On fire whirls and Popov’s equation

Abstract A fire whirl is a swirling flow with a burning core. A very simple model of a fire whirl is used, consisting of a linear second-order elliptic partial differential equation (PDE) for a stream function, extending established models for axisymmetric swirling flow of an incompressible inviscid fluid. The introduction of thermal effects means that the PDE cannot be solved by separating variables. The PDE contains a dimensionless parameter upper A A $A$ . When upper A A $A$ is large, parabolic approximations can be obtained, in much the same way that high-frequency approximations for diffraction problems are derived. The new PDE has similarities with Popov’s equation, which arises when a whispering gallery wave encounters a boundary inflection. To solve it, two approaches are developed, both leading to first-order PDEs. The simpler assumes that a second parameter in the PDE is also large. If this assumption is not made, progress can be made by using a Laplace transform with respect to a radial coordinate, but the result is seen to be much more complicated.