Synchronization of convective currents of a two-component fluid in adjacent porous cells

Abstract

In this work, we study the Soret-driven convection of a two-component fluid in adjacent rectangular cells of a porous medium heated from below in the presence of a gravitational field. In each cell, a long-wavelength oscillatory instability is possible; it can be described by means of phase reduction, which makes it possible to study the synchronization of currents. The horizontal boundaries of the cells are assumed to be impenetrable (including for impurities), the heat flux is assumed to be fixed. The vertical boundaries have low thermal conductivity. Fluid motions are described by the Darcy-Boussinesq approximation. Additionally, in this system there is a distributed heat source term describing the heat exchange between the cells. In this way, the equations become coupled via the temperature field. The equations within the long-wavelength approximation are derived using the standard method of multiple scales. An analytical description of the system within the framework of a weakly nonlinear analysis can only be constructed near the threshold of the convective instability of the system, and the issue of collective effects is relevant for oscillatory modes; therefore, we restrict our study to the case of an oscillatory instability. On the basis of equations for the amplitudes of the coupled oscillatory modes, we derived equations for the oscillation phases, which are the key equations of the phase description. For these equations, we found the values of the parameters at which the system has the regime of synchronization of currents; the solution for a stable synchronous regime is also reported.