The shape oscillation of a single two-dimensional nitrogen microbubble in an ultrasound field is numerically investigated. The Navier–Stokes equations are solved by using the finite-volume method combined with the volume-of-fluid model. The numerical results are in good accordance with experimental and theoretical results reported. According to the analyses of the shape oscillation process, the bubble deformation period is twice the driving acoustic pressure period and the shape oscillation is mainly caused by the change of interface velocity. The vortexes produced due to velocity variations lead to the deformation of the bubble interface.

Two-phase flow occurs in a wide range of natural phenomena and engineering applications [

So far, the characteristics of a single microbubble in the ultrasound field are mostly studied through experiments [

Therefore, the main objective of this work is to study the shape oscillations of a single microbubble in an ultrasound field based on numerical simulation, and the relationship between the bubble shape variation period and the driving pressure period is investigated.

In this study, a nitrogen microbubble in water in an ultrasound field is investigated numerically to analyze the characteristics of the bubble shape oscillation. The Navier–Stokes equations [

The liquid is considered to be incompressible, while the gas inside the bubble is compressible, conforming to the ideal gas law.

The gas phase is immiscible with the liquid phase, and the mass transfer between two phases is neglected.

The effect of gravity is ignored [

In the present study, a 2D simulation is carried out. The 2D microbubble is corresponding to the pancake bubble confined between two walls in the microchannel, as described in [

Simulation domain of a single microbubble in an ultrasound field.

In all simulations, the operating pressure is set to be 1 atm and the initial velocity is zero. All boundaries are set as the pressure inlet. The inlet pressures follow the ultrasound pressure equation:

To solve the time-dependent Navier–Stokes equations, the finite-volume method [^{−5} and for energy equation, are set to be 10^{−7}. The time step is chosen depending on the frequency of ultrasound.

The simulation results of different shape oscillation modes are first compared with experimental results of references [

Snapshots of a bubble oscillation under high amplitude of ultrasound [

The dynamic behaviors of a microbubble at specified ultrasound frequency and acoustic pressure amplitude are first investigated. The ultrasound frequency is set to be 130 kHz, and the acoustic pressure amplitude is 42 kPa. The shape oscillation of microbubbles at different initial bubble radii is shown in Figure

Shape oscillation of a microbubble at different oscillation modes, when _{d} = 42 kPa. The corresponding initial radii are 22

The natural frequency of two-dimensional bubble shape oscillation was studied by Mekki-Berrada et al. [

In general, the bubble oscillation mode increased monotonously with the radius within a certain frequency. As shown in Figure

The pattern distribution of the bubble shape oscillation mode, when _{d} = 42 kPa.

The shape oscillation modes of a 2D bubble in simulations are slightly different with that of a 3D bubble. However, the overall trend remains the same. It can be found from the figure that the same oscillation mode occurs over a certain bubble radius range. For example, when _{d} = 42 kPa, the oscillation mode is 4 when the bubble initial radius is in the range of 26 to 30

The simulation results and theoretical results at four frequencies are displayed in Figure

Mode distribution at different frequencies, when _{d} = 42 kPa.

The dynamic behaviors of bubble oscillation in the ultrasound field at different modes are similar. Without loss of generality, the shape oscillation processes of a single bubble at the fourth mode are discussed in detail.

A dimensionless time is first defined,

Shape oscillation of a microbubble in a single cycle at _{0} = 30 _{d} = 42 kPa. (a)

In order to further explore the shape variation during the oscillation process, the internal and external gage pressure variations and the velocity vector near the gas-liquid interface are analyzed. The external liquid pressure is probed near the boundary, while the internal gas pressure is probed at the bubble center. The variations of liquid and gas pressures from

The liquid and gas pressure at _{0} = 30 _{d} = 42 kPa.

According to the extreme outline of the bubble shape deformation, we presented the contrast chart of the instantaneous bubble morphology under the extreme state. As shown in Figure _{max} corresponds to _{max} corresponds to

Schematic diagram of the bubble’s shape at

The instantaneous velocity vectors and bubble shapes in a single shape oscillation period are shown in Figure _{1}. Eight vortexes appear at the gas-liquid surface. As a result, the interface in the _{1} direction moves inward, and the interface in the _{2} direction moves outward.

Instantaneous velocity vectors and bubble shapes during a single bubble oscillation period when

Later, the external liquid pressure starts to decrease, and the internal gas pressure starts to increase. But the liquid pressure is still larger than the gas pressure, so the inward flow velocity in the _{1} direction keeps increasing. This velocity reaches its maximum at _{1} direction begins to decrease. The interface in the _{1} direction keeps moving inward, and the interface in the _{2} direction keeps moving outward. The cross-shaped bubble begins to appear. The liquid pressure reaches its minimum, and the inner gas pressure is near the maximum at _{2} direction is larger than that in the _{1} direction.

The bubble reaches its extreme cross-shaped state at _{1} direction vanishes and these eight vortexes near the gas-liquid interface disappear. The gas and liquid pressures are nearly the same. The bubble reaches its maximum volume. After this moment, the outside liquid pressure is larger than the inner gas pressure. The bubble enters the compression stage. The interface in the _{2} direction begins to flow inward. As a result, it returns to the square shape at

Up to now, an ultrasound vibration period has finished. The bubble undergoes a compression and expansion process. In the next ultrasound vibration, the bubble undergoes another compression and expansion process and the bubble varies from circle to cross-shaped. The shape oscillation period is exactly twice the driving ultrasound period.

It can be concluded from Figure

In the present paper, the response of a single 2D microbubble in an ultrasound field was investigated numerically. The Navier–Stokes equations are used, and the VOF mode was applied to capture the bubble interface between gas nitrogen and liquid water. Bubble’s shape instability was investigated, and the effects of the initial radius and ultrasound frequency on the unstable mode of the microbubble were discussed. It can be found from the simulations that numerical results are in good accordance with experimental and theoretical results. The shape oscillation mode increases with the increase of the bubble’s initial radius and driving ultrasound frequency. The same oscillation mode occurs over a range of the bubble radius. In order to explore the mechanism of bubble shape oscillation in the ultrasound field, the detailed velocity variation of the bubble in a single shape oscillation period was presented. The bubble oscillation period is exactly twice the driving ultrasound pressure period, and the shape oscillation of the bubble is mainly caused by the change of interface velocity. Variations in the velocity result in vortexes, which will lead to the deformation of the bubble interface.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant nos. 11672284 and 11474259) and the National Key R&D Program of China (Grant nos. 2017YFB0603701 and 2016YFF0203302).

_{2}nanoparticle synthesis in a diffusion flame reactor