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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 $$\bar x^{\lambda}$$, where $${\bar x}$$ is measured from the leading edge of the plate, and $$\lambda$$ 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 $$\bar x^{\lambda}$$. The problem then depends on two parameters, namely $$\lambda$$ and $$\varepsilon$$, the ratio of the Rayleigh to Péclet numbers. It is found that when $$\lambda >0$$ ($$<0$$) there are (is) dual (unique) solution(s) when $$\varepsilon$$ is greater than some negative values of $$\varepsilon$$ (which depends on $$\lambda$$). When $$\lambda <0$$, there is a range of negative value of $$\varepsilon$$ (which depends on $$\lambda$$) for which dual solutions exist, and for both $$\lambda > 0$$ and $$\lambda <0$$ there is a negative value of $$\varepsilon$$ (which depends on $$\lambda$$) for which there is no solution. Finally, solutions for $$0<\varepsilon \ll 1$$ and $$\varepsilon \gg 1$$ have been obtained.

MSC:
 76R05 Forced convection 76R10 Free convection 76S05 Flows in porous media; filtration; seepage 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 80A20 Heat and mass transfer, heat flow (MSC2010)
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