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 PĂ©clet 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 |