The interaction of a gas bubble and a solid particle in a liquid under acoustic vibrations

The interaction of a spherical solid particle and a gas bubble in a liquid subjected to ultrasonic action is numerically studied. The ultrasonic parameters are chosen in such a way that the acoustic wavelength will be much larger than the sizes of both the bubble and the particle. The acoustic pressure field at a distance to the bubble is assumed to be uniform. In the absence of a particle, the flow is a spherically symmetric flow, and the velocity of the liquid-gas interface is found from the Rayleigh-Plesset equation. The problem considered in this paper is a generalization of the classical problem without a particle. The control of the movement of solid particles around the gas bubble is important for the flotation process, which is widely used in the mineral ore beneficiation technology. The problem is considered for high frequency and small or finite vibration velocity amplitude. In the leading order of smallness and taking into the viscosity of a liquid, a pulsating flow is found for the case of a heavy particle that remains its immobility. In the following order, the mechanisms of generation of an averaged flow in the liquid volume and near its boundaries are considered. Using the obtained averaged flow, the averaged vibrational force acting on the particle and its dependence on the distance to the bubble surface are determined. It is shown that this force is an attractive force. A comparison is made with the calculation data in the inviscid approximation. It has been found that, at small distances from the bubble, there is a deviation of the calculated value of vibrational force from the value known from the analytical expression, according to which this force is proportional to the square velocity gradient of pulsations. The results obtained demonstrate that, taking into account the viscosity of a liquid leads to a larger value of the averaged vibrational force near the bubble than the inviscid approach.