matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2025-2-65-80matmess-80Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕGeneralized functions of slow growth problem statement for capillary wave formation in gas–liquid interface under ultrasonic cavitationGeneralized functions of slow growth problem statement for capillary wave formation in gas–liquid interface under ultrasonic cavitationGolykhR. N.GolykhR. N.

Roman N. Golykh

street named after Hero of Soviet Union Trofimov, Biysk 659315

Roman N. Golykh

street named after Hero of Soviet Union Trofimov, Biysk 659315

romangl90@gmail.comBiysk Technological Institute (branch) of Polzunov Altai State Technical UniversityРоссияBiysk Technological Institute (branch) of Polzunov Altai State Technical UniversityRussian Federation2025300620253226580Copyright © Golykh R.N., 20252025Golykh R.N.Golykh R.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/80

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.

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.

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The authors declare that there are no conflicts of interest present.