Generalized functions of slow growth problem statement for capillary wave formation in gas–liquid interface under ultrasonic cavitation

The model of the formation of linear short capillary waves on the liquid-gas surface under the action of cavitation created by ultrasonic vibrations was proposed. The equations of propagation of capillary waves were constructed in the formulation of classical and generalized functions (of slow growth) that take into account: the viscosity of the liquid phase; attenuation of wave vibrations over time due to the viscosity of the liquid phase, which implies a limited amplitude of the waves (despite the fact that in the absence of attenuation, the wave can oscillate indefinitely over time). It was proved, that for equations in generalized functions for the case of collapse of a set of bubbles in a limited volume of liquid, the displacement profile (as generalized function, with is integral in the sense of the principal Cauchy value) of the interfacial surface is a regular generalized function of slow growth. The estimated dependences of the average increase in the interfacial surface on the parameters of ultrasonic action and the viscosity of the liquid are constructed. The dependences showed an increase in the interfacial surface up to 1.6 times or more for a liquid with a viscosity close to water. The obtained value is similar to the experimental data. The existence of a limiting viscosity has been established, starting from which the effect ceases to be noticeable. This indicates the need for research at different ambient temperatures. Since, on the one hand, with increasing temperature, the viscosity of the liquid phase decreases, and on the other hand, the degree of cavitation development decreases. Apparently, there may be an optimal temperature in this regard.

1. Novoselov A. G., Dujiy A. B., and Golikova E. Yu., “Molecular diffusion of gases in a liquid. Coefficients of molecular diffusion of carbon dioxide in water [in Russian],” Sci. J. Processes Food Prod. Equip., No. 2 (2014).

2. Podryga V. O., Vikhrov E. V., and Polyakov S. V., “Molecular dynamic calculation of the gas diffusion coefficient on the example of argon, nitrogen, hydrogen, oxygen, methane and carbon dioxide [in Russian],” Prepr. Keldysh Inst. Appl. Math., No. 96 (2019). doi:10.20948/prepr2019-96

3. Shadrin E. Y., Anufriev I. S., and Sharypov O. V., “Investigation of the process of spraying and burning coal-water fuel using a pneumatic nozzle [in Russian],” Appl. Mech. Tech. Phys., 62, No. 3, 165–171 (2021).

4. Khmelev V. N., Shalunov A. V., Golykh R. N., Nesterov V. A., Dorovskikh R. S., and Shalunova A. V., “Determination of the modes and the conditions of ultrasonic spraying providing specified productivity and dispersed characteristics of the aerosol,” J. Appl. Fluid Mech., 10, No. 5, 1409–1419 (2017).

5. Rozenberg L. D., Physical Foundations of Ultrasonic Technologies [in Russian], Nauka, Moscow (1970).

6. Golykh R. N., “Evaluation of optimum modes and conditions of cavitation and acoustic absorption intensification for increasing,” J. Appl. Fluid Mech., 10, No. 5, 1235–1246 (2017).

7. Rozenberg L. D., Powerful Ultrasonic Fields [in Russian], Nauka, Moscow (1968).

8. Morton J., Khavari M., Priyadarshi A., Kaur A., Grobert N., Mi J., Porfyrakis K., Prentice P., Eskin D., and Tzanakis I., “Dual frequency ultrasonic cavitation in various liquids: High-speed imaging and acoustic pressure measurements,” Phys. Fluids, 35 (2023).

9. Maltsev N. N., Absorption of Benzene and the Possibility of its Intensification by Ultrasound [in Russian], Dnepropetr. Chem. Technol. Inst., Dnepropetrovsk (1956).

10. Tan M., Friend J., Matar O., and Yeo L., “Capillary wave motion excited by high frequency surface acoustic waves,” Phys. Fluids, No. 22, 112112 (2023). doi:10.1063/1.3505044

11. Wallenberger P. and Lyzenga D. R., “Measurement of the surface tension of water using microwave backscatter from gravity-capillary waves,” IEEE Trans. Geosci. Remote Sensing, 28, No. 6, 1012–1016 (1990). 10.1109/36.62625.

12. Taller D. and Go D., “Modulated exponential films generated by surface acoustic waves and their role in liquid wicking and aerosolization at a pinned drop,” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys., No. 87, 53004 (2013).

13. Punzmann H., Shats M., and Xia H., “Phase randomization of three-wave interactions in capillary waves,” Phys. Review Lett., No. 103, 26946 (2009). doi:10.1103/PhysRevLett.103.064502

14. Xu J. and Attinger D., “Acoustic excitation of superharmonic capillary waves on a meniscus in a planar micro-geometry,” Phys. Fluids, No. 19 (2009). doi:10.1063/1.2790968

15. Sugondo A., Sutrisno T., Anggono W., and Anne O., “Effect of frequency on droplet characteristics in ultrasonic atomization process,” E3S Web Conf., 19, 1002 (2019). doi:10.1051/e3sconf/201913001002

16. Bonn D. and Wegdam G., “Capillary waves and ellipsometry experiments,” J. Phys. I, France, 2, N. 19, 1755–1764 (1992). doi:10.1051/jp1:1992242

17. Shen L., Denner F., Morgan N., Wachem B., and Dini D., “Capillary waves with surface viscosity,” J. Fluid Mech., No. 847, 644–663 (2018). doi:10.1017/jfm.2018.364

18. Rahimzadeh A., Ahmadian Y.M.R., and Eslamian M., “Experimental study on the characteristics of capillary surface waves on a liquid film on an ultrasonically vibrated substrate,” Fluid Dyn. Res., No. 50 (2018). doi:10.1088/1873-7005/aae446

19. Ehrhorn J. and Semke W., “Numerical prediction of vibration induced liquid atomization,” Int. J. Nov. Res. Eng. Pharm. Sci., 1, No. 3, 1–9 (2014).

20. Lugovskoy A. and Lyashok A., “Physical analogue of the process of ultrasonic liquid nebulisation in a thin layer,” J. Mech. Eng. Kyiv Polytech. Inst., 110–114 (2013).

21. Simon J. C., Sapozhnikov O. A., Khokhlova V. A., Crum L. A., and Bailey M. R., “Ultrasonic atomization of liquids in drop-chain acoustic fountains,” J. Fluid Mech., No. 766, 129–146 (2015).

22. Ostapenko V. V., “On the laws of conservation of shallow water theory [in Russian],” Dokl. Akad. Nauk, 464, No. 5, 558–561 (2015).

23. Schmidmayer K., Petitpas F., Daniel E., Favrie N., and Gavrilyuk S., “A model and numerical method for compressible flows with capillary effects,” J. Comput. Phys., No. 334, 468–496 (2017).

24. Ostapenko V. V., “Modified equations of shallow water theory allowing for the propagation of discontinuous waves along a dry riverbed [in Russian],” Appl. Mech. Tech. Phys., 48, No. 6, 22–43 (2007).

25. Abbasov I. B., “Numerical simulation of nonlinear surface gravity waves transformation under shallow-water conditions [in Russian],” Appl. Math., No. 3, 135–141 (2012).

26. Lannes D. and Marche F., “Nonlinear wave-current interactions in shallow water,” Stud. Appl. Math., 136, No. 4, 382–423 (2016).

27. Constantin A., Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, Philadelphia, PA (2011) (CBMS-NSF Reg. Conf. Ser. Appl. Math.; vol. 81).

28. D¨ull W. P., “On the mathematical description of water waves,” arXiv:1612.06242 (2016).

29. Lannes D., The Water Waves Problem: Mathematical Analysis and Asymptotics, Amer. Math. Soc., Providence, RI (2013).

30. Ogorodnikov I., “Reflection of sound pulses from an inhomogeneous bubble medium,” J. Phys., Conf. Ser., 2057, 12032 (2021). doi:10.1088/1742-6596/2057/1/012032

31. Ogorodnikov A., “The formation of nonlinear sound fields in the boundary region of the bubble medium,” J. Phys., Conf. Ser., 1677, 12144 (2020). doi:10.1088/1742-6596/1677/1/012144

32. Gimaltdinov I., Gizzatullina A., and Gimaltdinova A., “On the issue of initiation of bubble detonation by small-amplitude waves,” IOP Conf. Ser. Materials Sci. Eng., 919, 62060 (2020). doi:10.1088/1742-6596/1677/1/012144

33. Golykh R. N., Carrat J.-B., Khmelev V. N., Manyakhin I. A., Minakov V. D., Genne D. V., and Barsukov A. R., “Effect of ultrasonic cavitation on the gas-liquid interface under forced aeration ,” J. Appl. Mech. Tech. Phys. 65, No. 6, 1082–1095 (2024).

34. Golykh R., Shalunov A., Khmelev V., Lopatin R., Minakov V., and Shakura V., “Evaluation of optimum modes and conditions providing increasing ultrasonic cavitation area in high-viscous and non-newtonian fluids,” Rom. J. Acoustics Vibration, 17, No. 2, 101–108 (2020).