Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. IV. Case of modes n and m

This paper is the conclusion of work done in our previous papers [A. A. Doinikov et al., Phys. Rev. E 100, 033104 (2019)10.1103/PhysRevE.100.033104; Phys. Rev. E 100, 033105 (2019)10.1103/PhysRevE.100.033105]. The overall aim of the study is to develop a theory for modeling the velocity field of acoustic microstreaming produced by nonspherical oscillations of a gas bubble. In our previous papers, general equations were derived to describe the velocity field of acoustic microstreaming produced by modes m and n of bubble oscillations. Particular cases of mode interaction were derived, such as the 0-n, 1-1, 1-m, and n-n interactions. Here the general case of interaction between modes n and m, n>m, is solved analytically. Solutions are expressed in terms of complex mode amplitudes, meaning that the mode amplitudes are assumed to be known and serve as input data for the calculation of the velocity field of microstreaming. No restrictions are imposed on the ratio of the bubble radius to the viscous penetration depth. The n-m interaction results in specific streaming patterns: At large distance from the bubble interface the pattern exhibits 2|n-m| lobes, while 2min(m,n) lobes exist in the bubble vicinity. The spatial organization of the recirculation zones is unique for the interaction of two distinct nonspherical modes and therefore appears as a signature of the n-m interaction.