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 q0/U, θ0) 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) = ϕ0(x, y) + e-λtϕ1(x, y) + ..., where ϕ0 is the known steady-state solution and ϕ1 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.