Geodesic
Curl equation in viscous hydrodynamics in a channel of complex geometry
Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 4, pp. 5-15.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the Navier—Stokes equations for the plane steady motion of a viscous incompressible fluid in an orthogonal coordinate system in which the fluid streamlines coincide with the coordinate lines of one of the families of the orthogonal coordinate system. In this coordinate system, the velocity vector has only the tangential component and the system of three Navier—Stokes equations is an overdetermined system for two functions — the tangential component of velocity and pressure. In the present paper, the system is brought to involution, and the consistency conditions are obtained, which are the equations for the curl of the velocity in this coordinate system. The coefficients of these equations include the curvatures of the coordinate lines and their derivatives up to the second order. The equations obtained are significantly more complicated than the curl equations in a channel of simple geometry.
Keywords: Navier—Stokes equations, curvilinear coordinate system, streamline, streamline curvature, consistency condition, curl equation. 
@article{SJIM_2023_26_4_a0,
     author = {S. A. Vasyutkin and A. P. Chupakhin},
     title = {Curl equation in viscous hydrodynamics in a channel of complex geometry},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {5--15},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2023_26_4_a0/}
}
TY  - JOUR
AU  - S. A. Vasyutkin
AU  - A. P. Chupakhin
TI  - Curl equation in viscous hydrodynamics in a channel of complex geometry
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2023
SP  - 5
EP  - 15
VL  - 26
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2023_26_4_a0/
LA  - ru
ID  - SJIM_2023_26_4_a0
ER  - 
%0 Journal Article
%A S. A. Vasyutkin
%A A. P. Chupakhin
%T Curl equation in viscous hydrodynamics in a channel of complex geometry
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2023
%P 5-15
%V 26
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2023_26_4_a0/
%G ru
%F SJIM_2023_26_4_a0
S. A. Vasyutkin; A. P. Chupakhin. Curl equation in viscous hydrodynamics in a channel of complex geometry. Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 4, pp. 5-15. http://geodesic.mathdoc.fr/item/SJIM_2023_26_4_a0/

[1] Milne-Thomson L. N., Theoretical Hydrodynamics, The Macmillan And Company, London, 1962 | MR

[2] Schlichting H., Gersten K., Boundary-Layer Theory, Springer, Heidelberg, 2003 | MR

[3] Kochin N. E.,Kibel I. A.,Roze N. V., Theoretical Hydromechanics, Interscience Publishers, N. Y., 1964 | MR | Zbl

[4] Shikhmurzaev Y., Sisoev G., “Spiralling liquid jets: Verifiable mathematical framework, trajectories and peristaltic waves”, J. Fluid Mech., 819 (2017), 352–400 | DOI | MR | Zbl

[5] Chan C. H., Czubak M., Disconzi M. M., “The formulation of the Navier—Stokes equations on Riemannian manifolds”, J. Geom. Phys., 121 (2017), 335–346 | DOI | MR | Zbl

[6] Marusic-Paloka E., “The effects of flexion and torsion on a fluid flow through a curved pipe”, Appl. Math. Optim., 44:3 (2001), 245–272 | DOI | MR | Zbl

[7] Chen G. Q., Osborne D., Qian Z., “The Navier—Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries”, Acta Math. Sci., 29:4 (2009), 919–948 | DOI | MR | Zbl

[8] M. V. Korobkov, K. Pileckas, V. V. Pukhnachov, and R. Russo, “The flux problem for the Navier—Stokes equations”, Russ. Math. Surv., 69:6 (2014), 1065–1122 | DOI | DOI | MR | Zbl

[9] V. V. Pukhnachov, Symmetries in the Navier—Stokes Equations, Novosibirsk. Gos. Univ., Novosibirsk, 2022 (in Russian)

[10] A. A. Abrashkin and E. I. Yakubovich, Vortex Dynamics in the Lagrangian Description, Fizmatlit, M., 2006 (in Russian)

[11] Pommaret J. F., Differential Galois Theory, Mathematics and Its Applications: a Series of Monographs and Texts, 15, Gordon and Breach, Science Publishers, London, 1983 | MR | Zbl

[12] D. V. Parshin, R. A. Gaifutdinov, A. V. Koptyug, and A. P. Chupakhin, “Mechanics of ski sliding on snow: Current status and prospects”, J. Appl. Mech. Tech. Phys., 64:4 (2023), 693–706 | DOI

[13] A. I. Ageev and A. N. Osiptsov, “Macro- and microhydrodynamics of a viscous fluid on a superhydrophobic surface”, Colloid J., 84 (2022), 379–393 | DOI

[14] Plotnikov P. I., Sokolowski J., “Geometric Aspects of Shape Optimization”, J. Geom. Anal., 33 (2023), 206 | DOI | MR | Zbl

[15] Kobayashi S., Nomizu K., Foundations of differential geometry, Interscience Publishers, N. Y., 1963 | MR | Zbl

[16] Dall'Acqua A., Pozzi P., “A Willmore—Helfrich $L^2$-flow of curves with natural boundary conditions”, Comm. Anal. Geom., 22:4 (2014), 617–669 | DOI | MR | Zbl

[17] Dziuk G., Kuwert E., Schatzle R., “Evolution of elastic curves in $\mathbb{R}^n$: Existence and computation”, SIAM J. Math. Anal., 33:5 (2002), 1228–1245 | DOI | MR | Zbl

[18] Lin C. C., “$L^2$-flow of elastic curves with clamped boundary conditions”, J. Differ. Equ., 252:12 (2012), 6414–6428 | DOI | MR | Zbl

[19] Plotnikov P. I., Sokolowski J., “Gradient flow for Kohn-Vogelius functional”, Siberian Electron. Math. Rep., 20:1 (2023), 524–579 | MR

[20] Malkin A. Ya., Patlazhan S. A., Kulichikhin V. G., “Physicochemical phenomena leading to slip of a fluid along a solid surface”, Usp. Khim., 88:3 (2019), 319–349 | DOI

[21] Bazant M. Z., Vinogradova O. I., “Tensorial Hydrodynamic Slip”, J. Fluid Mech., 613 (2008), 125–134 | DOI | MR | Zbl