Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids

doi:10.1007/s10483-017-2280-9

Abstract

Abstract:

Linear and weakly nonlinear analyses are made for the Rayleigh-Bénard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret coefficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation. Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The influence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of appearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.

Key words: Newell-Whitehead-Segel, inverted, Nusselt number, elastic circular plate, Kirchhoff-Love assumption, generalized variational principle, Lorenz, Rayleigh-Bénard, Eckhaus, secondary instability, Sherwood number, zigzag, codimension-two, TakensBogdanov

2010 MSC Number:

76Rxx

R. RAVI, C. KANCHANA, P. G. SIDDHESHWAR. Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids. Applied Mathematics and Mechanics (English Edition), 2017, 38(11): 1579-1600.

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