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Rotating Brinkman–Lapwood Convection with Modulation

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Abstract

The purpose of this article is to analyze, theoretically, the effect of modulation on rotating Brinkman–Lapwood convection, i.e., buoyancy-driven convection in a sparse porous medium subjected to rotation. Darcy–Brinkman momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about an axis with non-uniform rotation speed. In particular, we assume that the rotation speed is varying sinusoidally with time. A linear stability analysis has been performed to find the critical Rayleigh number in modulated case. The effect of modulated rotation speed is found to have a stabilizing effect on the onset of convection for different values of modulation frequency and the other physical parameters involved.

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Abbreviations

a :

Horizontal wave number, (a x 2 a y 2 )1/2

a c :

Critical wave number

d :

Depth of the porous layer

ê z :

Unit vector along z-axis

g :

Gravitational acceleration

K :

Permeability of the porous medium

M:

Viscosity ratio (μ e/μ f)

V :

Mean filter velocity (u, v, w)

p :

Pressure

T :

Temperature

ΔT :

Temperature difference between the surfaces

Ra c :

Critical Rayleigh number

Da:

Darcy number, (K/d 2)

Pr:

Prandtl number, ν/κ

Ra:

Rayleigh number, (αgΔT K d/νκ)

Ta:

Taylor number, (2ΩK/δν)

Va:

Vadasz number, δ Pr/Da

\({\vec \Omega}\) :

Rotation speed vector (0, 0,Ω(t))

x, y, z:

Space coordinates

ρ :

Density

(ρc p)f :

Heat capacity of the fluid

(ρc p)S :

Heat capacity of the solid

(ρc p)m :

Relative heat capacity of the porous medium, δ(ρc p)f + (1 − δ)(ρ c p)S

ζ :

z-component of vorticity

α :

Coefficient of thermal expansion

\({\varepsilon}\) :

Amplitude of modulation

δ :

Porosity

γ :

Heat capacity ratio, (ρc p)m/(ρc p)f

κ f :

Thermal conductivity of the fluid

κ S :

Thermal conductivity of the solid

κ m :

Effective thermal conductivity of porous media, δκf + (1 − δS

κ :

Effective thermal diffusivity, κ m/(ρ c p)f

μ e :

Effective viscosity of the medium

μ f :

Fluid viscosity

ν :

Kinematic viscosity, μ f/ρ R

ω :

Modulation frequency

σ :

Growth rate (a complex number)

H:

Basic state

c:

Critical value

0:

Reference value

*:

Non-dimensional quantity

′:

Perturbed quantity

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Correspondence to Om.

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Om, Bhadauria, B.S. & Khan, A. Rotating Brinkman–Lapwood Convection with Modulation. Transp Porous Med 88, 369–383 (2011). https://doi.org/10.1007/s11242-011-9744-7

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Keywords

  • Rotation speed modulation
  • Brinkman Model
  • Lapwood convection
  • Coriolis force
  • Taylor number
  • Vadasz number
  • Viscosity ratio