Acoustic streaming outside and inside a fluid particle undergoing monopole and dipole oscillations

An analytical theory is developed for acoustic streaming induced by an acoustic wave field inside and outside a spherical fluid particle, which can be a liquid droplet or a gas bubble. The particle is assumed to undergo the monopole (pulsation) and the dipole (translation) oscillation modes. The dispersed phase and the carrier medium are considered to be immiscible, compressible, and viscous. The developed theory allows one to calculate the acoustic streaming both outside and inside the fluid particle. In contrast to earlier works, no restrictions are imposed on the thickness of the outer and inner viscous boundary layers with respect to the particle radius. A numerical implementation of the obtained analytical results is used to evaluate the acoustic streaming for different experimentally relevant configurations, such as an air bubble in water, a water droplet in oil, and a water droplet in air, considering both traveling and standing acoustic waves. The results show the richness of streaming pattern variations that arise in bubbles and droplets.

Figure 1

Geometry of the system under study. In our calculations, we use the spherical coordinate system with the coordinates $\left(r,\theta ,\varepsilon \right)$ and the origin at the equilibrium center of the particle. The particle is excited by an acoustic wave propagating in the positive direction of the $z$ axis.

Figure 2

Comparison of the present model for a freely moving bubble with the model of Longuet-Higgins [9] for a fixed bubble and oscillating fluid. (a,b) The radial and tangential components of the Lagrangian streaming velocity produced by the (01) mode. (c,d) The same for the (11) mode. The bubble is excited by a plane 1-MHz traveling wave, $R_0$ is 50 µm, the gas inside the bubble is air, and the surrounding fluid is water.

Figure 3

Acoustic streaming outside and inside an air bubble immersed in water. The bubble radius is 33 µm. The bubble is excited by a plane traveling wave at 0.1 MHz, which corresponds to the monopole resonance frequency of the bubble. The total streaming pattern shown in panel (a) is the sum of the contributions due to the pulsation-translation mode (01) from panel (b) and the translation-translation mode (11) from panel (c). The total streaming pattern is dominated by the contribution of the (01) mode, which is three orders of magnitude larger than that of the (11) mode. The legends show the amplitude of the streaming velocity in meters per second.

Figure 4

Streamlines outside and inside three air bubbles of different radii. The surrounding fluid is water. The bubbles are excited by a plane 1-MHz traveling wave. The excitation frequency corresponds to the resonance frequency of the 3.3-µm-radius bubble. The legends show the amplitude of the streaming velocity in meters per second.

Figure 5

A 70-μm-radius air bubble immersed in water and placed in a plane 1-MHz standing wave. The bubble is placed in different spatial points: (a) close to the pressure node, (b) between the pressure node and the pressure antinode, and (c) close to the pressure antinode. The legends show the amplitude of the streaming velocity in meters per second.

Figure 6

A 6-μm-radius air bubble in water. The bubble is located at the pressure node of a plane 1-MHz standing wave. The legend shows the amplitude of the streaming velocity in meters per second.

Figure 7

A 20-μm-radius water droplet in HFE oil. The droplet is placed in different spatial points of a plane 1-MHz standing wave: (a) close to the velocity antinode, (b) between the velocity antinode and the velocity node, and (c) close to the velocity node. The legends show the amplitude of the streaming velocity in meters per second.

Figure 8

A water droplet in HFE oil placed at the middle between the pressure and velocity node of a standing wave. Upper panel: Streamline patterns for three values of the droplet radius at the driving frequency 1 MHz. Lower panel: Streamline patterns at three values of the driving frequency for a 15-µm-radius droplet. The legends show the amplitude of the streaming velocity in meters per second.

Figure 9

A water droplet in silicone oil subjected to a plane 1-MHz standing wave. Upper panel: Streamline patterns for three values of the droplet radius at the viscosity of silicone oil $\eta =0.1$ Pa s. Lower panel: Streamline patterns at three values of the viscosity of silicone oil for a 20-µm-radius droplet. The legends show the amplitude of the streaming velocity in meters per second.

Figure 10

(a) The phase shift $\mathrm{\Delta }\varphi$ between the linear scattering coefficients $a_0$ (pulsation) and $a_1$ (translation) as a function of the droplet radius at $\eta =0.1\mathrm{Pa}\mathrm{s}$. (b) $\mathrm{\Delta }\varphi$ as a function of $\eta$ at $R_0=20\mu \mathrm{m}$. The other parameters are as in Fig. 9.

Figure 11

Comparison of our theory to the model of Rednikov et al. [5]. The case of a 35-µm-radius water droplet in air. The droplet is located at the pressure node of a plane 1-MHz standing wave. The velocities are calculated at $\theta =\pi /6.$ The ratio $\delta /R_0$ is 0.014 and 0.028 inside and outside the droplet, respectively.

Figure 12

Comparison of our theory to the model of Rednikov et al. [5]. The case of a 3.5-µm-radius water droplet in air. The droplet is located at the pressure node of a plane 1-MHz standing wave. The velocities are calculated at $\theta =\pi /6$. The ratio $\delta /R_0$ is 0.14 and 0.28 inside and outside the droplet, respectively.

Figure 13

Streamlines for a water droplet in air at different values of $R_0$. The droplet is located at the pressure node of a plane 1-MHz standing wave. The legends show the amplitude of the streaming velocity in meters per second.