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
$$**V** =
\overline{dw}/dz
($w\; =\; u\; +\; iv$Î **C**, $z\; =\; x\; +\; iy$) has
a singularity proportional to $arg\; (z$-
z_{k}^{*}).
In the case of a $K$-segment polygonal
obstacle and a (chosen in some way) number $L$ of point vortices taken
into account in calculations, the flow can be reconstructed by the
so-called *characteristic* values of the potential.
It occurs that, being the components of the required vector function
$$s:
t \mapsto (s_{1}(t),
¼,
s_{M}(t)) Î $$**R**^{M}, where
$M\; =\; M(K,L)$, they
are connected by certain functional equations corresponding to
geometric properties of the obstacle, intensity of vortices, frequency
of their breakdown from the obstacle, etc.
These equations involve the Helmholtz--Kirchhoff function $ln\; (dz/dw)$ specified on the
$L$-fold
Riemannian surface $Q\; =\; Q($s) '
w.
This surface and the boundary conditions for the function $ln\; (dz/dw)$ are parameterized
by the function $$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.

Location: http://mech.math.msu.su/~fpm/eng/k06/k064/k06405h.htm

Last modified: February 17, 2007