Uniqueness and the force formulas for plane subsonic flows

Abstract

Introduction. The first proof of uniqueness of a plane subsonic flow of a compressible fluid past an obstacle was given by Bers [1]. This proof utilizes an elaborate mathematical apparatus encompassing some of the most advanced tools of modern function theory. A conceptually simpler proof under somewhat weaker hypotheses and a proof of the Joukowsky force formula, due to the authors [3], make essential use of a general existence theorem and hence cannot properly be called elementary. In this note we show that the uniqueness of a compressible flow and the Joukowsky force formula can be obtained directly from a simple geometrical property of the velocity field defined by the flow. Specifically, we base our proof on the facts that the velocity components u, v of the flow define a quasi-conformal(') mapping of the (x, y) plane, and that a quasi-conformal mapping with dilatation ratio not exceeding K = 1/M satisfies at each point a Hilder condition with exponent ,. Until recently we would have considered this proof more difficult than our original one. We are, however, now able to refer to the preceding paper [5 ], in which a simple demonstration is given for the needed lemma. We note, further, that a simpler form of the arguments in [5] suffices for the present paper. The problem of proving uniqueness of a flow past an obstacle P can be divided into two parts; to prove the uniqueness if the velocity at infinity and circulation are both prescribed, and to prove that if P has a corner or cusp T the circulation is uniquely determined by the condition that the speed is finite at T. We shall settle both points here, but for the second we need the additional assumption, not used in [1] or [3], that the velocity components have bounded derivatives up to P. (This assumption can be avoided by a discussion analogous to that in [3, ?6.2].) By the same method we shall also obtain a new proof, simpler than that of [3], for the force formula of gas dynamics. 1. Uniqueness with prescribed circulation. As in [3 ] we define the velocity potential of a flow past P to be a solution k(x, y) of