## 振动条件下液体内部气泡下沉现象研究

*南京航空航天大学理学院,南京210016

## STUDY ON SUBSIDENCE OF THE BUBBLES INSIDE THE VIBRATING LIQUID UNDER THE CONDITION OF VIBRATION

ZHANG Xu*,, LI Jinbin,*,1)

*College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

College of Automation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Abstract

The mechanism of the bubble subsidence is studied based on hydrodynamics, focusing on the bubble subsidence behavior in vertically vibrating cylindrical vessels. With the consideration of additional mass and bubble compressibility, a mathematical model of the compressible bubble with subsidence effect is established, and the critical displacement and velocity of the bubble subsidence are obtained by the separating variable method. The results show that the amplitude and the frequency of the sinusoidal excitation are the important factors affecting the bubble subsidence, and they are closely related with the critical displacement and velocity, and the larger the amplitude and the frequency, the smaller the critical displacement, the easier the bubble sinks.

Keywords： compressibility bubble ; subsidence condition ; sinusoidal vibration ; separation of variables ; critical displacement

ZHANG Xu, LI Jinbin. STUDY ON SUBSIDENCE OF THE BUBBLES INSIDE THE VIBRATING LIQUID UNDER THE CONDITION OF VIBRATION. MECHANICS IN ENGINEERING[J], 2020, 42(3): 282-288 DOI:10.6052/1000-0879-19-450

## 1 气泡下沉动力学分析

### 1.2 气泡受力分析

\begin{align} m_{\rm att} = \chi \cdot m_{\rm f} = \frac{{2}}{3}\pi \rho R^3 \end{align}

\begin{align} & (m + m_{\rm att} )\ddot{x} + \dot {x}\frac{\partial m_{\rm att}}{\partial t} = \notag\\ & - \frac{{1}}{{2}}\rho \dot {x}^2C {\rm sgn}(\dot {x}) + (m + m_{\rm att} - \notag\\ & \rho V(x,t))(A\omega ^2\sin (\omega t) + g) \end{align}

### 1.3 气泡形态变化

Bleich等[7]的理论分析建立在球形气泡的理想状况下,忽略了气泡形变对下沉运动的影响。而实际情况下,在振动液体中运动的气泡,其体积和形状会随周围压力的改变而发生明显的变化,不再为理想球形状态,会呈现椭球形或球冠形。

\begin{align} P(x,t) = P(0,t) + \rho x(g + A\omega ^2\sin \omega t) + {2\sigma}/{R} \end{align}

\begin{align} V(x,t) \!=\! \frac{P(0,t)V(0,t)}{P(0,t) + \rho x(g + A\omega ^2\sin \omega t) + 2\sigma / R} \end{align}

\begin{align} We = {2\rho v^2R}/{\sigma} \end{align}

## 2 "快慢运动"模型及数值模拟

### 图2

\begin{align} x(t) = X(t) + \psi (t) \end{align}

### 2.1 "快运动"模型

\begin{align} & (m + m_{\rm att} )\ddot {\psi} = \notag\\ & - 4\rho R_0 ^2\psi (Re)(\dot {\psi}^2\text{sgn}\dot {\psi} - \langle \dot {\psi}^2\text{sgn}\dot {\psi}\rangle ) + \notag\\ & (m - \rho V(0,t))A\omega ^2\sin \omega t \end{align}

\begin{align} \psi (t) = B\text{sin}(\omega t + \varphi ) \end{align}

\begin{align} B^2 = \frac{2A^2}{\chi ^2 + \sqrt {\chi ^4 + \dfrac{16^2}{\pi ^4}\psi ^2_\infty \dfrac{A^{2}}{R_{0}^{2}}}}\end{align}

"快运动"为气泡下沉瞬间的脉动运动,即正弦运动。气泡的"快运动"过程受外界激励振幅、激励频率的影响较大,其运动振幅$B$接近于激励振幅$A$,其运动频率与激励频率相一致。

### 2.2 "慢运动"模型

\begin{align} & (m + m_{\rm att} )\ddot {X} + \langle F(\dot {x} + \dot {\psi})\rangle = \notag\\ & \gamma \omega ^2\frac{\rho V(0,t)g}{2}\frac{X}{H_0}\left[ 1 -\frac{\text{2}}{\text{3}}\left( {1 \!-\! \frac{m}{\rho V(0,t)}} \right)\sin ^2\varphi \right] -\notag\\ & (\rho V(0,t) - m)g \end{align}

\begin{align} \gamma \cdot w^2\frac{X}{\text{2}H_0}\left[ {{1 -}\dfrac{{2}}{{3}}\dfrac{\theta \dfrac{A^2}{R_0 ^2}}{2\left( {{1 +}\sqrt {1 + \theta \dfrac{A^2}{R_0 ^2}}} \right) \!+\! \theta \dfrac{A^2}{R_0 ^2}}} \right] \!>\! 1 \end{align}

\begin{align} X_0 = \frac{2H_0}{\gamma \cdot w^2} \cdot \frac{2\left( {{1 +}\sqrt {{1 +}\theta \dfrac{A^2}{R_0 ^2}}} \right) + \theta \dfrac{A^2}{R_0 ^2}}{2\left( {{1 +}\sqrt {{1 +}\theta \dfrac{A^2}{R_0 ^2}}} \right) + \dfrac{\theta}{3}\dfrac{A^2}{R_0 ^2}} \end{align}

\begin{align} \dot {X} \approx \upsilon \left( {\frac{X}{X_0} - 1} \right) \end{align}

\begin{align} \upsilon = \frac{\pi ^2}{12 \cdot \psi_\infty}\frac{R_0}{B}\frac{g}{\omega} \end{align}

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