A study of Darcy–Bénard regular and chaotic convection using a new local thermal non-equilibrium formulation

Abstract

The onset of Darcy–Bénard regular and chaotic convection in a porous medium is studied by considering phase-lag effects that naturally arise in the thermal non-equilibrium heat transfer problem between the fluid and solid phases. A new type of heat equation is derived for both the phases. Using a double Fourier series and a novel decomposition, an extended Vadasz–Lorenz model with three phase-lag effects is derived. New parameters arise due to the phase-lag effects between local acceleration, convective acceleration, and thermal diffusion. The principle of exchange of stabilities is found to be valid and the subcritical instability is discounted. The new perspective supports the finding of an analytical expression for the critical Darcy–Rayleigh numbers representing, respectively, the onset of regular and chaotic convection. The understanding of the transition from the local thermal non-equilibrium situation to the local thermal equilibrium one is also best explained through the new perspective. In its present elegant form, the extended Vadasz–Lorenz system with three phase-lag effects is analyzed using the largest Lyapunov exponent and the bifurcation diagram. It is found that the lag effects not only give rise to a quantitative difference in the above two metrics concerning chaos, but also present a qualitative difference as well in the form of the very nature of chaos.