Mode interactions in reaction-driven convection in a porous medium

Convective-mode interaction occurring in reaction-driven convection in a porous medium is analyzed using local bifurcation theory. The system considered is a porous medium in the shape of a rectangular box with nonflux boundary conditions at the side walls and the bottom. From linear stability analysis it is found that vertical mode 1 (with one roll in the vertical direction) was always the first to destabilize. Hence, mode interactions occurring among various horizontal modes and the first vertical mode are considered. Center manifold theory is used to derive formulae for evaluating the coefficients of the amplitude equations of the normal form for a double zero singularity, directly from the original system of PDEs. Three different kinds of bicritical points are considered. For the case of interaction between mode 1 and mode 2 solutions, pure mode 1 solution does not exist and the presence of a secondary Hopf bifurcation and hence, oscillatory convection is predicted. From the simple zero and limit point interaction, it is shown that when the convective solution branches from the middle (unstable) conduction branch, a secondary Hopf bifurcation point exists on the convective branch leading to stable oscillatory convective solutions. Similar analysis done for mode 2 and mode 3 interaction shows that three types of solutions, pure mode 2, pure mode 3 along with a mixed mode 23 solution exist. These results indicate that mode interactions occurring in reaction-driven convection are profoundly different from those in Lapwood or Benard convection. A simple lower-dimensional, discretized version of the original model was developed and shown to have the same local behavior as the infinite-dimensional model. Finally, global bifurcation diagrams were calculated for the full system of equations using the orthogonal collocation method. It was found that even though the collocation method breaks the symmetry present in the problem, spurious solutions can be avoided by taking enough terms in the direction where symmetry is present.