Advective removal of localized convective structures in a porous medium

Convection in a plane horizontal layer of a porous medium saturated with liquid, bounded by solid impermeable boundaries subjected to the given inhomogeneous heat flux, and liquid pumping along the layer are considered. In a range of physical systems, the first instability in Rayleigh-Bénard convection between thermally insulating horizontal plates is large scale. Large-scale thermal convection in a horizontal layer is governed by remarkably similar equations both in the presence of a porous matrix and without it, with only one additional term for the latter case, which vanishes under certain conditions (e.g., two-dimensional flows or infinite Prandtl number). In such systems, the occurrence of localized convective structures is possible in the region where the heat flux exceeds the critical value corresponding to the case of uniform heating from below. When the velocity of longitudinal pumping of a liquid through a layer varies, the system can be either in the state where the localized convective structures are stable and the monotonic or vibrational instability is observed, or in the state in which the localized convective flow is completely washed from the region of its excitation. Calculations based on amplitude equations in the long-wave approximation are carried out using the Darcy-Boussinesq model and the approximation of small deviations of values of the heat flux from critical values for the case of homogeneous heating. The results of numerical modeling of the process of removal of the localized flow from the region of its excitation with increasing rate of liquid pumping through the layer are presented. Stability maps for monotonic and oscillatory instabilities of the base state of the system are obtained.