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Theoretical and numerical analyses of convective instability in porous media with upward throughflow. (English) Zbl 0941.76029

Summary: We obtain exact analytical solutions for a hydrothermal system consisting of a horizontal porous layer with upward throughflow. The boundary conditions are constant temperature, constant pressure at the top, and constant vertical temperature gradient and constant Darcy velocity at the bottom of the layer. After deriving the exact analytical solutions, we examine the stability of the solutions using linear stability theory and the Galerkin method. We find that the exact solutions for such a hydrothermal system become unstable when the Rayleigh number of the system is equal to or greater than the corresponding critical Rayleigh number. For small and moderate Péclet numbers \((\text{Pe}\leq 6)\), an increase in upward throughflow destabilizes the convective flow in the horizontal layer. To confirm these findings, the finite element method with a progressive asymptotic procedure is used to compute the convective cells in such a hydrothermal system.

MSC:

76E06 Convection in hydrodynamic stability
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography
76M10 Finite element methods applied to problems in fluid mechanics
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