Flows involving natural convection are generally studied employing the Boussinesq approximation of the Navier–Stokes equations. This paper deals with large departures from this limit by considering the two-dimensional, periodic Rayleigh–Bénard problem as an example. The main effect investigated here is a strong variation in the density considered primarily as a function of temperature as it is accounted for by the low Mach number equations. Additionally, the effect of temperature-dependent viscosity and heat conductivity is considered. Both types of departure are measured by the nondimensional temperature gradient. They destroy the midplane symmetry of the problem and lead to quantitative and qualitative changes of the flow, in particular in the bifurcation from the conduction state. The present study illustrates how three different approaches—by linear stability analysis, weakly nonlinear analysis, and direct simulation—permit the investigation of complementary aspects of the problem. The first is used to determine the critical Rayleigh number for the onset of convection. The second is well suited for investigating the nature of the bifurcation and shows that the flow in the form of rolls becomes subcritical if the non-Boussinesq character is strong enough. Direct simulations of the flow field confirm this behavior and permit the study of situations far from criticality. It turns out that non-Boussinesq effects are more and more confined to small parts of the domain as the Rayleigh number increases, so that their influence on mean properties of the flow diminishes.

We analyse the force balance on a cylindrical drop in a Hele-Shaw cell, subjected to a Marangoni flow caused by a surface tension gradient. Depth-averaged Stokes equations, called Brinkman equations, are introduced and a general closed form solution is obtained. The validity of the averaging procedure is ascertained by considering a linear surface tension gradient acting on a cylindrical flattened drop. The Marangoni-driven flow field and resulting force predicted by the Brinkman model are seen to match well a full three-dimensional direct numerical simulation. A closed form expression of the force acting on the drop is obtained, calculated from contributions due to the normal viscous stress, tangential viscous stress, and pressure fields, integrated on the drop perimeter. This expression is used to predict the force balance when a stationary droplet is submitted to both a carrier flow and a Marangoni flow. We show that previous results in the literature had underestimated by a factor two the Marangoni-induced force.