Abstract
Here, we investigate weakly nonlinear hydrothermal two-dimensional convective flow in a horizontal aquifer layer with horizontal isothermal and rigid boundaries. We treat such a layer as a porous medium, where Darcy’s law holds, subjected to the conditions that the porous layer’s permeability and the thermal conductivity are variable in the vertical direction. This analysis is restricted to the case that the subsequent hydraulic resistivity and diffusivity have a small rate of change with respect to the vertical variable. Applying the weakly nonlinear approach, we derive various order systems and express their solutions. The solutions for convective flow quantities such as vertical velocity and the temperature that arise as the Rayleigh number exceeds its critical value are computed and presented in graphical form.
Highlights
A porous medium is a material that consists of a solid matrix with an interconnected void, and it is characterized by its porosity
We investigated the problem of weakly nonlinear two-dimensional convective flow in a horizontal aquifer layer with horizontal isothermal and rigid boundaries
As in the application of such problems in the case of groundwater flow, we treated such a layer as a porous layer, where Darcy’s law holds, subject to the conditions that that the porous layer’s permeability and the thermal conductivity are variable in the vertical direction
Summary
A porous medium is a material that consists of a solid matrix with an interconnected void, and it is characterized by its porosity. The present paper studies the problem of nonlinear thermal convection in a horizontal aquifer layer with horizontal rigid and isothermal boundaries, and the aquifer layer is assumed to have variable permeability and thermal conductivity. The analysis is based on the weakly nonlinear theory [4] for the particular case of convection in a porous medium with rigid and isothermal boundaries and is extended here to porous layers with variable permeability and thermal diffusivity. We restrict our analysis to the non-dimensional form of the governing equations in a porous layer subjected to Darcy’s law, and such a non-dimensional form of the governing system introduces three non-dimensional parameters, which are the Rayleigh number, hydraulic resistivity, which is the ratio of the viscosity to permeability, and non-dimensional thermal diffusivity, which is the ratio of diffusivity to a reference diffusivity at the lower boundary [3]. For the case where the vertical rate of change of either hydraulic resistivity or thermal diffusivity increases in the upward direction, the flow appears to be stabilizing, while the flow is destabilizing if such a vertical rate of change decreases in the upward direction
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