力学学报, 2021, 53(2): 352-361 DOI: 10.6052/0459-1879-20-300

流体力学

横向交流电场下液膜参数不稳定性分析1)

王铁晗*, 富庆飞,*,,2), 杨立军*,

*北京航空航天大学宇航学院, 北京 100191

北京航空航天大学大数据精准医疗高精尖创新中心, 北京 100191

PARAMETRIC INSTABILITY OF LIQUID SHEETS SUBJECTED TO A TRANSVERSE AC ELECTRIC FIELD1)

Wang Tiehan*, Fu Qingfei,*,,2), Yang Lijun*,

*School of Astronautics, Beihang University, Beijing 100191, China

Beijing Advanced Innovation Center for Big Date-based Precision Medicine, Beihang University, Beijing 100191, China

通讯作者: 2) 富庆飞, 研究员, 主要研究方向: 火箭发动机喷雾燃烧. E-mail:fuqingfei@buaa.edu.cn

收稿日期: 2020-08-25   接受日期: 2020-11-2   网络出版日期: 2021-02-07

基金资助: 1) 国家自然科学基金资助项目.  11922201

Received: 2020-08-25   Accepted: 2020-11-2   Online: 2021-02-07

作者简介 About authors

摘要

当将运动的平面液膜置于横向的交流电场之间时会产生参数振荡现象.为了得到交流电场下平面液膜的色散关系并为液膜的破碎行为分析提供理论基础,本文基于漏电介质模型对液体的电学特性进行假设,对平面液膜在直流和交流电场下的参数不稳定性进行了分析.由于主流是基于时间的流动, 在稳定性分析中引入了Floquet理论. 在文中,将电场定义为部分交流电场和部分直流电场共同耦合后的混合型电场. 最后,波数和不稳定增长率之间的无量纲色散方程可由矩阵的形式表示.本文考虑了多种参数对不稳定的影响, 包括气液密度比、韦伯数、雷诺数、欧拉数、 松弛时间以及衡量交流电场占比的参数及频率参数,并得知欧拉数同时影响毛细不稳定性及参数不稳定性,交流电场占比对不稳定性的影响体现在恒定电场力上, 而交流电场频率主要影响参数不稳定性. 为了在实验中更容易地寻求参数振荡现象,增大电欧拉数及减小交流电场频率是有效的方法.

关键词: 参数不稳定性 ; 平面液膜 ; 电场 ; 漏电介质模型

Abstract

Parametric resonance occurs at the gas-liquid interface when a liquid sheet moving in a transverse AC electrical field. In order to obtain the dispersion relation of liquid sheet under AC electric field and to provide the theoretical basis for the analysis of the breaking behavior of liquid sheet, in this paper, the temporal parametric instability under DC and AC electric field were both analyzed in the leaky dielectric model. The leaky dielectric model is used to characterize the electrical properties of liquid. Since the mean flow is time-dependent function, the Floquet theory is used to solve the stability problem. In this paper, the electric field is defined as a mixed electric field coupled by part of an ac electric field and part of a DC electric field. The dimensionless dispersion relation between wave number and temporal growth rate can be derived as a matrix. According to this relationship, the influence of various liquid properties on the parametric instability were discussed. The effects of the ratio of gas-to-liquid density ($\rho )$, the Weber number (We), the Reynolds number (Re), the electrical Euler number ($Eu$), the relative relaxation time ($\tau )$ and the characteristic of the proportion of AC electric field ($Pr$) and the frequency of the electric field ($\varOmega )$ was concluded in this paper. As a conclusion, the electrical Euler number ($Eu$) influence the instability of both capillary unstable and parametric unstable region, the proportion of electric field ($Pr$) effects as a constant electric field force, the frequency of the AC electric field ($\varOmega )$ mainly influence the parametric instability region. In the experiment, in order to obtain parameter oscillation phenomenon more easily, increasing electrical Euler number ($Eu$) and reducing the frequency of AC electric field ($\varOmega )$ are founded as effective methods.

Keywords: parametric instability ; liquid sheet ; electric field ; leaky dielectric model

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本文引用格式

王铁晗, 富庆飞, 杨立军. 横向交流电场下液膜参数不稳定性分析1). 力学学报[J], 2021, 53(2): 352-361 DOI:10.6052/0459-1879-20-300

Wang Tiehan, Fu Qingfei, Yang Lijun. PARAMETRIC INSTABILITY OF LIQUID SHEETS SUBJECTED TO A TRANSVERSE AC ELECTRIC FIELD1). Chinese Journal of Theoretical and Applied Mechanics[J], 2021, 53(2): 352-361 DOI:10.6052/0459-1879-20-300

引言

由于液体雾化在灭火系统、喷涂工艺、喷墨印刷、涡轮发动机和液体火箭发动机等领域的广泛应用, 气体介质中的液膜不稳定性分析的理论研究受到了持续的关注[1-3]. 在此方向上, 众多学者从不同角度进行了大量的研究. Squire[1]对液体薄层在静止空气中的流动进行了分析, 认为液体薄层在特定情况下可以达到稳定状态, 这一结论由Hagerty[4]通过实验证明, 并提出了两种不稳定模式, 即以反对称波为特征的弯曲模式和以对称波为特征的曲张模式, 并得出了更高的韦伯数和气液密度比使液膜更加不稳定的结论. Lin[5]和Li[6]对此进行了拓展, 对液膜周围气体赋以不同于液膜的速度进行了理论研究. Li和Kelly[7]对无黏液膜的不稳定性进行了研究, Cao和Li[8]以及Zhang[9]进行了黏性液膜的不稳定性分析.

随着对液膜的不稳定性分析理论趋于成熟, 学者们对液膜在电动力下的行为开始感兴趣. 在衡量液体的电学特性时, 起初大多数学者将液膜视为完全的电介质, 建立了完全导体模型来进行研究. EL-Sayed[10]对水平电场下无黏气体电介质中的液膜进行了不稳定性分析, 得到了韦伯数、气液密度比和电场对不稳定性的影响. 此外, 李广滨等[11]对无黏带电射流进行了时间稳定性分析, 李帅兵等[12]对同轴带电射流进行了稳定性分析, Yang等[13]对电场下静止气体电介质内的带电黏弹性液膜进行了研究, 证明电欧拉数会加速液膜的破裂. Taylor等[14]研究表明将液膜视为完全导体是不严谨的, 因为液膜中含有少量的自由电荷. Melcher等[15]和Taylor[16]在对液滴和液体薄层在交变电场和直流电场下的流动分析中总结得到了漏电介质模型, 随后, Fish[17]及Saville[18]对此模型进行了实验验证. Montanero等[19]也总结得到了漏电介质模型在非完全导体情况下的适用性, 且为锥射流模式下的电喷雾行为提供了准确的预测[20-21]. 漏电介质模型也被应用在电纺丝领域[22-23]以及交流电场下的电喷雾现象[24-25]的研究中. 电动力下的不稳定性问题是十分经典的科学问题, 而且由电场驱动的流动在微流体流动中应用广泛, 许多学者对此进行了研究. 在液体射流中, 施加交流电场被发现可以用于对液体行为进行控制[26-27]. 而由于交流电场的引入, 交流电场的频率同系统的固有频率相互作用, 将产生参数振荡[28]. 但对平面液膜在交流电场下的不稳定性分析研究较少, 而且随着电场的引入, 在液膜中势必会存在着自由电荷和电场切应力并与液膜的黏性力相平衡. 因此, 对交流电场作用下的液膜不稳定性分析进行研究意义重大.

因此本文基于漏电介质模型对平面液膜在交流电场下的线性不稳定性分析进行了研究, 在第一节通过控制方程和边界条件描述了理论模型, 并通过线性分析[29]及Floquet理论[30-35]建立了直流电场及交流电场下的色散方程, 在第二节基于色散方程讨论了电欧拉数、交流电场占比及频率等参数对液膜不稳定性的影响, 并在第三节对文中发现进行了总结. 本工作采用线性分析和Floquet理论, 推导了平面液膜在交流电场下的色散方程并研究了各参数对液膜不稳定性的影响, 以期为交流电场下的平面液膜不稳定性分析和实验提供理论基础和参考数据.

1 理论推导

1.1 模型建立

图1为本文采用的物理模型, 在横向的交流电场中, 黏性液膜在无黏不可压气体中流动. 液体密度为$\rho_{\rm l}$, 气体密度为$\rho_{\rm g}$, 液体表面张力为$\gamma $, 液膜厚度为$2a$, 主流速度为$U$. 在对液膜的电学特性分析时采用漏电介质模型, 液膜电导率为$\sigma$, 液体介电常数为$\varepsilon_{1}$, 气体介电常数为$\varepsilon_{2}$. 电极板至液膜表面距离为$d$, 施加电压为交流电压$V_{t} =V_{0} \left\{ 1+Pr\left[ \cos \left( \varOmega t \right)-1 \right] \right\}$, 其中$V_{0} $为电压幅值, $Pr$为衡量交流电场部分在整个电场中所占比例的参数, 且在0$\sim$1范围内变动, $\varOmega$为交流电场的振荡频率. 在本文的不稳定性分析中忽略重力、磁场力以及温度的影响.

图1

图1   交流电场中的运动液膜

Fig.1   Schematic diagram of moving liquid sheet in an AC electric field


1.2 直流电场作用下不稳定性分析

1.2.1 液相

液相的控制方程为

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\ \ -a+\eta <y<a+\eta$
$\begin{array}{r}\rho\left[\frac{\partial u}{\partial t}+(U+u) \frac{\partial u}{\partial x}+v \frac{\partial v}{\partial y}\right]=-\frac{\partial p}{\partial x}+ \\\mu\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}\right),-a+\eta<y<a+\eta\end{array}$
$\begin{array}{r}\rho\left[\frac{\partial v}{\partial t}+(U+u) \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}\right]=-\frac{\partial p}{\partial y}+ \\\mu\left(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right),-a+\eta<y<a+\eta\end{array}$

运动边界条件为

$v=\frac{\partial \eta }{\partial t}+\left( {U+u} \right)\frac{\partial \eta }{\partial x}+v\frac{\partial \eta }{\partial y},\ \ y=\pm a+\eta$

因存在电场切应力, 动力边界条件为

$\mu \left( {\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}} \right)=T_{\rm t1}^{\rm e} ,\ \ y=a+\eta$
$\mu \left( {\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}} \right)=T_{\rm t2}^{\rm e} ,\ \ y=-a+\eta$

1.2.2 气相

气体无黏无旋, 存在势函数$\phi_{\rm g}$. 与液相相似, 有控制方程

$\frac{\partial^{2}\phi_{\rm g} }{\partial x^{2}}+\frac{\partial^{2}\phi_{\rm g} }{\partial y^{2}}=0,\ \ y<-a+\eta\ \ {\rm or}\ \ y>a+\eta$
$\begin{aligned}p_{\mathrm{g}}=&-\rho_{\mathrm{g}}\left\{\frac{\partial \phi_{\mathrm{g}}}{\partial t}+\frac{1}{2}\left[\left(\frac{\partial \phi_{\mathrm{g}}}{\partial x}\right)^{2}+\frac{1}{2}\left(\frac{\partial \phi_{\mathrm{g}}}{\partial y}\right)^{2}\right]\right\} \\& y\langle-a+\eta \text { or } y\rangle a+\eta\end{aligned}$

气相边界条件为

$v_{\rm g} =\frac{\partial \eta }{\partial t}+u_{\rm g} \frac{\partial \eta }{\partial x}+v_{\rm g} \frac{\partial \eta }{\partial y},\ \ y=a\pm \eta$

1.2.3 电场相

引入电场势函数$V$, 存在

$\nabla^{2}V_{\rm g} =0$
$\nabla^{2}V_{\rm l} =0$

电场强度为

$E=-\nabla V$

电场的边界条件为

$V_{\rm g} =V_{0} ,\ \ y=a+\eta$
$V_{\rm g} =0,\ \ y=a+d$

综合电场力、压力、表面张力、黏性的作用, 有边界条件

$-p+2\mu \frac{\partial v}{\partial y}+\gamma \frac{1}{R}-p_{\rm g} -T_{\rm n1}^{\rm e} =0,\ \ y=a+\eta$
$-p+2\mu \frac{\partial v}{\partial y}-\gamma \frac{1}{R}-p_{\rm g} -T_{\rm n2}^{\rm e} =0,\ \ y=-a+\eta$

其中

$\frac{1}{R}=\dfrac{\dfrac{\partial^{2}\eta }{\partial x^{2}}}{\left( {1+\dfrac{\partial^{2}\eta }{\partial x^{2}}} \right)^{{2}/{3}}}$

1.2.4 线性分析

将变量表示为波动形式

$\begin{array}{c}\left(u, v, p, \phi_{\mathrm{g}}, p_{\mathrm{g}}, V_{\mathrm{l}}, V_{\mathrm{g}}, \eta\right)=[u(y), v(y), p(y) \\\left.\phi_{\mathrm{g}}(y), p_{\mathrm{g}}(y), V_{1}(y), V_{\mathrm{g}}(y), \eta_{0}\right] \mathrm{e}^{\mathrm{i} k x+\omega t}\end{array}$

其中, $k$为波数, $\omega $为表现不稳定增长的频率, $\eta $为扰动的幅值.

将式(18)代入式(2)和式(3)有

$p\left( y \right)=A_{1}{\rm e}^{ky}+A_{2}{\rm e}^{-ky}$

其中$A_{1} ,A_{2} $为常数, 同理电势函数可表示为

$V_{\rm l} \left( y \right)=P_{1} {\rm e}^{ky}+P_{2} {\rm e}^{-ky}$
$V_{\rm g} \left( y \right)=P_{3} {\rm e}^{ky}+P_{4} {\rm e}^{-ky}$

图2给出了对平面液膜所施加的电场力坐标图, 电场强度可表示为

$E_{\rm t} =E_{x}\cos\theta -E_{y}\sin\theta$
$E_{\rm n} =E_{x}\sin\theta +E_{y}\cos\theta$

图2

图2   电场力坐标图其中

Fig.2   Schematic diagram of electric force vector


其中

$\sin\theta =\frac{\partial \eta }{\partial x}\cos\theta =1$

将式(20)$\sim$式(24)结合式(12)、式(17), 代入麦克斯韦电学公式可算得电场力

$\begin{aligned}T_{\mathrm{t} 1}^{\mathrm{e}}=&\left[\varepsilon_{2} \mathrm{i} k \frac{V_{0}^{2}}{d^{2}} \eta_{0}+\varepsilon_{2} \mathrm{i} k \cdot \frac{V_{0}}{d} P_{3}\left(\mathrm{e}^{k a}+\mathrm{e}^{-k a}\right)\right] \\\mathrm{e}^{\mathrm{i} k x+\omega t}, \quad y=a\end{aligned}$
$\begin{aligned}T_{\mathrm{t} 2}^{\mathrm{e}}=&\left[-\varepsilon_{2} \mathrm{i} k \frac{V_{0}^{2}}{d^{2}} \eta_{0}+\varepsilon_{2} \mathrm{i} k \cdot \frac{V_{0}}{d} P_{3}\left(\mathrm{e}^{k a}+\mathrm{e}^{-k a}\right)\right] \\\mathrm{e}^{\mathrm{i} k x+\omega t}, \quad y=-a\end{aligned}$
$\begin{aligned}T_{\mathrm{n} 1}^{\mathrm{e}} &=\left[\varepsilon_{2} \frac{V_{0}^{2}}{d^{3}} \eta_{0}-\varepsilon_{1} k \frac{V_{0}}{d} P_{1}\left(\mathrm{e}^{k a}-\mathrm{e}^{-k a}\right)\right] \\& \mathrm{e}^{\mathrm{i} k x+\omega t}, \quad y=a\end{aligned}$

将式(25)、式(26)结合式(19), 代入液相边界条件(4)$\sim$(6)可得

$\begin{aligned}u(y) &=B_{1} \mathrm{e}^{l y}+B_{2} \mathrm{e}^{-l y}+\\& \frac{\mathrm{i} k}{\mu\left(k^{2}-l^{2}\right)}\left(A_{1} \mathrm{e}^{k y}+A_{2} \mathrm{e}^{-k y}\right)\end{aligned}$
$\begin{aligned}v(y) &=-\frac{\mathrm{i} k}{l} B_{1} \mathrm{e}^{l y}+\frac{\mathrm{i} k}{l} B_{2} \mathrm{e}^{-l y}+\\& \frac{k}{\mu\left(k^{2}-l^{2}\right)}\left(A_{1} \mathrm{e}^{k y}-A_{2} \mathrm{e}^{-k y}\right)\end{aligned}$

其中

$l^{2}=k^{2}+\frac{\rho}{\mu}(\omega+U \mathrm{i} k)$
$\begin{aligned}A_{1}=&-\frac{\mu}{k}\left(k^{2}+l^{2}\right)(\omega+U \mathrm{i} k) \eta_{0} \frac{1}{2 \cosh (k a)}+\\& \varepsilon_{2} k \frac{V_{0}^{2}}{d^{2}} \eta_{0} \frac{1}{2 \sinh (k a)}+\varepsilon_{2} k \frac{V_{0}}{d} P_{3} \operatorname{coth}(k a)\end{aligned}$
$\begin{aligned}A_{2}=& \frac{\mu}{k}\left(k^{2}+l^{2}\right)(\omega+U \mathrm{i} k) \eta_{0} \frac{1}{2 \cosh (k a)}+\\& \varepsilon_{2} k \frac{V_{0}^{2}}{d^{2}} \eta_{0} \frac{1}{2 \sinh (k a)}+\varepsilon_{2} k \frac{V_{0}}{d} P_{3} \operatorname{coth}(\end{aligned}$
$\begin{aligned}B_{1}=& \frac{\mathrm{i} k l}{k^{2}-l^{2}}(\omega+U \mathrm{i} k) \eta_{0} \frac{1}{\cosh (l a)}-\\& \frac{\mathrm{i} k l \eta_{0} \varepsilon_{2}}{\mu\left(k^{2}-l^{2}\right)} \frac{V_{0}^{2}}{d^{2}} \eta_{0} \frac{1}{2 \sinh (l a)}-\\& \frac{1}{\mu} \varepsilon_{2} \frac{\mathrm{i} k V_{0}}{d\left(k^{2}-l^{2}\right)} P_{3} \frac{\operatorname{coth}(k a)}{\sinh (l a)}\end{aligned}$
$\begin{aligned}B_{2}=&-\frac{\mathrm{i} k l}{k^{2}-l^{2}}(\omega+U \mathrm{i} k) \eta_{0} \frac{1}{\cosh (l a)}-\\& \frac{\mathrm{i} k \ln _{0} \varepsilon_{2}}{\mu\left(k^{2}-l^{2}\right)} \frac{V_{0}^{2}}{d^{2}} \eta_{0} \frac{1}{2 \sinh (l a)}-\\& \frac{1}{\mu} \varepsilon_{2} \frac{\mathrm{i} k V_{0}}{d\left(k^{2}-l^{2}\right)} P_{3} \frac{\operatorname{coth}(k a)}{\sinh (l a)}\end{aligned}$

求解气相方程(7)$\sim$(9), 可得

$\phi_{\rm g1} =-\frac{1}{k}\omega {\rm e}^{ka-ky}{\rm e}^{{\rm i}kx+\omega t}\eta_{0}$
$p_{\rm g1} =\frac{1}{k}\rho_{\rm g} \omega^{2}{\rm e}^{ka-ky}{\rm e}^{{\rm i}kx+\omega t}\eta_{0}$
$\phi_{\rm g2} =\frac{1}{k}\omega {\rm e}^{-ka+ky}{\rm e}^{{\rm i}kx+\omega t}\eta_{0}$
$p_{\rm g1} =-\frac{1}{k}\rho_{\rm g} \omega^{2}{\rm e}^{-ka+ky}{\rm e}^{{\rm i}kx+\omega t}\eta_{0}$

最后, 结合气相液相的运动边界条件和电场界面应力平衡边界条件, 由式(15)、式(25)、式(27), 弯曲模式下的色散方程可由$3\times 3$矩阵表示

$\begin{eqnarray} &&{\rm det}(A)=\\&&\quad\left|\begin{array}{c@{\ \ }c@{\ \ }c} \sigma-\varepsilon_{1}(\omega+U{\rm i}k) & \varepsilon_{2}(\omega+U{\rm i}k) & \dfrac{V_{0}}{2 k d^{2}} \sinh (k a) \\ 2 \cosh (k a) & -2 \cosh (k a) & -\dfrac{V_{0}}{d} \\ 0 & D_{32} & D_{33} \end{array}\right|=0\\&& \end{eqnarray}$

其中

$\begin{aligned}D_{32} &=-\frac{k^{2}+l^{2}}{k^{2}-l^{2}} \varepsilon_{2} \frac{k V_{0}}{d} 2 \cosh (k a) \operatorname{coth}(k a)+\\& 4 \varepsilon_{2} \frac{k^{2} l V_{0}}{d\left(k^{2}-l^{2}\right)} \frac{\cosh (k a)}{\tanh (l a)}+\varepsilon_{2} \frac{2 k V_{0}}{d} \sinh (k a)\end{aligned}$

$\begin{aligned}D_{33} &=-\frac{\left(k^{2}+l^{2}\right)^{2}}{k^{2}-l^{2}} \frac{\mu}{k}(\omega+U i k) \tanh (k a)-\\& \frac{k^{2}+l^{2}}{k^{2}-l^{2}} \varepsilon_{2} \frac{k V_{0}^{2}}{d^{2}} \eta_{0} \operatorname{coth}(k a)-\\& 4 \frac{k^{2} l}{k^{2}-l^{2}} \mu(\omega+U i k) \tanh (l a)+\\& 2 \frac{k^{2} l}{k^{2}-l^{2}} \varepsilon_{2} \frac{V_{0}^{2}}{d^{2}} \operatorname{coth}(l a)-\gamma k^{2}-\\& \frac{1}{k} \rho_{\mathrm{g}} \omega^{2}-\varepsilon_{2} \frac{V_{0}^{2}}{d^{3}}\end{aligned}$

为了使结果更具有普适性, 将色散方程无量纲化, 引入无量纲韦伯数$We=\rho_{\rm l}U^{2}a/\gamma $, 雷诺数$Re=\rho_{\rm l} Ua/\mu $, 气液密度比$\bar{\rho}=\rho_{\rm g}/\rho_{\rm l}$, 电欧拉数$Eu=\varepsilon_{1} V_{0}^{2} /\rho_{\rm l} U^{2}a^{2}$, 电松弛时间$\tau =\sigma U/\varepsilon_{1} a$, 介电常数比$\varepsilon =\varepsilon_{2} /\varepsilon_{1} $, 无量纲距离$D=d/a$可得

$\begin{aligned}\operatorname{det} A &=\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \cosh K) D_{33}+\\& \varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\cosh K}{K D \sinh K} \frac{V_{0}}{d} D_{32}+\\&\left(\tau-\frac{L^{2}-K^{2}}{R e}\right) \frac{V_{0}}{d} D_{32}-\\& \varepsilon \frac{L^{2}-K^{2}}{R e} 2 \cosh K D_{33}=0\end{aligned}$

其中

$\begin{array}{c}\frac{V_{0}}{d} D_{32}=-\frac{L^{2}+K^{2}}{K^{2}-L^{2}} 2 \operatorname{coth} K \cosh K \frac{K E u}{D^{2}}+ \\\frac{4 K^{2} L}{K^{2}-L^{2}} \frac{\cosh K}{\tanh L} \frac{E u}{D^{2}}-2 \sinh K \frac{K E u}{D^{2}}\end{array}$

$\begin{aligned}D_{33} &=-\frac{\left(L^{2}+K^{2}\right)^{2}}{K} \frac{1}{R e^{2}} \tanh K+\frac{4 K^{2} L}{R e^{2}} \tanh L-\\& \frac{K^{2}}{W e}-\bar{\rho} \frac{\omega^{2}}{K}-\frac{\left(L^{2}+K^{2}\right)^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \operatorname{coth} K+\\& \frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \operatorname{coth} L+\frac{E u}{D^{3}}\end{aligned}$

在曲张模式中

$\begin{aligned}\operatorname{det} A &=\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \sinh K) D_{33}+\\& \varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\sinh K}{K D \cosh K} \frac{V_{0}}{d} D_{32}+\\&\left(\tau-\frac{L^{2}-K^{2}}{R e}\right) \frac{V_{0}}{d} D_{32}-\\& \varepsilon \frac{L^{2}-K^{2}}{R e} 2 \sinh K D_{33}=0\end{aligned}$

其中

$\frac{V_{0}}{d} D_{32}=-\frac{L^{2}+K^{2}}{K^{2}-L^{2}} 2 \tanh K \sinh K \frac{K E u}{D^{2}}+\frac{4 K^{2} L}{K^{2}-L^{2}} \frac{\sinh K}{\operatorname{coth} L} \frac{E u}{D^{2}}-2 \cosh K \frac{K E u}{D^{2}}$

$\begin{aligned}D_{33} &=-\frac{\left(L^{2}+K^{2}\right)^{2}}{K} \frac{1}{R e^{2}} \operatorname{coth} K+\\& \frac{4 K^{2} L}{R e^{2}} \operatorname{coth} L-\frac{K^{2}}{W e}-\bar{\rho} \frac{\omega^{2}}{K}-\\& \frac{\left(L^{2}+K^{2}\right)^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \tanh K+\\& \frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \tanh L+\frac{E u}{D^{3}}\end{aligned}$

1.3 交流电场作用下不稳定性分析

当交流电场作用时, 运用Floquet理论[30], 将原本的直流电压项$V_{0}$由交流电压$V_{\rm t}=V_{0}$ $\big\{1+Pr\big[\cos \big(\varOmega t \big)-1\big]\big\}$代替, 同时, 频率$\omega$由$\omega+{\rm i}n\varOmega $代替.

进行傅里叶展开时, 有

$\left.\begin{array}{l}\cos \omega t=\frac{1}{2}\left(\mathrm{e}^{\mathrm{i} \Omega t}+\mathrm{e}^{-\mathrm{i} \Omega t}\right) \\\cos ^{2} \omega t=\frac{1}{2}+\frac{1}{4}\left(\mathrm{e}^{2 \mathrm{i} \Omega t}+\mathrm{e}^{-2 \mathrm{i} \Omega t}\right)\end{array}\right\}$

原本直流电场下的色散方程中$V_{0}^{2}$项变为$V_{\rm t}^{2}=V_{0}^{2}\left\{{1+Pr\left[ {\cos \left( {\varOmega t} \right)-1} \right]} \right\}^{2}$其中

$\begin{array}{c}\{1+\operatorname{Pr}[\cos (\Omega t)-1]\}^{2}=(1-P r)^{2}+\frac{1}{2} P r^{2}+ \\\frac{1}{2} \operatorname{Pr}(1-\operatorname{Pr})\left(\mathrm{e}^{\mathrm{i} \Omega t}+\mathrm{e}^{-\mathrm{i} \Omega t}\right)+\frac{1}{4} \operatorname{Pr}^{2}\left(\mathrm{e}^{2 \mathrm{i} \Omega t}+\mathrm{e}^{-2 \mathrm{i} 2 t}\right)\end{array}$

代入直流电场下的色散方程, 可得到以下形式

$D_{n} \hat{\eta}_{n}+x\left(\hat{\eta}_{n+1}+\hat{\eta}_{n-1}\right)+y\left(\hat{\eta}_{n+2}+\hat{\eta}_{n-2}\right)=0$

其中

$\begin{aligned}D_{n}=&\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \cosh K)-\right.\\&\left.\varepsilon \frac{L^{2}-K^{2}}{R e} 2 \cosh K\right]\left\{\frac{4 K^{2} L}{R e^{2}} \tanh L-\frac{K^{2}}{W e}-\right.\\& \bar{\rho} \frac{w+i n \Omega}{K}+\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \operatorname{coth} L+\frac{E u}{D^{3}}-\right.\\&\left.\left.\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \operatorname{coth} K\right) \cdot\left[(1-P r)^{2}+\frac{1}{2} P r^{2}\right]\right\}+\\&\left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\cosh K}{K D \sinh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\&\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \operatorname{coth} K \cosh K \frac{K E u}{D^{2}}+\frac{4 K^{2} L}{K^{2}-L^{2}}\right.\\&\left.\frac{\cosh K}{\tanh L} \frac{E u}{D^{2}}-2 \sinh K \frac{K E u}{D^{2}}\right) \\&\left[(1-P r)^{2}+\frac{1}{2} P r^{2}\right]\end{aligned}$

$\begin{aligned}x=\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \cosh K)-\right.\\ \left.\varepsilon \frac{L^{2}-K^{2}}{R e} 2 \sinh K\right] \cdot\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \operatorname{coth} L+\right.\\ \left.\frac{E u}{D^{3}}-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \cosh K\right) \cdot \frac{1}{2} \operatorname{Pr}(1-P r)+\\ \left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\cosh K}{K D \sinh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \operatorname{coth} K \cosh K \frac{K E u}{D^{2}}+\frac{4 K^{2} L}{K^{2}-L^{2}}\right.\\\left.\frac{\cosh K}{\tanh L} \frac{E u}{D^{2}}-2 \sinh K \frac{K E u}{D^{2}}\right) \cdot \frac{1}{2} \operatorname{Pr}(1-P r)\end{aligned}$

$\begin{aligned}y=&\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \cosh K)-\right.\\&\left.\varepsilon \frac{L^{2}-K^{2}}{\operatorname{Re}} 2 \sinh K\right] \cdot\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \operatorname{coth} L+\right.\\&\left.\frac{E u}{D^{3}}-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \cosh K\right) \frac{1}{4} P r^{2}+\\&\left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\cosh K}{K D \sinh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\&\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \operatorname{coth} K \cosh K \frac{K E u}{D^{2}}+\frac{4 K^{2} L}{K^{2}-L^{2}}\right.\\&\left.\frac{\cosh K}{\tanh L} \frac{E u}{D^{2}}-2 \sinh K \frac{K E u}{D^{2}}\right) \frac{1}{4} P r^{2}\end{aligned}$

在曲张模式中

$\begin{aligned}D_{n}=&\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \sinh K)-\right.\\&\left.\varepsilon \frac{L^{2}-K^{2}}{R e} 2 \sinh K\right]\left\{\frac{4 K^{2} L}{R e^{2}} \operatorname{coth} L-\frac{K^{2}}{W e}-\right.\\& \bar{\rho} \frac{w+i n \Omega}{K}+\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \tanh L+\frac{E u}{D^{3}}-\right.\\&\left.\left.\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \cdot \frac{K E u}{D^{2}} \tanh K\right)\left[(1-P r)^{2}+\frac{1}{2} P r^{2}\right]\right\}+\\&\left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\sinh K}{K D \cosh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\&\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \tanh K \sinh K \frac{K E u}{D^{2}}+\right.\\&\left.\frac{4 K^{2} L}{K^{2}-L^{2}} \frac{\sinh K}{\operatorname{coth} L} \cdot \frac{E u}{D^{2}}-2 \cosh K \frac{K E u}{D^{2}}\right) \\&\left[(1-P r)^{2}+\frac{1}{2} P r^{2}\right]\end{aligned}$

$\begin{aligned}x=&\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \sinh K)-\right.\\&\left.\varepsilon \frac{L^{2}-K^{2}}{R e} 2 \cosh K\right] \cdot\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \tanh L+\right.\\&\left.\frac{E u}{D^{3}}-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \frac{K E u}{D^{2}} \sinh K\right) \cdot \frac{1}{2} \operatorname{Pr}(1-P r)+\\&\left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\sinh K}{K D \cosh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\&\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \tanh K \sinh K \frac{K E u}{D^{2}}+\right.\\&\left.\frac{4 K^{2} L}{K^{2}-L^{2}} \frac{\sinh K}{\operatorname{coth} L} \frac{E u}{D^{2}}-2 \cosh K \frac{K E u}{D^{2}}\right) \\& \frac{1}{2} \operatorname{Pr}(1-P r)\end{aligned}$

$\begin{aligned}y=&\left[\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)(-2 \sinh K)-\right.\\&\left.\varepsilon \frac{L^{2}-K^{2}}{R e} 2 \cosh K\right]\left(\frac{2 K^{2} L}{K^{2}-L^{2}} \frac{E u}{D^{2}} \tanh L+\right.\\&\left.\frac{E u}{D^{3}}-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} \cdot \frac{K E u}{D^{2}} \sinh K\right) \frac{1}{4} P r^{2}+\\&\left[\varepsilon \frac{L^{2}-K^{2}}{R e} \frac{\sinh K}{K D \cosh K}+\left(\tau-\frac{L^{2}-K^{2}}{R e}\right)\right] \\&\left(-\frac{K^{2}+L^{2}}{K^{2}-L^{2}} 2 \tanh K \sinh K \frac{K E u}{D^{2}}+\frac{4 K^{2} L}{K^{2}-L^{2}}\right.\\&\left.\frac{\sinh K}{\operatorname{coth} L} \frac{E u}{D^{2}}-2 \cosh K \frac{K E u}{D^{2}}\right) \frac{1}{4} P r^{2}\end{aligned}$

色散方程可用矩阵表示为

$\left(\begin{array}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c} \ddots & \vdots & \vdots & 0 & 0 & 0 & 0 \\ \ldots & D_{-2} & x_{-1} & y_{0} & 0 & 0 & 0 \\ \ldots & x_{-2} & D_{-1} & x_{0} & y_{1} & 0 & 0 \\ 0 & y_{-2} & x_{-1} & D_{0} & x_{1} & y_{2} & 0 \\ 0 & 0 & y_{-1} & x_{0} & D_{1} & x_{2} & \ldots \\ 0 & 0 & 0 & y_{0} & x_{1} & D_{2} & \ldots \\ 0 & 0 & 0 & 0 & \vdots & \vdots & \ddots \end{array}\right)\left(\begin{array}{c} \hat{\eta}_{-2} \\ \hat{\eta}_{-1} \\ \hat{\eta}_{0} \\ \hat{\eta}_{1} \\ \hat{\eta}_{2} \\ \vdots \end{array}\right)=0$

显然, 系数矩阵行列式为0为方程的解

$\left|\begin{array}{ccccccc}\ddots & \vdots & \vdots & 0 & 0 & 0 & 0 \\\ldots & D_{-2} & x_{-1} & y_{0} & 0 & 0 & 0 \\\ldots & x_{-2} & D_{-1} & x_{0} & y_{1} & 0 & 0 \\0 & y_{-2} & x_{-1} & D_{0} & x_{1} & y_{2} & 0 \\0 & 0 & y_{-1} & x_{0} & D_{1} & x_{2} & \ldots \\0 & 0 & 0 & y_{0} & x_{1} & D_{2} & \ldots \\0 & 0 & 0 & 0 & \vdots & \vdots & \ddots\end{array}\right|=0$

2 结果与分析

在进行分析前还需确定色散矩阵的阶数, 阶数由计算的节点数$n$产生, 从理论上分析, 当$n$趋于无穷大时计算结果最为准确, 但从计算上的速度考虑阶数不能取到无穷, 因此需要选取合适的节点数$n$, 使其兼顾计算精度与快速性.

图3做出了弯曲模式下不同节点数$n$作用时的最大不稳定增长率.

图3

图3   最大不稳定增长率随节点数$n$的变化$(\varepsilon =0.012\,5$, $D=10.45$, $Re=1000$, $We=400$, $Eu=5$, $\bar{\rho}=0.001\,2$, $Pr=2/3$, $\varOmega =1$)

Fig.3   $\omega_{\max}$ versus increasing the node number $(\varepsilon =0.012\,5$, $D=10.45$, $Re=1000$, $We=400$, $Eu=5$, $\bar{\rho}=0.001\,2$, $Pr=2/3$, $\varOmega =1$)}


图3可以看出, 选取节点数$n=5$是合适的, 在保障计算精度的同时提供了计算的快速性, 此时色散矩阵阶数$N=2n+1=11$.

图4做出了弯曲模式和曲张模式下的色散曲线. 如图4所示, 本文发现了在交流电场作用下的曲张模式中, 液膜的色散曲线同弯曲模式下近乎相同, 这与直流电场及无电场作用时有很大区别.

图4

图4   弯曲模式和曲张模式下的色散曲线($\varepsilon =0.012 5$, $D=10.45$, $Re=1000$, $We=400$, $Eu=5$, $\bar{\rho}=0.001 2$, $Pr=2/3$, $\varOmega =1$)

Fig.4   Dispersion relation in sinuous and varicose mode ($\varepsilon =0.012 5$, $D=10.45$, $Re=1000$, $We=400$, $Eu=5$, $\bar{\rho}=0.001 2$, $Pr=2/3$, $\varOmega =1$)


为验证本文的准确性, 取$Pr=0$, 即退化为直流电场作用情况, 同Cui[29]的结果进行了对比, 图5为对比结果, 从图中可以看出色散曲线拟合良好, 可认为本文的理论推导没有错误.

图5

图5   直流情况下色散方程的对比验证 $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=0$, $\varOmega =1)$

Fig.5   Comparison of dispersion relation in DC case $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=0$, $\varOmega =1)$


图6做出了对交流电场下液膜弯曲模式下的色散曲线. 由图6可见, 此时在交流电场作用下, 发生了参数振荡并出现了多个不稳定区域, 其中小波数下的不稳定区域为毛细不稳定区域, 而随其后出现的不稳定区域为参数不稳定区域, 是由交流电场的振荡所引起的.

图6

图6   色散曲线$(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=0.4$, $\varOmega =1)$

Fig.6   Dispersion relation $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=0.4$, $\varOmega =1)$


在探究各参数对横向电场下平面液膜的线性不稳定影响的过程中, 常规的气液密度比($\rho )$, 韦伯数(We), 雷诺数(Re)与松弛时间($\tau)$等参数对不稳定性影响已在Cui等[29]对平面液膜的研究中考虑过. 因此, 本文主要考虑电学参数对不稳定性的影响.

图7表示了弯曲模式下交流电场占比$Pr$对液膜不稳定性的影响. 由图7可以看出交流电场占比$Pr$逐渐增大时, 弯曲模式下毛细不稳定区域和参数不稳定区域内主导波数和其对应的最大不稳定增长率的变化情况. 随着$Pr$的增大, 两个不稳区域的主导波数和最大不稳定增长率总体上都是一个先下降后上升的过程. 其中当$Pr$较小时难以观测到明显的参数不稳定区域, 从$Pr=1/3$开始可以观测到较为明显的参数不稳定区域.

图7

图7   弯曲模式下交流电场占比$Pr$的影响$(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $\varOmega =1)$

Fig.7   Effects of $Pr$ in sinuous mode $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $\varOmega =1)$


图8为随着交流电场占比$Pr$增大时, 电场作用力中恒定电场力部分占比的变化趋势. 结合图7图8可得, 交流电场作用下的毛细不稳定性主导波数和最大不稳定增长率的变化趋势同恒定电场项变化趋势基本一致, 而在$Pr=2/3$时, 参数不稳定区域的主导波数相较$Pr=1/2$时有显著的提升, 甚至超过了$Pr=5/6$时的主导波数, 此时是因为参数不稳定区域的最大增长率发生在参数不稳定区域的位置出现了变化, 此前最大不稳定增长率出现在参数不稳定区域的第一个峰, 而从$Pr=2/3$之后最大不稳定增长率出现在参数不稳定区域的第二个峰. 实际上, 同一个不稳定区域主导波数和最大不稳定增长率的变化趋势同恒定电场项的变化趋势是一致的. 此时, 不稳定主要是由恒定电场力和表面张力的对抗所导致的.

图8

图8   恒定电场项$(1-Pr)^{2}+Pr^{2}/2$作用趋势

Fig.8   Behavior of the DC term $(1-Pr)^{2}+Pr^{2}/2$}


图9表示了在弯曲模式下交流电场频率$\varOmega $对液膜不稳定性的影响. 从图9可以看出, 参数$\varOmega$的增大略微增大了毛细不稳定区域的主导波数和最大不稳定增长率, 但会造成参数不稳定区域的最大不稳定增长率急剧下降, 在$\varOmega=3$之后难以观测到参数不稳定区域. 可见参数$\varOmega$的增大略微增大毛细不稳定性, 但急剧地减小了参数不稳定性, 其对参数不稳定的影响要远远大于毛细不稳定. 在实验中, 为了更容易产生参数不稳定, 因尽量将交流电场频率维持在较低水平.

图9

图9   弯曲模式下交流电场频率$\varOmega $的影响 $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=2/3)$

Fig.9   Effects of $\varOmega $ in sinuous mode $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $Eu=2$, $\bar{\rho}=0.001\,2$, $Pr=2/3)$


图10表示了在弯曲模式下电欧拉数$Eu$对液膜不稳定性的影响. 由图10可见无论是在毛细不稳定区域还是参数不稳定区域, $Eu$的增大都会增大液膜的不稳定性. 在实验中, 为了更容易产生参数不稳定现象, 因尽量加大电场强度.

图10

图10   弯曲模式下欧拉数$Eu$的影响 $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $\bar{\rho}=0.001\,2$, $Pr=0.4$, $\varOmega =1)$

Fig.10   Effects of $Eu$ in sinuous mode $(\varepsilon =0.012\,5$, $D=40$, $Re=1000$, $We=400$, $\bar{\rho}=0.001\,2$, $Pr=0.4$, $\varOmega =1)$


3 结论

本文通过分析不同电学参数对横向交流电场作用下的平面液膜的参数不稳定性的影响, 得出了以下几点结论:

(1) 当交流电场占比$Pr$改变时, 对液膜不稳定性的影响是由于恒定电场比例发生变化所引起的, 此时毛细不稳定性的主导波数和最大不稳定增长率同恒定电场项比例变化趋势相一致, 此时不稳定体现为恒定电场力和表面张力的对抗.

(2) 当交流电场频率$\varOmega $改变时, 交流电场频率的增大急剧地减小液膜的参数不稳定性而略微增大毛细不稳定性.

(3) 当电欧拉数$Eu$发生改变时, 欧拉数的增大同时增大液膜的毛细不稳定性以及参数不稳定性.

(4) 为了在实验中寻求参数振荡现象, 增大电欧拉数$Eu$及减小交流电场频率$\varOmega$是有效的方法.

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