Transitions and chaos for free convection in a rotating porous layer

The non-linearity which is inherently present in centrifugally driven free convection in porous media raises the problem of multiple solutions existent in this particular type of system. The solution to the non-linear problem is obtained by using a truncated Galerkin method to obtain a set of ordinary differential equation for the time evolution of the Galerkin amplitudes. It is demonstrated that Darcy's model when extended to include the time derivative term yields, subject to appropriate scaling, the familiar Lorenz equations although with different coefficients, at a similar level of Galerkin truncation. The system of ordinary differential equations was solved by using Adomian's decomposition method. Below a certain critical value of the centrifugally related Rayleigh number the obvious unique motionless conduction solution is obtained. At slightly super-critical values of the centrifugal Rayleigh number a pitchfork bifurcation occurs, leading to two different steady solutions. For highly supercritical Rayleigh numbers transition to chaotic solutions occurs via a Hopf bifurcation. The effect of the time derivative term in Darcy's equation is shown to be crucial in this truncated model as the value of Rayleigh number when transition to the non-periodic regime occurs goes to infinity at the same rate as the time derivative term goes to zero. Examples of different convection solutions and the resulting rate of heat transfer are provided.