Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-dif
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ORIGINAL PAPERS
Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid P. G. Siddheshwar . B. R. Revathi . C. Kanchana
Received: 18 February 2020 / Accepted: 11 September 2020 Ó Springer Nature B.V. 2020
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 freefree 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 nondimensional parameters (Lewis number, solutal Rayleigh number, electrical Rayleigh number and Prandtl number), amplitude and frequency of gravity P. G. Siddheshwar Department of Mathematics, CHRIST (Deemed to be University), Bangalore 560029, India e-mail: [email protected] B. R. Revathi (&) Department of Mathematics, Nitte Meenakshi Institute of Technology, Bangalore 560064, India e-mail: [email protected] C. Kanchana Instituto de Alta Investigacio´n, Universidad de Tarapaca´, Casilla 7 D, Arica, Chile e-mail: [email protected]
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. Keywords Dielectric liquid Double-diffusive convection Gravity modulation Ginzburg–Landau equation Lorenz model Nusselt number Sherwood number Mathematics Subject Classification 76E06 76E30
76A99
List of symbols Latin symbols A, B, C, L, M Amplitudes D Electric displacement E Electric field E0 Root mean square value of the electric field at the lower surface g Acceleration due to gravity (0,0,-g) h Depth of the fluid layer Le Lewis number Nu Nusselt number P Dielectric polarisation Pr Prandtl number
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Meccanica
p q RE RT RS T Sh S t
Pressure Velocity vector Electrical Rayleigh number Thermal Rayleigh number Solutal Rayleigh number Temperature Sherwood number Solute concentration Time
Greek symbols aT Thermal diffusivity in vertical direction aS Solute diffusivity in vertical direction ve Electric susceptibility b1 Thermal expansion coefficient b2 Coefficient of solute e
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