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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Volume 20, Number 1, Pages 43–51
(Mi timm1028)
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Description of a helical motion of an incompressible nonviscous fluid
V. P. Vereshchaginab, Yu. N. Subbotinac, N. I. Chernykhac a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Russian State Professional Pedagogical University
c Institute of Mathematics and Computer Science, Ural Federal University
Abstract:
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, Gromeka's problem.
Received: 12.04.2013
Citation:
V. P. Vereshchagin, Yu. N. Subbotin, N. I. Chernykh, “Description of a helical motion of an incompressible nonviscous fluid”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 1, 2014, 43–51; Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 202–210
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https://www.mathnet.ru/eng/timm1028 https://www.mathnet.ru/eng/timm/v20/i1/p43
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Abstract page: | 315 | Full-text PDF : | 90 | References: | 48 | First page: | 21 |
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