Труды Института механики им. Р.Р. Мавлютова

Aganin A.A., Davletshin A.I. Hydrodynamic interaction of weakly nonspherical gas bubbles in a liquid in the three-dimensional problem statement. Proceedings of the Mavlyutov Institute of Mechanics. 2017. 12(1). 51–58.

2017. Vol. 12. Issue 1, Pp. 51–58
URL: http://proc.uimech.org/uim2017.1.008
DOI: 10.21662/uim2017.1.008

Hydrodynamic interaction of weakly nonspherical gas bubbles in a liquid in the three-dimensional problem statement

Aganin A.A., Davletshin A.I.

Institute of Mechanics and Engineering, Kazan

Abstract

The influence of the relative location of weakly nonspherical gas (air) bubbles in liquid (water) on their hydrodynamic interaction at the pressure antinode of an ultrasonic standing wave is studied. Three essentially different configurations of the mutual arrangement of the bubbles are considered: the linear, planar, and spatial ones. In the linear configuration, three bubbles are located on one straight line. In the planar configuration, five bubbles are situated on two mutually orthogonal straight lines. In the spatial configuration, seven bubbles are located on three mutually orthogonal straight lines. In all the cases, one of the bubbles is central, the others are equally-spaced from it on those lines. All the bubbles are initially spherical with a radius of 3 ?m. Room conditions are considered, the frequency of the wave is 20 kHz, its amplitude is 1.2 times the static liquid pressure. A mathematical model is used, in which the liquid is assumed weakly compressible, the bubbles being homobaric. It is shown that in each configuration of the bubbles the influence of their interaction on their radial dynamics is insignificant. In each configuration, the central bubble remains stationary, while the others move to it at a speed equal for a fixed configuration. At transition from a linear configuration to a planar one and from a planar configuration to a spatial one, the bubble motion speed increases. In all the cases, the deformations of the central stationary bubbles are smaller than those of the rest. Among the central bubbles, the bubble in a planar configuration is deformed most.

Keywords

hydrodynamic interaction of bubbles,
acoustic field,
deformation of bubbles

Article outline

Purpose. Numerical study of radial oscillations of bubbles, their spatial motion and deformations at the pressure antinode of a standing wave as a function of mutual arrangement of the bubbles. Comparison of three main configurations of the mutual arrangement of the bubbles. The first configuration consists of three bubbles located on one straight line, the second configuration includes five bubbles situated in a plane on two mutually orthogonal straight lines, and the third configuration involves seven bubbles located in space on three mutually orthogonal straight lines. In each configuration, one of the bubbles is central, and the others, lateral, are equally-spaced from the central one on those straight lines.

Methodology. The dynamics of the bubbles is described by a system of ordinary differential equations obtained by the method of spherical functions under the assumption that the liquid is weakly compressible, weakly viscous, the bubbles are homobaric, the bubbles are located not very close to one another, and their surfaces are weakly nonspherical. The system of equations is solved numerically by a high-accurate Runge-Kutta method with an automatically chosen time step.

Findings. In all the configurations considered, the influence of the interaction of the bubbles on their radial dynamics is insignificant. In all the cases, the central bubble remains stationary, and the lateral bubbles move to the central one at an equal speed. With increasing the number of interacting bubbles (i.e. with changing their configuration) the speed of the bubble motion increases. In all the cases, the deformations of the central stationary bubbles are smaller than those of the lateral bubbles moving to the central one. Among the central bubbles, the maximum deformations are experienced by the bubble in a planar configuration. It is found that, at the distances between the bubbles corresponding to the utilized model, the central bubble in the planar configuration remains axisymmetric (with the axis of symmetry orthogonal to the plane of the bubble centers), and in the spatial configuration it is purely spherical.