various parameters, including Soret and Dufour parameters, on stationary and oscillatory convection, have been obtained, and shown graphically.

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Abbreviations

\(l,m\) :

Horizontal wave numbers

\(a\) :

Wave number

\(a_{c}\) :

Critical wave number

\(d\) :

Depth of the porous layer

\(\mathrm{{Du}}\) :

Dufour Parameter \((\alpha _{T}\kappa _{12}/\alpha _{S}\kappa _{11})\)

\(K\) :

Permeability of the porous medium

\(p\) :

Pressure

q :

Velocity \((u, v, w)\)

\(\mathrm{{Ra}}_{T}\) :

Thermal Rayleigh number \((\mathrm{{Ra}}_{T}=\alpha _{T}gK_{z}(\Delta T)d/\nu {\kappa _{11}})\)

\(\mathrm{{Ra}}_{S}\) :

Concentration Rayleigh number (\(\mathrm{{Ra}}_{S}=\alpha _{S}gK_{z}(\varDelta S)d/\nu {\kappa _{22}})\)

\(\mathrm{{Sr}}\) :

Soret parameter \((\alpha _{S}\kappa _{21}/\alpha _{T}\kappa _{11})\)

\({\mathrm{{Ra}}_{T}}_{c}\) :

Critical Rayleigh number

g :

Gravitational acceleration \((0, 0, -g)\)

\(t\) :

Time

\(K^{2}\) :

\(\pi ^{2}(a^{2}+1)\)

\(\mathrm{{Ta}}\) :

Taylor number, \((2\Omega K_{z}/\nu \delta )^{2}\)

\(T\) :

Temperature

\(\Delta T\) :

Temperature difference between the walls

\(S\) :

Concentration

\(\Delta S\) :

Concentration difference between the walls

\(\bar{H}\) :

Rate of Heat transport per unit area

\(\bar{J}\) :

Rate of mass transport per unit area

\(\mathrm{{Nu}}\) :

Thermal Nusselt number

\(\mathrm{{Nu}}^{S}\) :

Concentration Nusselt number

\(x,y,z\) :

Space co-ordinate

\(\alpha _{T}\) :

Thermal expansion coefficient

\(\alpha _{S}\) :

Concentration expansion coefficient

\(\kappa _{11}\) :

Thermal diffusivity of the fluid

\(\kappa _{12}\) :

Cross diffusion due to S component

\(\kappa _{21}\) :

Cross diffusion due to T component

\(\kappa _{22}\) :

Concentration diffusivity of the fluid

\({\varvec{\Omega }}\) :

Angular velocity vector\((0, 0, \Omega )\)

\(\omega \) :

Vorticity vector, \(\nabla \times q\)

\(\tau \) :

Diffusivity ratio, \(\kappa _{22}/\kappa _{11}\)

\(\xi \) :

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

\(\delta \) :

Porosity

\(\rho \) :

Density

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity, \(\mu /\rho _{0}\)

\(\sigma \) :

Growth rate of fluid

\(\psi \) :

Stream function

\(b\) :

Basic state

\(c\) :

Critical

\(0\) :

Reference state

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

Unit normal vector in x-direction

\({\hat{j}}\) :

Unit normal vector in y-direction

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

Unit normal vector in z-direction

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

\(\displaystyle \frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}\), Horizontal Laplacian

\(\nabla ^{2}\) :

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

\(D\) :

\(d/dz\)

\(i\) :

\(\sqrt{-1}\)

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Acknowledgments

Author BSB is grateful to Banaras Hindu University, Varanasi for sanctioning the lien to work as Professor of Mathematics at Department of Applied Mathematics, BB Ambedkar University, Lucknow-226025, India.

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Authors and Affiliations

  1. Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

    B. S. Bhadauria

  2. School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 , Bangi, Selangor, Malaysia

    I. Hashim

  3. School of Basic and Applied Sciences, Galgotias University, Greater Noida, 201306, UP, India

    Jogendra Kumar

  4. Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi, 221005, India

    Alok Srivastava

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  1. B. S. Bhadauria
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  2. I. Hashim
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  3. Jogendra Kumar
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  4. Alok Srivastava
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Hashim, I., Kumar, J. et al. Cross Diffusion Convection in a Newtonian Fluid-Saturated Rotating Porous Medium. Transp Porous Med 98, 683–697 (2013). https://doi.org/10.1007/s11242-013-0166-6

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