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Transport in Porous Media

, Volume 130, Issue 3, pp 819–845 | Cite as

Impact of Thermal Non-equilibrium on Weak Nonlinear Rotating Porous Convection

  • R. N. DayanandaEmail author
  • I. S. Shivakumara
Article
  • 86 Downloads

Abstract

The consequences of local thermal non-equilibrium (LTNE) on both stationary and oscillatory weak nonlinear stability of gravity-driven porous convection in an incompressible fluid-saturated rotating porous layer are investigated. A stability map is drawn in the Darcy–Taylor and scaled Vadasz number plane to demarcate the regions of stationary and oscillatory convection, and thereby, co-dimension-2 points are determined. It is found that the effect of increasing interphase heat transfer coefficient is to enhance the region of stationary convection and decrease the region of oscillatory convection. The complex Ginzburg–Landau equations are derived using the multi-scale method, and pitchfork and Hopf bifurcations occur at stationary and oscillatory critical Darcy–Rayleigh numbers, respectively. The linear and nonlinear oscillatory neutral curves are illustrated, and at the quartic point, the transition from supercritical to subcritical bifurcations is identified for the governing parameters. The impact of LTNE model is to enhance the region of forward bifurcation and post-transient amplitude compared to LTE case. Heat transfer is obtained in terms of Nusselt number for both stationary and oscillatory convection. The region of enhancement in heat flux for oscillatory convection in the smaller scaled Vadasz number domain with increasing Darcy–Taylor number increases with increasing interphase heat transfer coefficient and the porosity-modified conductivity ratio.

Keywords

Nonlinear convection Local thermal non-equilibrium Porous medium Rotation 

List of Symbols

\(a\)

Horizontal wave number

c

Specific heat

\(Da\)

Darcy number, \({K \mathord{\left/ {\vphantom {K {d^{2} }}} \right. \kern-0pt} {d^{2} }}\)

\(d\)

Depth of the porous layer

\(\vec{g}\)

Gravitational acceleration

H

Non-dimensional interphase heat transfer coefficient, \({{hd^{2} } \mathord{\left/ {\vphantom {{hd^{2} } {\varepsilon k_{\text{f}} }}} \right. \kern-0pt} {\varepsilon k_{\text{f}} }}\)

h

Interphase heat transfer coefficient

\(K\)

Permeability of the porous medium

k

Thermal conductivity

\(Pr\)

Prandtl number, \({\nu \mathord{\left/ {\vphantom {\nu {\kappa_{\text{f}} }}} \right. \kern-0pt} {\kappa_{\text{f}} }}\)

\(R_{\text{TD}}\)

Darcy–Rayleigh number, \({{\beta_{\text{T}} g(T_{\text{L}} - T_{\text{U}} )Kd} \mathord{\left/ {\vphantom {{\beta_{\text{T}} g(T_{\text{L}} - T_{\text{U}} )Kd} {\nu \kappa_{\text{f}} \varepsilon }}} \right. \kern-0pt} {\nu \kappa_{\text{f}} \varepsilon }}\)

T

Temperature

\(Ta\)

Darcy–Taylor number, \(({{2\varOmega K} \mathord{\left/ {\vphantom {{2\varOmega K} {\nu \varepsilon )^{2} }}} \right. \kern-0pt} {\nu \varepsilon )^{2} }}\)

\(t\)

Time

\(Va\)

Vadasz number, \({{Pr\varepsilon } \mathord{\left/ {\vphantom {{Pr\varepsilon } {Da}}} \right. \kern-0pt} {Da}}\)

\(x,y,z\)

Space coordinates

Greek Symbols

\(\alpha\)

Ratio of diffusivities, \({{\kappa_{\text{f}} } \mathord{\left/ {\vphantom {{\kappa_{\text{f}} } {\kappa_{\text{s}} }}} \right. \kern-0pt} {\kappa_{\text{s}} }}\)

\(\beta_{\text{T}}\)

Thermal expansion coefficient

\(\delta\)

Scaled Vadasz number, \(Va/\pi^{2}\)

\(\varepsilon\)

Porosity

\(\gamma\)

Porosity-modified conductivity ratio, \({{\varepsilon k_{\text{f}} } \mathord{\left/ {\vphantom {{\varepsilon k_{\text{f}} } {(1 - \varepsilon )k_{\text{s}} }}} \right. \kern-0pt} {(1 - \varepsilon )k_{\text{s}} }}\)

\(\kappa_{\text{f}}\)

Thermal diffusivity of the fluid, \({{k_{\text{f}} } \mathord{\left/ {\vphantom {{k_{\text{f}} } {(\rho C)_{\text{f}} }}} \right. \kern-0pt} {(\rho C)_{\text{f}} }}\)

\(\kappa_{\text{s}}\)

Thermal diffusivity of the solid, \({{k_{\text{s}} } \mathord{\left/ {\vphantom {{k_{\text{s}} } {(\rho C)_{\text{s}} }}} \right. \kern-0pt} {(\rho C)_{\text{s}} }}\)

\(\mu\)

Dynamic viscosity of the fluid

\(\nu\)

Kinematic viscosity of the fluid, \(({\mu \mathord{\left/ {\vphantom {\mu {\rho_{0} }}} \right. \kern-0pt} {\rho_{0} }})\)

\(\overrightarrow {\varOmega }\)

Angular velocity (0, 0, \(\varOmega\))

\(\omega\)

Frequency

\(\psi\)

Stream function

Notes

Acknowledgement

The authors wish to thank the reviewers and the associate editor for their useful comments which helped in improving the paper considerably.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBengaluruIndia

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