Acoustic streaming produced by a cylindrical bubble undergoing volume and translational oscillations in a microfluidic channel

A theoretical model is developed for acoustic streaming generated by a cylindrical bubble confined in a fluid channel between two planar elastic walls. The bubble is assumed to undergo volume and translational oscillations. The volume oscillation is caused by an imposed acoustic pressure field and generates the bulk scattered wave in the fluid gap and Lamb-type surface waves propagating along the fluid-wall interfaces. The translational oscillation is induced by the velocity field of an external sound source such as another bubble or an oscillatory fluid flow. The acoustic streaming is assumed to result from the interaction of the volume and the translational modes of the bubble oscillations. The general solutions for the linear equations of fluid motion and the equations of acoustic streaming are calculated with no restrictions on the ratio between the viscous penetration depth and the bubble size. Approximate solutions for the limit of low viscosity are provided as well. Simulations of streamline patterns show that the geometry of the streaming resembles flows generated by a source dipole, while the vortex orientation is governed by the driving frequency, bubble size, and the distance of the bubble from the source of translational excitation. Experimental verification of the developed theory is performed using data for streaming generated by bubble pairs.

Figure 1

A cylindrical bubble is squeezed between two planar elastic walls in a microfluidic channel. (a) Side view. (b) Top view.

Figure 2

Dependence of the streaming velocity on the phase shift $\mathrm{\Delta }\varphi$ between the radial and translational oscillations. Normalized (a) radial and (b) azimuthal velocity components vs $sin\left(\mathrm{\Delta }\varphi \right)$ for three values of $R_0$. The numbers in brackets are the values of the ratio ${\delta }_v/R_0$.

Figure 3

Relative position of two bubbles: top view.

Figure 4

The radial component of the streaming velocity at $\theta =0$ vs distance $r_1$ from the center of bubble 1. The results were obtained for two equal bubbles with radii $R_0=30\mu \mathrm{m}$, separated by a distance $d=10R_0$ and excited at a frequency $f=50\mathrm{kHz}$. (a) Surface waves are absent. (b) Surface waves are present.

Figure 5

Streamlines for two equal bubbles undergoing radial and translational oscillations between two walls when surface waves are absent: (a) overall view, (b) enlarged view near the surface of bubble 1 (solid line). $R_0=30\mu \mathrm{m}, d=10R_0, f=50\mathrm{kHz}, sin\left(\mathrm{\Delta }\varphi \right)=0.003$. The dashed line indicates the thickness of the viscous boundary layer ${\delta }_v=0.08R_0$. The streaming is of the fountain type.

Figure 6

Streamlines for two equal bubbles undergoing radial and translational oscillations between two walls when surface waves are present: (a) overall view, (b) enlarged view near the surface of bubble 1 (solid line). $R_0=30\mu \mathrm{m}, d=10R_0, f=50\mathrm{kHz}, sin\left(\mathrm{\Delta }\varphi \right)=-0.67$. The dashed line indicates the thickness of the viscous boundary layer ${\delta }_v=0.08R_0$. The streaming is of the antifountain type.

Figure 7

Behavior of the radial component of the streaming velocity at different bubble radii. The calculations were made for two bubbles of equal radii $R_0$ at $\theta =0, d=5R_0$ and $f=$ (a) 30 kHz and (b) 50 kHz. $r_1$ is the distance from the center of bubble 1. The numbers in brackets are the values of $sin\left(\mathrm{\Delta }\varphi \right)$.

Figure 8

Behavior of the radial component of the streaming velocity at different interbubble distances $d$. The calculations were made for two bubbles with equal radii $R_0=$(a) 30 μm and (b) 50 μm at $\theta =0$ and $f=50\mathrm{kHz}$. $r_1$ is the distance from the center of bubble 1. The numbers in brackets are the values of $sin\left(\mathrm{\Delta }\varphi \right)$.

Figure 9

Comparison of experimental (circles) and theoretical (solid and dashed lines) results. The radial component of the streaming velocity is plotted as a function of distance $r_1$ from the center of the right bubble (bubble 1). (a) Bubbles with radii $R_{10}=38.3\mu \mathrm{m}$ and $R_{20}=33.6\mu \mathrm{m}$, separated by a distance $d=175\mu \mathrm{m}$, excited at 40 kHz. (b) Bubbles in contact, with radii $R_0=30\mu \mathrm{m}$, excited at 80 kHz. The solid and dashed lines show the results given by the present model and the model of Mekki-Berrada et al. [32], respectively.