Jets with two fluids. I. One free boundary. (English) Zbl 0551.76013

Alt, Hans Wilhelm; Caffarelli, Luis A.; Friedman, Avner

Indiana Univ. Math. J. 33, 213-247 (1984).

In this work the authors consider the irrotational flow of two ideal fluids; specifically, the problem of two jets with one free boundary between the two fluids. In the case of plane flows, the problem (in term of the stream function u, when suitably normalized) is to find an harmonic function satisfying suitable Dirichlet conditions on the fixed and free boundaries and such that \(| \nabla u|^ 2\) has jump \(\lambda\) across the free boundary, for some constant \(\lambda\). Existence and uniqueness of the solution is proved by means of a variational formulation, previously studied by the authors in e.g. Trans. Am. Math. Soc. 282, 431-462 (1984)]. Such results are then extended to three-dimensional axially symmetric flows in the case of monotone decreasing nozzles. The jet problem for one fluid was studied by the authors, e.g. in Arch. Ration Mech. Anal. 81, 97-149 (1983; Zbl 0515.76017) and Commun. Pure Appl. Math. 35, 29-68 (1982; Zbl 0515.76018)], based on the general treatment of the corresponding variational principles in: the first and the second author, J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105).

Reviewer: P.Secchi

MSC:

76B10	Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

stream function; Dirichlet conditions; fixed and free boundaries; Existence and uniqueness of the solution; variational formulation; three- dimensional axially symmetric flows

Citations:

Zbl 0515.76017; Zbl 0515.76018; Zbl 0449.35105