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Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

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Abstract

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.

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Abbreviations

d :

Height of the porous layer

Da :

Darcy number, K / d 2

g :

Gravitational acceleration

K :

Permeability

l,m :

Wavenumbers in x, y-directions

p :

Pressure

Pr :

Prandtl number, ν/k

Pr D :

Darcy–Prandtl number, δPr / Da

q :

Velocity vector, (u, v, w)

R :

Darcy–Rayleigh number, βgΔTdK / ν k

t :

Time

T :

Temperature

Ta :

Taylor number, (2KΩ / δν)2

(x, y, z):

Space coordinates

α :

Horizontal wavenumber

β :

Thermal expansion coefficient

γ :

Specific heat ratio, (ρ c p ) m / (ρ c p ) f

δ :

Porosity

\({\varepsilon}\) :

Amplitude of modulation

ϕ :

Phase angle

κ :

Thermal diffusivity

μ :

Dynamic viscosity

ν :

Kinematic viscosity, μ / ρ0

ρ :

Density

Ω :

Angular velocity, (0, 0, Ω)

\({\bar{\omega}}\) :

Dimensional frequency of modulation

ω:

Non-dimensional frequency of modulation, \({d^{2}\bar{\omega} / \kappa}\)

b :

Basic state

c :

Critical

Osc:

Oscillatory

St:

Stationary

0:

Reference value

*:

Non-dimensional quantity

/:

Perturbed quantity

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Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Jnana Ganga, Gulbarga, 585 106, India

    M. S. Malashetty & Mahantesh Swamy

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  1. M. S. Malashetty
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  2. Mahantesh Swamy
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Correspondence to M. S. Malashetty.

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Malashetty, M.S., Swamy, M. Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer. Transp Porous Med 69, 313–330 (2007). https://doi.org/10.1007/s11242-006-9087-y

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