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The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer

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Abstract

The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and a dispersion relation are obtained analytically using linear theory. The effect of anisotropy parameters, Taylor number, Darcy number, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory and finite amplitude convection is shown graphically. It is found that contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter show contrasting effect on the onset criterion at moderate and high rotation rates. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The effect of various parameters on heat and mass transfer is shown graphically. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.

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Abbreviations

a :

Wavenumber

c :

Specific heat of solid

c p :

Specific heat of fluid

Da :

Darcy number, \({\frac{\mu_{\rm e}}{\mu_{\rm f}}\frac{K_z}{{\rm d}^{2}}}\)

d :

Height of the porous layer

g :

Gravitational acceleration, (0, 0, −g)

K :

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

Le :

Lewis number, κ Tz /κ S

l, m:

Horizontal wavenumbers

Nu :

Nusselt number

Pr D :

Darcy–Prandtl number, γενd2/K z κ Tz

p :

Pressure

q :

Velocity vector, (u, v, w )

Ra T :

Thermal Rayleigh number, β Tg ΔT dK z /νκ Tz

Ra S :

Solute Rayleigh number, β Sg ΔS dK z /νκ Tz

S :

Solute concentration

Sh :

Sherwood number

ΔS :

Salinity difference between the walls

Ta :

Taylor number, \({\left({\frac{2\Omega K_{z}}{\varepsilon \nu}}\right)^{2}}\)

t :

Time

T :

Temperature

ΔT :

Temperature difference between the walls

x, y, z:

Space coordinates

β T :

Thermal expansion coefficient

β S :

Solute expansion coefficient

ε :

Porosity

Φ :

Dimensionless amplitude of concentration perturbation

γ :

Ratio of specific heat, \({\frac{(\rho c)_{\rm m}}{(\rho c_{\rm p})_{\rm f}}}\)

η :

Thermal anisotropy parameter, κ Tx /κ Tz

κT:

Anisotropic thermal diffusion tensor, κ Tx ii + κ Ty jj + κ Tz kk

κ S :

Solute diffusivity

λ:

Normalized porosity, \({\frac{\varepsilon}{\gamma}}\)

μ f :

Fluid viscosity

μ e :

Effective viscosity

ν :

Kinematic viscosity

Θ :

Dimensionless amplitude of temperature perturbation

ρ :

Density

σ :

Growth rate

Ω :

Angular velocity of rotation, (0, 0, Ω)

ξ :

Mechanical anisotropy parameter, K x /K z

ψ :

Stream function

\({\nabla_1^2}\) :

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

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

\({\nabla_1^2 +\frac{\partial^{2}}{\partial z^{2}}}\)

b:

Basic state

c:

Critical

f:

Fluid

m:

Porous medium

0:

Reference value

s:

Solid

*:

Dimensionless quantity

′:

Perturbed quantity

F:

Finite amplitude

Osc:

Oscillatory state

St:

Stationary state

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Correspondence to M. S. Malashetty.

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Malashetty, M.S., Begum, I. The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer. Transp Porous Med 88, 315–345 (2011). https://doi.org/10.1007/s11242-011-9741-x

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Keywords

  • Double diffusive convection
  • Rotation
  • Brinkman model
  • Porous layer
  • Anisotropy
  • Heat mass transfer