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Instabilities of convection patterns in a shear-thinning fluid between plates of finite conductivity

Abstract : Rayleigh-Bénard convection in a horizontal layer of a non-Newtonian fluid between slabs of arbitrary thickness and finite thermal conductivity is considered. The first part of the paper deals with the primary bifurcation and the relative stability of convective patterns at threshold. Weakly non-linear analysis combined with Stuart-Landau equation is used. The competition between squares and rolls, as a function of the shear-thinning degree of the fluid, the slabs' thickness and the ratio of the thermal conductivity of the slabs to that of the fluid, is investigated. Computations of heat transfer coefficients are in agreement with the maximum heat transfer principle. The second part of the paper concerns the stability of the convective patterns towards spatial perturbations and the determination of the band width of the stable wavenumber in the neighborhood of the critical Rayleigh number. The approach used is based on the Ginzburg-Landau equations. The study of rolls stability shows that: (i) for low shear-thinning effects, the band of stable wavenumbers is bounded by zigzag instability and cross-roll instability. Furthermore, the marginal cross-roll stability boundary enlarges with increasing shear-thinning properties; (ii) for high shear-thinning effects, Eckhaus instability becomes more dangerous than cross-roll instability. For square patterns, the wavenumber selection is always restricted by zigzag instability and by 'rectangular Eckhaus' instability. In addition, the width of the stable wavenumber decreases with increasing shear-thinning effects. Numerical simulations of the planform evolution are also presented to illustrate the different instabilities considered in the paper.
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Submitted on : Thursday, January 16, 2020 - 1:03:35 PM
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Thomas Varé, Chérif Nouar, Christel Metivier. Instabilities of convection patterns in a shear-thinning fluid between plates of finite conductivity. Physical Review E , American Physical Society (APS), 2017. ⟨hal-02442198v1⟩



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