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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{TIMM_2014_20_1_a4,
     author = {V. P. Vereshchagin and Yu. N. Subbotin and N. I. Chernykh},
     title = {Description of a~helical motion of an incompressible nonviscous fluid},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {43--51},
     year = {2014},
     volume = {20},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a4/}
}
TY  - JOUR
AU  - V. P. Vereshchagin
AU  - Yu. N. Subbotin
AU  - N. I. Chernykh
TI  - Description of a helical motion of an incompressible nonviscous fluid
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2014
SP  - 43
EP  - 51
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a4/
LA  - ru
ID  - TIMM_2014_20_1_a4
ER  - 
%0 Journal Article
%A V. P. Vereshchagin
%A Yu. N. Subbotin
%A N. I. Chernykh
%T Description of a helical motion of an incompressible nonviscous fluid
%J Trudy Instituta matematiki i mehaniki
%D 2014
%P 43-51
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a4/
%G ru
%F TIMM_2014_20_1_a4
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/

[1] Vereschagin V. P., Subbotin Yu. N., Chernykh N. I., “Preobrazovanie, izmenyayuschee geometricheskoe stroenie vektornogo polya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15, no. 1, 2009, 111–121

[2] Gromeka I. S., Nekotorye sluchai dvizheniya neszhimaemoi zhidkosti, Dis. $\dots$ d-ra fiz.-mat. nauk, Kazan, 1881, 107 pp.; И. С. Громека, Собрание сочинений, Изд-во АН СССР, М., 1952, 296 с. | MR

[3] Vereschagin V. P., Subbotin Yu. N., Chernykh N. I., “K mekhanike vintovykh potokov v idealnoi neszhimaemoi nevyazkoi sploshnoi srede”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18, no. 4, 2012, 120–134

[4] Korn G., Korn T., Spravochnik po matematike, Dlya nauchnykh rabotnikov i inzhenerov, Nauka, M., 1977, 832 pp. | MR