Weakly nonlinear stability analysis and study of chaotic Darcy-Benard convection of a combusting fluid

Abstract

Linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid undergoing combustion are investigated in the paper using a zeroth-order chemical reaction model. The Frank-Kamenetskii-Vadasz-Lorenz model that has in it the influence of combustion is derived and transformed into a Ginzburg–Landau equation using a method of multiscales. An analytical expression for the Frank-Kamenetskii-Darcy-Rayleigh number and the Hopf-Frank-Kamenetskii-Darcy-Rayleigh number is reported. In the limiting case, these expressions are validated with the results of previous investigations. Using an analytical solution of the Ginzburg–Landau equation, heat transport in the combusting fluid is studied. The clear blow-up of the Nusselt number in the post-ignition regime is shown. The overall effect of combustion is to advance the onset of regular convective and chaotic motions and to enhance the heat transport. The symmetry in the Frank-Kamenetskii-Vadasz-Lorenz model and its Hamiltonian nature are shown. The appearance of chaotic/periodic motion in the system for large values of the eigenvalue (with the butterfly diagram being trapped in an ellipsoidal region) are highlighted in the paper. One new feature in the problem is the favouring of prolonged periodic convection in a larger range of values of the scaled Darcy-Frank-Kamenetskii-Rayleigh number compared with no combustion. The exact nature of the influence of combustion on Darcy–Bénard convection is visualized with the help of a bivariate, least-squares surface fit of the data and some important conclusions are drawn.