Abstract
In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.
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Abbreviations
- a :
-
Wave number
- a c :
-
Critical wave number
- d :
-
Depth of the porous layer
- Da :
-
Darcy number, \({K_{z}/d^{2}}\)
- Pr :
-
Prandtl number, \({\nu/{\kappa_{\rm T}}_{z}}\)
- Pr D :
-
Vadasz number, δPr/Da
- K :
-
Permeability, \({K_{x}(\hat{i}\hat{i}+\hat{j}\hat{j})+K_{z}(\hat{k}\hat{k})}\)
- p :
-
Pressure
- q :
-
Velocity (u, v, w)
- Ra :
-
Rayleigh number, \({(Ra=\alpha_{\rm T}gK_{z}(\Delta T)d/\nu {\kappa_{\rm T}}_{z})}\)
- Ra c :
-
Critical Rayleigh number
- t :
-
Time
- Ta :
-
Taylor number \({(2\Omega K_{z}/\nu \delta)^{2}}\)
- T :
-
Temperature
- ΔT :
-
Temperature difference between the walls
- \({\bar{H}}\) :
-
Rate of Heat transport per unit area
- Nu :
-
Nusselt number
- (x, y, z):
-
Space co-ordinate
- α T :
-
Thermal expansion coefficient
- κ T :
-
Thermal diffusivity \({\kappa_{{\rm T}_{x}}(\hat{i}\hat{i}+\hat{j}\hat{j})+\kappa_{{\rm T}_{z}}(\hat{k}\hat{k})}\)
- \({\vec{\Omega }}\) :
-
Angular velocity \({{vector}(0, 0, \Omega)}\)
- ω :
-
Vorticity vector, \({\nabla \times q}\)
- δ :
-
Porosity
- η :
-
Thermal anisotropy parameter, \({{\kappa_{\rm T}}_{x}/{\kappa_{\rm T}}_{z}}\)
- γ :
-
Ratio of heat capacities
- ρ :
-
Density
- μ :
-
Dynamic viscosity
- ν :
-
Kinematic viscosity, μ/ρ 0
- ξ :
-
Mechanical anisotropy parameter, K x /K z
- σ :
-
Growth rate
- α :
-
Rescaled wave number, \({a^{2}/\pi^{2}}\)
- ψ :
-
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
- \({\nabla^{2}_{1}}\) :
-
\({\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}}\) , horizontal Laplacian
- \({\nabla^{2}}\) :
-
\({\nabla^{2}_{1}+ \frac{\partial^{2}}{\partial z^{2}}}\)
- D:
-
d/dz
- i :
-
\({\sqrt{-1}}\)
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Bhadauria, B.S., Kumar, A., Kumar, J. et al. Natural convection in a rotating anisotropic porous layer with internal heat generation. Transp Porous Med 90, 687–705 (2011). https://doi.org/10.1007/s11242-011-9811-0
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DOI: https://doi.org/10.1007/s11242-011-9811-0