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Rayleigh number in a stability problem for a micropolar fluid. (English) Zbl 1128.76022
The author is concerned with the onset of thermal convection in a heat-conducting micropolar fluid placed in a horizontal unbounded layer between two rigid walls in the presence of an applied constant magnetic field. Thus, the author numerically determines the approximate Rayleigh number on the neutral curve and on neutral surfaces for some values of the governing micropolar parameters $R, A, \overline\delta$ and the magnetic parameter $Q$. Assuming that the principle of exchange of stability holds (the linear stability against normal mode perturbation), the governing two-point problem has been obtained which has been solved using Fourier expansion. Both the hydrodynamic case ($Q=0$) and the presence of a magnetic field ($Q\neq0$) have been studied. It has been found that in the hydrodynamic case ($Q=0$), when the micropolar parameter $\overline\delta$ is not null, the viscosity parameter $k$ has a stabilizing influence on the flow. For $\overline\delta=Q=0$ the completed neutral curves and neutral surfaces show the following influence of the micropolar parameter $R$: the domain of stability enlarges as $R$ increases. When $A$ and $\overline\delta$ increases, large values of the wave number seems to have a stabilizing effect on the flow.

76E06Convection (hydrodynamic stability)
76E05Stability of parallel shear flows
76A05Non-Newtonian fluids
76M25Other numerical methods (fluid mechanics)