Asymmetric solutions of the reaction-diffusion equation

Abstract

A class of new, non-symmetric solutions are computed for the reaction-diffusion equation (the Bratu problem) governed by the partial differential equation Δ θ + Є e θ /(1+ µθ ) = 0 in a two-dimensional, rectangular domain where µ is the Arrhenius factor and Є is the Frank–Kamenetskii parameter. When µ = 0 previously computed solutions that conform to the expected symmetries are confirmed. In addition, certain hitherto unknown solution branches are found. These appear to be disconnected from the primary branch with in the parameter region investigated here. Each of these asymmetric solution branch contains an additional limit point. These additional solutions do not conform to the expected symmetries and hence they occur with multiplicities of four. Grid refinement tests and extrapolation to zero grid size indicate that these non-symmetric solutions remain distinct, indicating that they are not spurious solutions of a poorly resolved grid. Next a perturbed Bratu problem is examined with convection as an additional heat transport mechanism and Rayleigh number, Ra and tilt angle, ϕ as additional parameters. This variation is examined using both a regular perturbation series expansion in Ra around the conduction state and, a full numerical solution. All of the newly computed solutions at Ra = 0 remain smoothly connected as Ra is increased, which is viewed as an additional evidence in support of the claim that the non-symmetric solutions are real. Even for extremely small convection ( Ra = 5), the quenching effect of convection is significant in both extending the range of stable solutions in Є and in reducing the mean temperature of the region. The quenching effect of the convection in turn reduces the sharp temperature gradients thus making the limit points less sensitive to grid refinement in the presence of weak convection.