FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 4, PAGES 65-77

## Helmholtz--Kirchhoff method and boundary control of a plane flow

A. S. Demidov

Abstract

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We consider the problem of the description of the eddy singularities behind an obstruction $S$ flowed around and their dependence (and hence, the dependence of flow functionals, e.g., the resistance force) on parameters determining the boundary S of the obstacle and/or flow characteristics on S. We propose a new approach to these problems for a flat potential flow of an incompressible liquid, which is based on ideas of the Helmholtz--Kirchhoff method and the Euler equation $d$V/dt = Ñp under the assumption that the flow has a point vortices concentrated at the required centers $z$k*, where the potential $u$ of the velocity V = \overline{dw}/dz ($w = u + iv$Î C, $z = x + iy$) has a singularity proportional to $arg \left(z$- zk*). In the case of a $K$-segment polygonal obstacle and a (chosen in some way) number $L$ of point vortices taken into account in calculations, the flow can be reconstructed by the so-called characteristic values of the potential. It occurs that, being the components of the required vector function s: t \mapsto (s1(t), ¼, sM(t)) Î RM, where $M = M\left(K,L\right)$, they are connected by certain functional equations corresponding to geometric properties of the obstacle, intensity of vortices, frequency of their breakdown from the obstacle, etc. These equations involve the Helmholtz--Kirchhoff function $ln \left(dz/dw\right)$ specified on the $L$-fold Riemannian surface $Q = Q\left($s) ' w. This surface and the boundary conditions for the function $ln \left(dz/dw\right)$ are parameterized by the function s and by a control defined on S. As for the pressure $p$, it is defined by the Cauchy--Lagrange equation for the Euler equation.

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