Acoustically excited interaction of gas bubbles in a liquid with centers located on a plane

The hydrodynamic interaction of weakly non-spherical gas bubbles in a liquid was made in the case of the bubbles located in the antinode of an intense ultrasonic standing wave, where the pressure varies harmonically. The influence of the interaction on the radial pulsations of the bubbles, their spatial displacements, and the deformations of their surfaces were investigated. Various mutual arrangements of bubbles, in which one of the bubbles is in their center and the others are evenly distributed on a series of concentric circles in a plane, were under consideration.
A system of ordinary second-order differential equations in the radii of the bubbles, the position-vectors of their centers, the vectors characterizing small arbitrary deviation of the shape of the bubbles from a spherical one, and the temperatures in the bubbles was applied. The effects of the surface tension, the liquid viscosity and compressibility, the heat transfer between the bubbles, and the liquid were taken into account. The gas in the bubbles was assumed ideal with a uniform pressure.
A number of features of influence of the interaction between the bubbles on their dynamics were found, depending on the number of the bubbles, their relative position, as well as the frequency and the amplitude of the acoustic excitation. In particular, it was shown that the rate of compression of the central bubble and its nonspherical deformations increase both with an increase in the number of the surrounding bubbles and with a decrease in the number of the circles, on which the surrounding bubbles are located. It was revealed that in the central bubble, the maximum pressures over one period of the acoustic excitation monotonously depend on the amplitude and the frequency of the acoustic excitation, whereas similar dependences of the maximum values of the deformation of the central bubble are nonmonotonic.