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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 ${\overline{x}}^{\lambda }$, where $\overline{x}$ 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 ${\overline{x}}^{\lambda }$. The problem then depends on two parameters, namely $\lambda$ and $\epsilon$, the ratio of the Rayleigh to Péclet numbers. It is found that when $\lambda >0$ ($<0$) there are (is) dual (unique) solution(s) when $\epsilon$ is greater than some negative values of $\epsilon$ (which depends on $\lambda$). When $\lambda <0$, there is a range of negative value of $\epsilon$ (which depends on $\lambda$) for which dual solutions exist, and for both $\lambda >0$ and $\lambda <0$ there is a negative value of $\epsilon$ (which depends on $\lambda$) for which there is no solution. Finally, solutions for $0<\epsilon \ll 1$ and $\epsilon \gg 1$ have been obtained.
##### MSC:
 76R05 Forced convection (fluid mechanics) 76R10 Free convection (fluid mechanics) 76S05 Flows in porous media; filtration; seepage 76D10 Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids) 76M45 Asymptotic methods, singular perturbations (fluid mechanics) 80A20 Heat and mass transfer, heat flow