Analysis of coupling between hydrodynamic and thermal instabilities in non-Boussinesq convection

High-temperature convection arises in many technical applications such as thermal insulation systems, chemical vapour deposition reactors, etc. Under high-temperature conditions, fluid density and transport property variations can reach up to 30% of the average values across the flow region. Associated symmetry breaking nonlinearities are responsible for a wide spectrum of flow instabilities not found in low-temperature flows that are typically described by the Boussinesq approximation of the Navier-Stokes equations. In this work we use a set of low-Mach-number equations suggested by Paolucci in the early 1980s to describe a high-temperature mixed convection flow between two vertical plates. We find that non-Boussinesq instabilities have either a hydrodynamic (shear, common to both low- and high-temperature flows) or thermal (buoyancy, purely non-Boussinesq) character and they can occur simultaneously at certain values of the governing physical parameters (the so-called codimension-2 points). We use a weakly nonlinear analysis to show that such situations can be successfully modelled by two coupled cubic complex Landau equations. Subsequently the unfoldings of the double Hopf bifurcations involving shear modes detected in weakly non-Boussinesq mixed convection are investigated, and the complete set of resulting flow patterns is then studied as functions of the governing parameters. The spatio-temporal competition between shear and buoyancy disturbances in a strongly non-Boussinesq regime is also modelled by two coupled complex Ginzburg-Landau equations. The results obtained for these model equations are then interpreted from the physical point of view and the nature and asymptotic outcomes of instability mode competition at large times are discussed.