The Bernoulli Equation in PDE form modelling Interfacial Fluid Flows

We consider two inviscid, immiscible, and incompressible uids separated by a sharp interface. The outer uid ows in a so-called straining pattern about the inner uid, which contains a source. When the inner uid exhibits cylindrical symmetry (line source) or spherical symmetry (point source), the shape of the interface is described by a non-linear rst order PDE. For both geometries the PDE resembles the famous non-linear rst order ODE, the Bernoulli Equation. Remarkably, the power law variable substitution technique used in the single variable case is also e ective in this multivariable case, and allows us to obtain closed-form solutions to these nonlinear PDEs. The examples presented may have applications in astrophysics.