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 solutio ns existent in this particular type of system. The solution to the non -linear problem is obtained by using a truncated Galerkin method to ob tain a set of ordinary differential equation for the time evolution of the Galerkin amplitudes. It is demonstrated that Darcy's model when e xtended to include the time derivative term yields, subject to appropr iate scaling, the familiar Lorenz equations although with different co efficients, at a similar level of Galerkin truncation. The system of o rdinary differential equations was solved by using Adomian's decomposi tion method. Below a certain critical value of the centrifugally relat ed Rayleigh number the obvious unique motionless conduction solution i s obtained. At slightly super-critical values of the centrifugal Rayle igh number a pitchfork bifurcation occurs, leading to two different st eady solutions. For highly supercritical Rayleigh numbers transition t o chaotic solutions occurs via a Hopf bifurcation. The effect of the t ime derivative term in Darcy's equation is shown to be crucial in this truncated model as the value of Rayleigh number when transition to th e non-periodic regime occurs goes to infinity at the same rate as the time derivative term goes to zero. Examples of different convection so lutions and the resulting rate of heat transfer are provided. (C) 1998 Elsevier Science Ltd. All rights reserved.