Description of a helical motion of an incompressible nonviscous fluid

Abstract

We consider the problem of describing the motion of a fluid filling at any specific time t ≥ 0 a domain D ⊂ R ³ in terms of velocity v and pressure p. We assume that the pair of variables (v, p) satisfies a system of equations that includes Euler’s equation and the incompressible fluid continuity equation. For the case of an axially symmetric cylindrical layer D, we find a general solution of this system of equations in the class of vector fields v whose lines for any t ≥ 0 coincide everywhere in D with their vortex lines and lie on axially symmetric cylindrical surfaces nested in D. The general solution is characterized in a theorem. As an example, we specify a family of solutions expressed in terms of cylindrical functions, which, for D = R ³, includes a particular solution obtained for the first time by I.S. Gromeka in the case of steady-state helical cylindrical motions.