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Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium. (English) Zbl 1033.76055
Summary: We investigate existence and uniqueness of a vertically flowing fluid passed a thin vertical fin in a saturated porous medium. We assume the two-dimensional mixed convection from the fin, which is modelled as a fixed semi-infinite vertical surface embedded in the fluid-saturated porous medium. The temperature, in excess of the constant temperature in the ambient fluid on the fin, varies as x ¯ λ , where x ¯ is measured from the leading edge of the plate, and λ is a fixed constant. The Rayleigh number is assumed to be large, so that the boundary-layer approximation may be made, and the fluid velocity at the edge of the boundary layer is assumed to vary as x ¯ λ . The problem then depends on two parameters, namely λ and ε, the ratio of the Rayleigh to PĂ©clet numbers. It is found that when λ>0 (<0) there are (is) dual (unique) solution(s) when ε is greater than some negative values of ε (which depends on λ). When λ<0, there is a range of negative value of ε (which depends on λ) for which dual solutions exist, and for both λ>0 and λ<0 there is a negative value of ε (which depends on λ) for which there is no solution. Finally, solutions for 0<ε1 and ε1 have been obtained.
MSC:
76R05Forced convection (fluid mechanics)
76R10Free convection (fluid mechanics)
76S05Flows in porous media; filtration; seepage
76D10Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)
76M45Asymptotic methods, singular perturbations (fluid mechanics)
80A20Heat and mass transfer, heat flow