A trial-free-boundary method for computing Batchelor flows
Batchelor (or Prandtl-Batchelor) flows are steady inviscid flows in which a bounded domain of constant vorticity \(\omega\) is separated from an external irrotational flow by a vortex sheet. In two-dimensional case, these flows are described by the following free-boundary problem: determine a curve \(\Gamma\) and a function \(u\) such that \(\Delta u=-\omega\) inside \(\Gamma\), \(\Delta u=0\) outside \(\Gamma\), and there is a constant jump in \(|\nabla u|^2\) across \(\Gamma\). Here a family \(T_\varepsilon\) of extended operator methods (called by the authors trial-free-boundary method) is employed to solve the problem approximately, giving for \(\varepsilon\to 0\) the exact solution. For every fixed \(\varepsilon>0\), the authors establish the convexity and monotonicity of the operator \(T_\varepsilon\), and prove the existence of the approximate solution \(u_\varepsilon\). The convergence \(u_\varepsilon\to u\) is demonstrated, however, only numerically in some examples.
scientific article; zbMATH DE number 1034165
scientific article