Advective Removal of Localized Convective Structures in a Porous Medium

Abstract

Thermal convection in a planar horizontal layer of a porous medium, which is saturated with a viscous incompressible liquid pumped along the layer and has solid impermeable boundaries with a set heat flux at them, is considered. In certain physical systems, the first instability in Rayleigh–Bénard convection between thermally insulated horizontal plates is long-wave. The equations characterizing large-scale thermal convection in a horizontal layer in a homogeneous liquid and in a porous medium are similar: they differ in just one term, which vanishes under certain conditions (e.g., for two-dimensional flows or an infinite Prandtl number). In the system in question with a vertical heat flux that is nonuniform along the layer, localized convective structures may emerge in the region where the heat flux exceeds the critical value corresponding to uniform heating from below and to the onset of convection within the layer. When the rate of the longitudinal pumping of a liquid through the layer changes, the system may enter either a state with stable localized convective structures and a monotonic (or oscillatory) instability or a state in which the localized convective flow is removed completely from the region of its excitation. Calculations are carried out based on the amplitude equations in the long-wave approximation using the Darcy–Boussinesq model and the approximation of small deviations of the heat flux across the boundaries from the critical values under uniform heating. The results of the numerical modeling of the process of the removal of a localized flow from the region of its excitation at higher rates of the longitudinal pumping of the liquid through the layer are detailed. Stability maps for monotonic and oscillatory instabilities of the base state of the system are presented.