Equations for generalized coordinates of non-linear motions interface surfaces of liquids

Аuthors

Win K. K.

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

e-mail: win.c.latt@gmail.com

Abstract

Nonlinear problems of the dynamics of a rigid body with a cavity filled with several fluids are of considerable applied and theoretical interest. The article shows that with the help of the variational principle, written in a form different from the traditional one, it is possible to obtain a complete set of equations of nonlinear motions of liquids, including nonlinear kinematic and dynamic conditions on the interfaces of liquids filling the cavity of a solid body that performs a given movement. The variational formulation of the problem of dynamics has certain advantages, for example, from the point of view of substantiating the necessity and sufficiency of the derived equations and boundary conditions, and considering the body and fluid as one system allows one to achieve a certain multiplicity.

The article is devoted to the definition of differential equations for the generalized coordinate motions of a two-layer liquid in the cavity of a solid body performing a given motion in space. In the article, the formulation of a nonlinear problem about the motions of immiscible incompressible ideal liquids that completely fill a cylindrical cavity is formulated, and velocity potentials are given for each liquid. When obtaining differential equations for generalized coordinates of non-linear movements of liquid interface surfaces, the variational principle of Hamilton – Ostrogradsky is used, in which a modified Lagrange function is used. As a result, infinite systems of nonlinear differential equations were obtained for the generalized coordinates of the problem under consideration in the complex motion of a rigid body, as well as differential equations in particular cases.

Keywords:

variational principle, immiscible liquids, perturbed interface surface, nonlinear boundary value problem, generalized coefficients

References

1. La Rocca, G. Sciortino, C. Adduce, M.A. Boniforti. Experimental and theoretical investigation on the sloshing of a two-liquid system with free surface, Physics of Fluids, 2005, no. 17 (6), pp. 062101. DOI: 10.1063/1.1922887

2. La Rocca. Interfacial gravity waves in a two-fluid system, Fluids Dynamics Research, 2002, no. 30, pp. 31-66. DOI: 10.1016/S0169-5983(01)00039-9

3. Win Ko Ko, Temnov A.N. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2021, no. 69. DOI: 10.17223/19988621/69/8

4. Win Ko Ko, Temnov A.N. Trudy MAI. 2023. № 130. URL: https://trudymai.ru/published.php?ID=174607. DOI: 10.34759/trd-2023-130-11

5. Moiseev G.A. Dvizhenie tverdogo tela, imeyushchego polost’, tselikom zapolnennuyu dvumya nesmeshivayushchimisya zhidkostyami. V kn.: Matematicheskaya fizika (The motion of a rigid body having a cavity completely filled with two immiscible liquids. In the book: Mathematical Physics), Kiev, Naukova dumka, issue 13. 1973.

6. Lukovskii I.A. Vvedenie v nelineinuyu dinamiku tverdogo tela polostyami, soderzhashchimi zhidkost’ (Introduction to the nonlinear dynamics of a rigid body with cavities containing a liquid), Kiev, Naukova dumka, 1990, 296 p.

7. Grishanina T.V., Shklyarchuk F.N. Primenenie metoda otsekov k raschetu kolebanii zhidkostnykh raket-nositelei Application of the compartment method to the calculation of oscillations of liquid-propellant launch vehicles), Moscow, MAI, 2017, 100 p.

8. Shklyarchuk F.N. Dinamika konstruktsii letatel’nykh apparatov (Dynamics of aircraft structures), Moscow, MAI, 1983, 79 p.

9. Blinkova A.Yu., Ivanov S.V., Kuznetsova E.L., Mogilevich L.I. Trudy MAI, 2014, no. 78. URL: https://trudymai.ru/eng/published.php?ID=53486

10. Pozhalostin A.A., Goncharov D.A. O parametricheskikh osesimmetrichnykh kolebaniyakh zhidkosti v tsilindricheskom sosude, Trudy MAI, 2017, no. 95. URL: http://trudymai.ru/eng/published.php?ID=84412

11. Pak Songi, Grigor’ev V.G. Trudy MAI, 2021, no. 119. URL: https://trudymai.ru/eng/published.php?ID=159785. DOI: 34759/trd-2021-119-08

12. Sharfarets B.P. Nauchnoe priborostroenie, 2017, vol. 27, no. 1, pp. 102-112.

13. Makarov P.A. Vestnik Syktyvkarskogo universiteta. Seriya 1: Matematika. Mekhanika. Informatika, 2017, no. 2 (23), pp. 46-59.

14. Akulenko L.D., Nesterov S.V. Mekhanika tverdogo tela, 1986, no. 1, pp. 27–36.

15. Akulenko L.D., Nesterov S.V. Mekhanika tverdogo tela, 1987, no. 2, pp. 52–58.

16. Ganichev A.I., Kachura V.P., Temnov A.N. Malye kolebaniya dvukh nesmeshivayushchikhsya zhidkostei v podvizhnom tsilindricheskom sosude. V kn.: Kolebaniya uprugikh konstruktsii s zhidkost’yu (Small oscillations of two immiscible liquids in a movable cylindrical vessel. In the book: Vibrations of elastic structures with liquid, Novosibirsk, NETI, 1974, pp. 82-88.

17. Zhukovskii N.E. O dvizhenii tverdogo tela, imeyushchego polosti, napolnennye odnorodnoi kapel’noi zhidkost’yu (On the motion of a solid body having cavities filled with a homogeneous droplet liquid), Moscow, Gostekhizdat, 1948, 143 p.

18. Mikishev G.N, Rabinovich B.I. Dinamika tverdogo tela s polostyami, chastichno zapolnennymi zhidkost’ (Dynamics of a rigid body with cavities partially filled with liquid), Moscow, Mashinostroenie, 1968, 532 p.

19. Kolesnikov K.S. Dinamika raket (Rocket dynamics), Moscow, Mashinostroenie, 2003, 520 p.

20. Mikishev G.N. Eksperimental’nye metody v dinamike kosmicheskikh apparatov. (Experimental methods in the dynamics of spacecraft), Moscow, Mashinostroenie, 1987, 248 p.

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