The effect of imperfectly insulated sidewalls on natural convection in porous media

The paper presents the results of an investigation of the effect of imperfectly insulated sidewalls on natural convection in porous media at slightly supercritical Rayleigh numbers. An analytical solution for a rectangular domain with imperfectly insulated sidewalls and heated from below, was obtained through the weak nonlinear theory. The solution enables the determination of the amplitude of the convection and the direction of the flow. The amplitude results from an ordinary nonhomogeneous differential equation, with a forcing term representing the heat leakage through the lateral walls. The steady state amplitude solution shows that the transition through the critical Rayleigh number is smooth, differing from the case of perfectly insulated sidewalls where a bifurcation usually appears at the critical Rayleigh number. As a result, within a certain range of slightly supercritical Rayleigh number values, the amplitude and the direction of the convection currents are uniquely determined by the heat leakage through the lateral walls and they are independent of the initial conditions. A subcritical convection occurs as a result of the imperfectly insulated sidewalls, enabling the smooth transition through the critical Rayleigh value. A three-branch bifurcation develops at a higher Rayleigh number. A stability analysis of the solutions, corresponding to these branches, shows that the amplitudes which correspond to the two highest values are stable, while the third is unstable.