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Küppers–Lortz Instability in the Rotating Brinkman–Bénard Problem

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Abstract

We investigate the Küppers–Lortz (KL) instability in the rotating Brinkman–Bénard convection problem by assuming that there is local thermal non-equilibrium (LTNE) between the Newtonian liquid and the high-porosity medium that it has occupied to the point of saturation. The effects of local thermal non-equilibrium parameters on the threshold value of the Taylor number and the angle between the rolls at which KL-instability sets in are presented. The four routes through which the local thermal equilibrium situation can be approached are presented with the help of asymptotic analyses. The corresponding results of the rotating Darcy–Bénard problem are extracted as a limiting case from the present problem with the help of another asymptotic analysis. The problem identifies the specific range of values of parameters within which LTNE effect is discernible and also clearly shows that the onset of KL-instability is delayed by the ratio of thermal conductivities. The heat transfer coefficient, however, has a dual effect on \({\mathrm{Ta}}_{\mathrm{c}}\). Such a dual nature is seen, perhaps, due to the heat transport equations being of the hyperbolic type when local thermal non-equilibrium effect is significant. The results show that LTNE in the presence of rotation favors hexagonal pattern.

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Abbreviations

\(c_{\mathrm{p}}\) :

Specific heat at constant pressure (\({\mathrm{J\, kg}}^{-1}\,{\mathrm{K}}^{-1}\))

d :

Channel depth (m)

\({\mathbf {g}}\) :

Acceleration due to gravity (\({\mathrm{m\, s}}^{-2}\))

h :

Interphase heat transfer coefficient (\({\mathrm{W\, m}}^{-2}\,{\mathrm{K}}^{-1}\))

H :

Dimensionless interphase heat transfer coefficient

k :

Horizontal wave number (\({\mathrm{m}}^{-1}\))

\(k_1\) :

Wave number in the x-direction (\({\mathrm{m}}^{-1}\))

\(k_2\) :

Wave number in the y-direction (\({\mathrm{m}}^{-1}\))

K :

Permeability (\({\mathrm{m}}^{2}\))

p :

Growth parameter

t :

Time (s)

P :

Pressure (\({\mathrm{kg\, m}}^{-1}{\mathrm{s}}^{-2}\))

Pr:

Prandtl number

\({\mathbf {q}}\) :

Filtration velocity or Darcy velocity (\({\mathrm{m\,s}}^{-1})\)

T :

Temperature (K)

Ta:

Taylor number

x, y, z :

Cartesian coordinate

X, Y, Z :

Dimensionless coordinates

\(\alpha\) :

Thermal expansion coefficient (\({\mathrm{K}}^{-1}\))

\(\gamma\) :

Porosity-modified ratio of thermal conductivities

\(\varGamma\) :

Ratio of thermal diffusitivities

\(\kappa\) :

Thermal conductivity (\({\mathrm{W\,m}}^{-1}{\mathrm{K}}^{-1}\))

\(\varLambda\) :

Viscosity ratio (Givler and Altobelli 1994)

\(\mu\) :

Dynamic viscosity (\({\mathrm{kg\,m}}^{-1}{\mathrm{s}}^{-1}\))

\({{\varvec{\Omega }}}\) :

Angular velocity

\(\phi\) :

Porosity (\(0<\phi <1\))

\(\rho\) :

Density (\({\mathrm{kg\,m}}^{-3}\))

\(\sigma ^2\) :

Inverse Darcy number or porous parameter

\(\tau\) :

Dimensionless time

\(\Theta\) :

Dimensionless temperature

0:

Reference value

b:

Basic state

BBC:

Brinkman–Bénard convection

c:

Critical

DBC:

Darcy–Bénard convection

f:

Fluid

LTE:

Local thermal equilibrium

LTNE:

Local thermal non-equilibrium

RBC:

Rayleigh–Bénard convection

s:

Solid

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Acknowledgements

DL acknowledges partial financial support from FONDECYT 1180905 and Centers of Excellence with BASAL/CONICYT financing, Grant AFB180001, CEDENNA.

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Authors and Affiliations

  1. Department of Mathematics, Jnana Bharathi Campus, Bangalore University, Bangalore, 560056, India

    P. G. Siddheshwar

  2. Department of Mathematics, M. S. Ramaiah University of Applied Sciences, Bangalore, 560058, India

    C. Siddabasappa

  3. Instituto de Alta Investigación, CEDENNA, Universidad de Tarapacá, Casilla 7D, Arica, Chile

    D. Laroze

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  1. P. G. Siddheshwar
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  3. D. Laroze
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Correspondence to P. G. Siddheshwar.

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Siddheshwar, P.G., Siddabasappa, C. & Laroze, D. Küppers–Lortz Instability in the Rotating Brinkman–Bénard Problem. Transp Porous Med 132, 465–493 (2020). https://doi.org/10.1007/s11242-020-01401-4

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