Finite-amplitude instability of the convection in a porous vertical slab with horizontal heterogeneity in permeability

The finite-amplitude instability of the natural convection in a vertical porous slab filled with variable permeability porous medium is investigated analytically. The side walls of the slab are kept at different temperatures, and the permeability in the horizontal direction is assumed to be exponential heterogeneous models. Two-dimensional, finite-amplitude solutions for the thermal buoyant flow are obtained for Darcy-Rayleigh numbers close to the critical values by using the amplitude expansion method. The dependence of the fundamental mode, the distortion of the mean flow, and the second harmonic upon the variable permeability constant are discussed. By calculating the first Landau coefficient, the primary bifurcations in the vicinity of the neutral stability curves are identified. The results show that only supercritical bifurcations are found to occur, rather than subcritical instabilities. In terms of the well-known Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.