Solutions of a System of Two-Dimensional Euler Equations and Stationary Structures in an Ideal Fluid
O. V. Kaptsov
Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
Keywords: Euler equations for an ideal fluid, -function, elliptical solutions
Abstract
A system of the Euler equations that describe two-dimensional steady flows of an ideal fluid is considered. This system is reduced to a nonlinear Laplace equation for the stream function. With the use of the Hirota τ-function, solutions of three elliptical equations (sin-Gordon, sinh-Gordon, and Titeica equations) are found. A simple method of deriving solutions in the form of rational expressions in elliptical functions is proposed. The resultant solutions describe sources in a swirled fluid, jet flows, chains of sources and sinks, and vortex structures. It is shown that the fluid flow rate over a closed curve is quantized in the case of the elliptical sin-Gordon equation.
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