Unsteady two-dimensional flows with free boundaries. - I. General theory

The paper contains a summary of relevant earlier work on free-boundary problems (§1) and then considers the initial development of steady two-dimensional flows. The motion of an incompressible in viscid fluid with free boundaries is considered (§2) by transforming into the hodograph plane (In q 0 / U , θ 0 ) of the steady flow. The equation of the free boundary and the velocity potential are expanded in powers of e- λt . Thus ϕ ( x, y, t ) = ϕ 0 ( x, y ) + e - λt ϕ 1 ( x, y ) + ..., where ϕ 0 is the known steady-state solution and ϕ 1 is to be determined. The exact boundary condition, which is the unsteady form of Bernoulli’s equation, is applied on the free boundary which is not taken as a streamline. A general discussion of the validity of the approach is given (§3). It is foreshadowed that for jet flow through a slit the predicted shape of the jet will probably have a kink at the nose; this is consistent with the assumptions made in the analysis.