Cavitational flows and global injectivity of conformal maps

Lanza de Cristoforis, Massimo

doi:10.1090/S0002-9947-1991-1040041-6

Cavitational flows and global injectivity of conformal maps
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Trans. Amer. Math. Soc. 323 (1991), 509-527 Request permission

Abstract:

This paper treats some new mathematical aspects of the two-dimensional cavitational problem of the flow of a perfect fluid past an obstacle. Natural regularity conditions of very general type are found to ensure the global injectivity of the complex-potential and the presence of at most one zero of its derivative on the boundary of the flow. This derivative is the complex velocity. Previous authors have hypothesized the properties obtained here. The same regularity conditions are then shown to be satisfied by the classical solutions found via Villat’s integral equation. A simple counterexample in $\S 4$ shows that the global injectivity of a holomorphic map defined on an unbounded Jordan domain cannot be deduced solely from its injectivity on the boundary. This simple fact raises new questions on the relation between cavitational flows and Villat’s integral equation, which are discussed in $\S 3$.

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