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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 PĂ©clet 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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