Geodesic
Description of a~helical motion of an incompressible nonviscous fluid
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 43-51

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We consider a problem of describing the motion of a fluid filling at any specific instant $t\ge0$ a domain $D\subset R^3$ in terms of velocity $\mathbf v$ and pressure $p$. We assume that the pair of variables $(\mathbf v,p)$ 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 $\mathbf v$ whose lines for any $t\ge0$ 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.
Keywords: scalar and vector fields, curl, helical motion
Mots-clés : Gromeka's problem. 
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V. P. Vereshchagin; Yu. N. Subbotin; N. I. Chernykh. Description of a~helical motion of an incompressible nonviscous fluid. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 43-51. http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a4/