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Zbl 1033.76055
Aly, E. H.; Elliott, L.; Ingham, D. B.
Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium.
(English)
[J] Eur. J. Mech., B, Fluids 22, No. 6, 529-543 (2003). ISSN 0997-7546

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 2000:
*76R05 Forced convection
76R10 Free convection
76S05 Flows in porous media
76D10 Boundary-layer theory (incompressible fluids)
76M45 Asymptotic methods, singular perturbations
80A20 Heat and mass transfer

Keywords: existence; uniqueness; fluid-saturated porous medium; Dual solutions

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