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Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid

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Abstract

The paper deals with the study of effect of gravity modulation on double-diffusive convection in a dielectric liquid for the cases of rigid-rigid and free-free boundaries. Using a modified Venezian approach, expressions for the Rayleigh number and its correction are determined. Fourier–Galerkin expansion is employed for a weakly nonlinear stability analysis and this results in a fifth-order Lorenz system that retains the structure of the classical one in the limiting case. A local nonlinear stability analysis using the method of multiscales leads to the time-periodic Ginzburg–Landau equation from the time-periodic generalized Lorenz system and the numerical solution of this simpler equation helps in quantifying unsteady heat and mass transports. Influence of various non-dimensional parameters (Lewis number, solutal Rayleigh number, electrical Rayleigh number and Prandtl number), amplitude and frequency of gravity modulation on onset of convection and heat and mass transports is discussed. The study reveals that the influence of gravity modulation is to stabilize the system and enhance heat and mass transports. The results from free-free boundaries are qualitatively similar to that of rigid-rigid boundaries. Further, it is shown that in the case of free-free boundaries the heat and mass transports are less compared to those of rigid-rigid boundaries.

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Abbreviations

A, B, C, L, M :

Amplitudes

\(\mathbf {D}\) :

Electric displacement

\(\mathbf {E}\) :

Electric field

\(\mathbf {E_{0}}\) :

Root mean square value of the electric field at the lower surface

\(\mathbf {g}\) :

Acceleration due to gravity (0,0,-g)

h :

Depth of the fluid layer

Le :

Lewis number

Nu :

Nusselt number

\(\mathbf {P}\) :

Dielectric polarisation

Pr :

Prandtl number

p :

Pressure

\(\mathbf {q}\) :

Velocity vector

\(R_{E}\) :

Electrical Rayleigh number

\(R_T\) :

Thermal Rayleigh number

\(R_{S}\) :

Solutal Rayleigh number

T :

Temperature

Sh :

Sherwood number

S :

Solute concentration

t :

Time

\(\alpha _T\) :

Thermal diffusivity in vertical direction

\(\alpha _S\) :

Solute diffusivity in vertical direction

\(\chi _{e}\) :

Electric susceptibility

\(\beta _1\) :

Thermal expansion coefficient

\(\beta _2\) :

Coefficient of solute expansion

\(\delta\) :

Amplitude of gravity modulation

\(\varDelta S\) :

Solute difference across the fluid layer

\(\nabla ^{2}\) :

Laplacian operator \((=\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial z^{2}})\)

\(\psi\) :

Stream function

\(\epsilon\) :

Amplitude of convection

\(\varepsilon _{0}\) :

Electric permittivity of free space

\(\varepsilon _{r}\) :

Relative permittivity

\(\varOmega\) :

Frequency

\(\kappa _{S}\) :

wave number

\(\mu _{1}\) :

Reference viscosity

\(\kappa _{T}\) :

Effective thermal diffusivity in horizontal direction

\(\nabla\) :

Differential operator

\(\varPhi\) :

Electric scalar potential

\(\varPsi\) :

Dimensionless stream function

\(\rho _{0}\) :

Reference density

\(\rho\) :

Fluid density

c :

Critical value

b :

Basic value

\(*\) :

Dimensionless quantity

\(\prime\) :

Perturbed quantity

Tr :

Transpose

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Acknowledgements

The author BRR expresses her gratitude to the management of NMIT for encouragement and to the Bangalore University for research facilities during her Ph.D. program.

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Authors and Affiliations

  1. Department of Mathematics, CHRIST (Deemed to be University), Bangalore, 560029, India

    P. G. Siddheshwar

  2. Department of Mathematics, Nitte Meenakshi Institute of Technology, Bangalore, 560064, India

    B. R. Revathi

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

    C. Kanchana

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  1. P. G. Siddheshwar
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  3. C. Kanchana
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Correspondence to B. R. Revathi.

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Siddheshwar, P.G., Revathi, B.R. & Kanchana, C. Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid. Meccanica 55, 2003–2019 (2020). https://doi.org/10.1007/s11012-020-01241-y

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