Abstract
The motion of a viscous incompressible capillary liquid with the free boundary in the absence of external forces is considered. Suppose that initially the liquid is bounded by a cylindrical free surface and that the longitudinal component of the initial velocity field depends linearly on a longitudinal coordinate, while the other components and the pressure are independent of this coordinate. Then the Navier-Stokes equations have a solution where the velocity field keeps the same structure and the free surface remains a cylindrical one. The solution gives a pithy example of the partially invariant solution for the Navier-Stokes equations describing a free boundary flow. As a corollary, the primary 3-D problem is reduced to a 2-D one.
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