Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. II. Case of modes 1 and $m$

This paper continues a study that was started in our previous paper [A. A. Doinikov et al., Phys. Rev. E 100, 033104 (2019)]. The overall aim of the study is to develop a theory for modeling the velocity field of acoustic microstreaming produced by nonspherical oscillations of an acoustically driven gas bubble. In the previous paper, general equations were derived that describe the velocity field of acoustic microstreaming produced by modes $n$ and $m$ of bubble oscillations. In the present paper, the above equations are solved analytically in the case that acoustic microstreaming is the result of the interaction of the translational mode (mode 1) with a mode of arbitrary order $m\ge 1$. Solutions are expressed in terms of complex mode amplitudes, which means that the mode amplitudes are assumed to be known and serve as input data for the calculation of the velocity field of acoustic microstreaming. No restrictions are imposed on the ratio of the bubble radius to the viscous penetration depth. Analytical results are illustrated by numerical examples.

Figure 1

Case $1-1$. Comparison of the streaming velocity components given by different theories inside the viscous boundary layer.

Figure 2

Case $1-1$. Comparison of the streaming velocity components given by Longuet-Higgins’ and our theories outside the viscous boundary layer.

Figure 3

Evolution of the radial (left column) and tangential (right column) components of the Lagrangian streaming velocity given by the theory of Spelman and Lauga [7] and the present model for modes 1 and 4. The velocity components are plotted for three angles: (a), (b) $\theta =0$; (c), (d) $\theta =\pi /\phantom{\pi 4}4$; (e), (f) $\theta =\pi /\phantom{\pi 2}2$.

Figure 4

Numerical examples of streamline patterns produced by various mode pairs.

Figure 5

Case 1 – 3. Dependence of the magnitude of the Eulerian streaming velocity on the distance from the bubble surface at various values of the phase shift $\varphi$ between the modes. The velocity variation along three directions is shown: (a) $\theta =0$, (b) $\theta =\pi /\phantom{\pi 4}4$, (c) $\theta =\pi /\phantom{\pi 2}2$.