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Rayleigh number in a stability problem for a micropolar fluid

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Approximate numerical evaluations of the Rayleigh number are obtained for a stability problemof thermal convection in a heat-conducting micropolar fluid layer between two rigid boundaries. The influences of all the physical parameters on the values of the Rayleigh number are studied. | Turk J Math 31 (2007) , 123 – 137. ¨ ITAK ˙ c TUB Rayleigh Number in a Stability Problem for a Micropolar Fluid Ioana Florica Dragomirescu Abstract Approximate numerical evaluations of the Rayleigh number are obtained for a stability problem of thermal convection in a heat-conducting micropolar fluid layer between two rigid boundaries [7]. The influences of all the physical parameters on the values of the Rayleigh number are studied. Also, approximate neutral curves and neutral surfaces are represented in various parameters spaces. Key Words: Fourier series methods, micropolar fluid, thermal convection, Rayleigh number. 1. Introduction The general theory of fluid microcontinua is attributed to AC Eringen. His work was concerned with a nonlinear theory of microelastic solids, but his treatment of motion, balance of moments, conservation of energy, and entropy production is applicable to all continuous media consisting of microelements, e.g. micropolar fluids. In this paper, we are concerned with the onset of thermal convection in a heat conducting micropolar fluid situated in a horizontal unbounded layer between two rigid walls. In a particular case this stability problem [12] was solved theoretically in [3] using the Chandrasekhar - Galerkin method. In this method, the unknown functions are expanded upon complete sets of functions which satisfy all the boundary conditions. For the case Q = 0, δ = 0, where Q and δ are two physical parameters, the Budianski Mathematics Subject Classification: 65L15,34K20,34K28 123 DRAGOMIRESCU -DiPrima (B-D) method was proposed and a variational formulation of the problem was presented too leading us to the same secular equation as the direct B-D method. Herein, the problem is treated analytically and numerically by the B- D method, also for the cases Q = 0 and/or δ = 0. We calculate the values of the Rayleigh number for various values of the micropolar parameters. In [6] a simple general direct method for solving two-point .