Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Shagapov V. S., Galimzyanov M. N., Agisheva U. O. Single Waves in a Gas-Liquid Bubble Mixture. Izv. Sarat. Univ. Math. Mech. Inform., 2020, vol. 20, iss. 2, pp. 232-240. DOI: 10.18500/1816-9791-2020-20-2-232-240

Published online: 
01.06.2020
Full text:
(downloads: 49)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
532.529
DOI: 
10.18500/1816-9791-2020-20-2-232-240

Single Waves in a Gas-Liquid Bubble Mixture

Autors: 
Shagapov Vladislav Shyhulagzamovich, Institute of Mechanics, Ufa Science Center
Galimzyanov M. N., Institute of Mechanics, Ufa Science Center
Agisheva Uliana Olegovna, Institute of Mechanics, Ufa Science Center
Abstract: 

Nonlinear wave processes in a two-phase medium (bubbly liquid) do not lose their relevance asan object of study due to their wide use in various fields of physics, engineering, chemical and petroleum industries. Last decades the jump in the development of computing has expanded the possibilities for the study of significantly nonlinear problems. The aim of this work was to obtain a stationary solution of equations describing the motion of a solitary wave in a gas-liquid mixture without taking into account dissipative processes. A one-dimensional stationary flow of a liquid with gas bubbles was considered under the following assumptions: the mixture is monodisperse, i.e. in each elemental volume all bubbles are spherical and of the same radius; viscosity and thermal conductivity are considerable only in the process of interfacial interaction and during bubble pulsations. Moreover, it is assumed that there is no mass transfer between the phases, and the liquid temperature is constant unlike the gas temperature in a bubble. This is always fulfilled under not very high pressures due to the bigger mass content of the liquid (therefore it can be considered as a thermostat). It greatly simplifies the task since there is no need to consider the equation of the energy in the liquid. The pressure in the bubble was assumed to be uniform. It is ensured if the radial velocity of the bubble walls is significantly less than the speed of sound in the gas. Phase pressure and bubble size were bound by the condition of combined deformation. The Rayleigh equation corresponding to the pulsations of a single spherical bubble in an infinite incompressible fluid was taken as the condition in this case. Properties of the gas in bubbles were described by the polytropic law. Based on one-dimensional stationary equations of fluid flow with gas bubbles, a solution of the “solitary wave” type is constructed. This solution in a special case of weak solitons is equal to the results taken on the basis of the Korteweg – de Vries equation for bubble media.

References: 
  1. Nigmatulin R. I. Dinamika mnogofaznykh sred [Dynamics of multiphase media: in 2 vols.]. Moscow, Nauka, 1987. Vol. 1, 360 p.; vol. 2, 464 p. (in Russian).
  2. Kutateladze S. S., Nakoriakov V. E. Teplomassoobmen i volny v gazozhidkostnykh sistemakh [Heat and mass transfer and waves in gas-liquid systems]. Novosibirsk, Nauka, 1984. 301 p. (in Russian).
  3. Kedrinskii V. K. Gidrodinamika vzryva: eksperiment i modeli [Explosion hydrodynamics: experiment and models]. Novosibirsk, Publishing House SB RAS, 2000. 435 p. (in Russian).
  4. Kim D. Ch. Physical nature of acoustical solitons in liquid with distributed gas bubbles. Doklady Physics, 2008, vol. 53, iss. 2, pp. 66–70. DOI: https://doi.org/10.1007/s11446-008-2004-9
  5. Kudriashov N. A., Sinelshchikov D. I. Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer. Fluid Dynamics, 2010, vol. 45, iss. 1, pp. 96–112. DOI: https://doi.org/10.1134/S0015462810010114
  6. Zemlyanukhin A. I., Bochkarev A. V. New Exact Solutions to the Generalized Konno – Kameyama – Sanuki Equation. Vestnik Saratov State Technical University, 2015, vol. 2, iss. 1, pp. 5–9 (in Russian).
  7. Zemlianukhin A. I., Bochkarev A. V. Exact Solitary-wave Solutions of the Burgers – Huxley and Bradley – Harper Equations. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2017, vol. 17, iss. 1, pp. 62–70 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2017-17-1-62-70
  8. Galimzyanov M. N. Propagation of Pressure Waves in Finite-Size Bubbles Zones. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2010, vol. 10, iss. 4, pp. 27–35 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2010-10-4-27-35
  9. Agisheva U. O. Shock wave impact on bubble and foam structures in two-dimensionalaxisymmetric volumes. Vestnik Bashkirskogo Universiteta, 2013, vol. 18, no. 3, pp. 640– 645 (in Russian).
  10. Agisheva U. O., Bolotnova R. Kh., Buzina V. A., Galimzyanov M. N. Parametric analysis of the regimes of shock-wave action on gas-liquid media. Fluid Dynamics, 2013, vol. 48, iss. 2, pp. 151–162. DOI: https://doi.org/10.1134/S0015462813020038
  11. Shagapov V. S., Galimzyanov M. N., Vdovenko I. I., Khabeev N. S. Characteristic Features of Sound Propagation in a Warm Bubble-Laden Water. Journal of Engineering Physics and Thermophysics, 2018, vol. 91, no. 4, pp. 854–863. DOI: https://doi.org/10.1007/s10891-018-1809-9
  12. Galimzyanov M. N., Shagapov V. Sh. Analytical studies of suspension acoustics. Multiphase Systems, 2019, vol. 14, no. 1, pp. 27–35 (in Russian). DOI: https://doi.org/10.21662/mfs2019.1.004
  13. Galimzyanov M. N., Agisheva U. O. Wave equation for bubble liquid in Lagrangian variables. Lobachevskii Journal Mathematics, 2019, vol. 40, no. 11, pp. 1922–1928. DOI: https://doi.org/10.1134/S199508021911009X
  14. Nigmatulin R. I. Osnovy mekhaniki geterogennykh sred [Fundamentals of Heterogeneous Media Mechanics]. Moscow, Nauka, 1978. 306 p. (in Russian).
  15. Nigmatulin R. I. Small-scale flows and surface effects in the hydrodynamics of multiphase media. Journal of Applied Mathematics and Mechanics, 1971, vol. 35, iss. 3, pp. 409–423.
  16. Goncharov V. V., Naugol’nykh K. A., Rybakh S. A. Steady-state perturbations in a liquid containing gas bubbles. Journal of Applied Mechanics and Technical Physics, 1976, vol. 17, iss. 6, pp. 824–829. DOI: https://doi.org/10.1007/BF00858105
  17. Nakoriakov V. E., Pokusaev B. G., Shreiber I. R. Rasprostranenie voln v gazo- i parozhidkostnykh sredakh [Wave propagation in gas and vapor-liquid environments]. Novosibirsk, Institut teplofiziki, 1983. 237 p. (in Russian).
  18. Boguslavskii Yu. Ya., Grigor’ev S. B. On arbitrary amplitude sound propagation in gasliquid mixture. Akusticheskij Zhurnal, 1977, vol. 23, iss. 4, pp. 636–639 (in Russian).
  19. Lyapidevskey V. Yu., Plaksin S. I. The structure of shock waves in a gas-liquid medium with nonlinear state equation. Dinamika sploshnoi sredy [Continuous Media Dynamics]. Novosibisk, Institut gidrodinamiki, 1983, iss. 62, pp. 75–92 (in Russian).
Received: 
12.03.2019