International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 11, Pages 541-570

Hydrodynamic equations for incompressible inviscid fluid in terms of generalized stream function

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Vernadskii Avenue, Moscow 119526, Russia

Received 5 August 2003

Copyright © 2004 Yuri A. Rylov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Hydrodynamic equations for ideal incompressible fluid are written in terms of generalized stream function. Two-dimensional version of these equations is transformed to the form of one dynamic equation for the stream function. This equation contains arbitrary function which is determined by inflow conditions given on the boundary. To determine unique solution, velocity and vorticity (but not only velocity itself) must be given on the boundary. This unexpected circumstance may be interpreted in the sense that the fluid has more degrees of freedom than it was believed. Besides, the vorticity is a less observable quantity as compared with the velocity. It is shown that the Clebsch potentials are used essentially at the description of vortical flow.