Analytical investigation of steady convection of a near-critical Van der Waals gas in a porous thin annular cylinder embedded in a heat-conducting medium

Investigation of filtration convection in a near-critical Van-der-Waals gas for substantial temperature differences

“…The dependence of the critical Rayleigh number on the parameters of problem (1.9)-(1.10) was found on the basis of numerical calculations using the analytical formulas derived in [8]. The solution of the nonlinear problem inside the contour was found using the Galerkin method.…”

“…In the framework of our model, an analytic solution of this problem can be found for an arbitrary equation of state. A detailed calculation of the solution is given in [8]. As shown in the same paper, for low temperature differences the model has an analytical solution over the stability threshold as well.…”

“…Therefore, in this approximation we can ignore the original motion and regard the initial state as the mechanical equilibrium. Nevertheless, as in [8], we will call a solution of (1.9)-(1.10) for Q = 0 the hydraulic equilibrium, rather than the mechanical one. The word convection will mean the motion averaged over the cross section of the ring, i.e.…”

“…In this context, a reasonable simplification of the problem with the preservation of its main features is of great importance. This approach used in [8] yielded a number of results relating to the general case. In particular, it was possible to find an analytical expression for the critical Rayleigh number.…”

“…The nonlinear problem was considered only for weak spatial variations of the gas properties. This work is a continuation of [8]. Its aim is to study the general features of the behavior of the critical Rayleigh number, which determines the threshold of the onset of motion (averaged over the cross section of the ring), as well as the parameters of convection over this threshold, as 534 RAMAZANOV functions of the non-Boussinesq parameters.…”