Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. III. Case of self-interacting modes $n\text{−}n$

This paper is the continuation of work done in our previous papers [A. A. Doinikov et al., Phys. Rev. E 100, 033104 (2019); Phys. Rev. E 100, 033105 (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 our previous papers, general equations have been derived to describe the velocity field of acoustic microstreaming produced by modes $m$ and $n$ of bubble oscillations. After solving these general equations for some particular cases of modal interactions (cases 0-$n$, 1-1, and 1-$m$), in this paper the general equations are solved analytically for the case that acoustic microstreaming results from the self-interaction of an arbitrary surface mode $n\ge 1$. 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 acoustic microstreaming. No restrictions are imposed on the ratio of the bubble radius to the viscous penetration depth. The self-interaction results in specific streaming patterns: a large-scale cross pattern and small recirculation zones in the vicinity of the bubble interface. Particularly the spatial organization of the recirculation zones is unique for a given surface mode and therefore appears as a signature of the $n\text{−}n$ interaction. Experimental streaming patterns related to this interaction are obtained and good agreement is observed with the theoretical model.

Figure 1

Geometry of the system under study. (a) Three-dimensional representation of the zonal spherical harmonics (here mode 4 is shown), where $z$ is the axis of axial symmetry. (b) Axial symmetry allows using polar coordinates ($r,\theta$) to parametrize the bubble interface $r_s$.

Figure 2

Numerical examples of streamline patterns produced by the self-interaction mode in cases: (a) 1-1, (b) 2-2, (c) 3-3, (d) 4-4.

Figure 3

Comparison of the present model to the theory of Spelman and Lauga [11] for the self-interaction of mode 2. Upper line: Streamline pattern according to (a) the theory of Spelman and Lauga and (b) the present model. Evolution of the normalized (c) radial and (d) tangential components of the Lagrangian streaming velocity given by both theories and calculated for the angle $\theta =\pi /5$, as a function of the normalized distance $r/R_0$.

Figure 4

Comparison of the present model to the theory of Maksimov [16] for the self-interaction of nonspherical mode $n=10$. Evolution of the normalized (a) radial and (b) tangential components of the Lagrangian streaming velocity given by both theories and calculated for the angle $\theta =\pi /5$, as a function of the normalized distance $r/R_0$.

Figure 5

Experimental and numerical investigation of the microstreaming patterns induced by a self-interacting mode. First line: A bubble of equilibrium radius $48\mu \mathrm{m}$ is driven by a $31.25\mathrm{kHz}$ ultrasound field at the acoustic pressure amplitude $20\mathrm{k}\mathrm{Pa}$. (a) Streak photography of the resulting microstreaming pattern (image size $620\times 620\mu \mathrm{m}$). (b) Numerical streaming pattern of the 2-2 interaction. Second line: A bubble of equilibrium radius $64.5\mu \mathrm{m}$ is driven by a $31.25\mathrm{kHz}$ ultrasound field at the acoustic pressure amplitude $12\mathrm{k}\mathrm{Pa}$. (c) Streak photography of the resulting microstreaming pattern (image size $700\times 700\mu \mathrm{m}$). (d) Numerical streaming pattern of the 3-3 interaction.