A viscous buoyancy-driven flow exhibiting finite-time blow-up

We consider 2D viscous flow driven by buoyancy forces resulting from a quadratic horizontal density variation in an unbounded domain between horizontal walls. The density is a quadratic function of the concentration of a tracer, so we solve the vorticity equation under the Boussinesq approximation, together with an advection–diffusion equation for the tracer. Stagnation-point similitude eliminates dependence on the horizontal coordinate. For the case of small Grashof number (large viscosity), the flow passes through three stages. A transient adjusts from the initial condition of static fluid to a regime in which buoyancy and viscous forces are in balance. The flow and temperature gradient slowly intensify until eventually the non-linear advection terms become dominant. The flow then enters its final stage, in which a more rapid intensification leads to a singularity in finite time. While no rigorous proof has been found that this blow-up occurs, the combination of asymptotic analysis and numerical computation provides strong support for its occurrence.