Exact Solutions for Interfacial Outflows with Straining

Abstract

The Bernoulli equation is a famous ordinary differential equation of first order. Although it is non-linear, it can be transformed into a linear differential equation by a power-law change of variable. This paper presents a corresponding non-linear partial differential equation that can be solved in closed form, with an analogous transformation. It is therefore referred to here as the "Bernoulli Partial Differential Equation". Two applications in interfacial hydrodynamics are then presented. They both involve inviscid outflows from sources in straining flows, and in both cases, the Bernoulli partial differential equation governing the shape of the interface is solved in closed form, and the evolution of the interface with time is illustrated.