毛细管出口表面活化剂包覆液滴形成过程研究 1)

图1 表面活化剂包覆液柱的几何示意图

1.1 控制方程

假设流动为轴对称流动,沿毛细管道轴向建立柱坐标系,如图1所示,其中$z^\ast$轴正方向竖直向下。重力$g^\ast $沿坐标轴$z^\ast $正方向。在$t^\ast$时刻,液滴自由面位置记作$r^\ast =h^\ast \left( {z^\ast ,t^\ast }\right)$。液滴下落运动的质量和动量守恒方程如下

(1) $\begin{eqnarray}\frac{1}{r^\ast }\frac{\partial \left( {ru_r^\ast } \right)}{\partial r^\ast }+\frac{\partial u_z^\ast }{\partial z^\ast }=0\end{eqnarray}$

(2) $\begin{eqnarray}\rho ^\ast \left( {\frac{\partial u_r^\ast }{\partial t^\ast }+u_r^\ast \frac{\partial u_r^\ast }{\partial r^\ast }+u_z^\ast \frac{\partial u_r^\ast }{\partial z^\ast }} \right)=-\frac{\partial p^\ast }{\partial r^\ast }+ \mu ^\ast \left( {\Delta ^\ast u_r^\ast -\frac{u_r^\ast }{r^{\ast 2}}} \right) \end{eqnarray}$

(3) $\begin{eqnarray}\rho ^\ast \left( {\frac{\partial u_z^\ast }{\partial t^\ast }+u_r^\ast \frac{\partial u_z^\ast }{\partial r^\ast }+u_z^\ast \frac{\partial u_z^\ast }{\partial z^\ast }} \right)=-\frac{\partial p^\ast }{\partial z^\ast }+ \mu ^\ast \Delta ^\ast u_z^\ast +\rho ^\ast g^\ast \end{eqnarray}$

其中

(4) $\begin{eqnarray} \label{eq4} \Delta ^\ast =\frac{\partial ^2}{\partial r^{\ast 2}}+\frac{1}{r^\ast }\frac{\partial }{\partial r^\ast }+\frac{\partial ^2}{\partial z^{\ast 2}} \end{eqnarray}$

在液滴自由面$r^\ast =h^\ast \left( {z^\ast ,t^\ast }\right)$处,根据光滑流体自由面的保持性,有运动学边界条件

(5) $\begin{eqnarray} \label{eq5} u_r^\ast =h_t^\ast +u_z^\ast h_z^\ast \end{eqnarray}$

式中,$h_t^\ast =\partial h^\ast /\partial t^\ast $,$h_z^\ast =\partial h^\ast /\partial z^\ast $。与此同时,在液滴自由面上的法向和切向应力平衡条件为

(6) $\begin{eqnarray} \label{eq6} &&p^\ast -\frac{2\mu ^\ast }{1+h_z^{\ast 2} }\left[ \frac{\partial u_r^\ast }{\partial r^\ast }+h_z^{\ast 2} \frac{\partial u_z^\ast }{\partial z^\ast }- \right. \\&&\qquad \left.h_z^\ast \left( {\frac{\partial u_r^\ast }{\partial z^\ast }+\frac{\partial u_z^\ast }{\partial r^\ast }} \right) \right]=\gamma ^\ast \kappa ^\ast \end{eqnarray}$

(7) $\begin{eqnarray} \frac{\mu ^\ast }{1+h_z^{\ast 2} }\left[ 2h_z^\ast \left( \frac{\partial u_r^\ast }{\partial r^\ast }-\frac{\partial u_z^\ast }{\partial z^\ast } \right)+\left( {1-h_z^{\ast 2} } \right)\cdot \right.\left.\left( {\frac{\partial u_r^\ast }{\partial z^\ast }+\frac{\partial u_z^\ast }{\partial r^\ast }} \right) \right]=\frac{1}{\sqrt {1+h_z^{\ast 2} } }\frac{\partial \gamma ^\ast }{\partial z^\ast }\mbox{ } \end{eqnarray}$

式中,$\gamma ^\ast $为表面张力系数,$\kappa ^\ast $为液滴自由面上的局部曲率,其具体表达式为

(8) $\begin{eqnarray} \label{eq8} \kappa ^\ast =\frac{1}{h^\ast \left( {1+h_z^{\ast 2} } \right)^{1/2}}-\frac{h_{zz}^\ast }{\left( {1+h_z^{\ast 2} } \right)^{3/2}} \end{eqnarray}$

式中,$h_{zz}^\ast =\partial ^2h^\ast /\partial z^{\ast 2}$。由于液滴自由面上有表面活化剂存在,其浓度变化会引起液滴自由面上局部表面张力系数的变化, 两者之间的关系可以由线性Szyszkowski方程^[29]给出

(9) $\begin{eqnarray} \label{eq9} \gamma ^\ast =\gamma _0^\ast +\varGamma _{\rm m}^\ast R_{\rm g}^\ast T^\ast \ln \left( {1-\varGamma ^\ast /\varGamma _{\rm m}^\ast } \right) \end{eqnarray}$

式中,$\gamma _0^\ast $为不含表面活化剂时液滴自由面表面张力系数,$\varGamma^\ast $为液滴自由面上局部表面活化剂浓度,$\varGamma _{\rm m}^\ast $为表面活化剂饱和吸附时的表面过剩浓度,$R_{\rm g}^\ast $为气体常数,{$T^\ast$}为活化剂的温度。在本问题中,假定液滴是在恒温环境下自由下落,即$T^\ast$为给定常数。

由于表面活化剂的不溶性,其只能在液滴自由面上进行输运。在轴对称假设下, 表面活化剂在液滴自由面上的对流和扩散方程^[30]为

(10) $\begin{eqnarray} \label{eq10} &&\frac{\partial }{\partial t^\ast }\left( {\varGamma ^\ast h^\ast \sqrt {1+h_z^{\ast 2} } } \right)+\frac{\partial }{\partial z^\ast }\left( {u_z^\ast \varGamma ^\ast h^\ast \sqrt {1+h_z^\ast } } \right)=\\&&\qquad D_{\rm s}^\ast \frac{\partial }{\partial z^\ast }\left( {\frac{h^\ast }{\sqrt {1+h_z^\ast } }\frac{\partial \varGamma ^\ast }{\partial z^\ast }} \right) \end{eqnarray}$

式中,$D_{\rm s}^\ast $为表面活化剂沿液滴自由面的浓度扩散系数。至此,方程式(1)$\sim$式(10)构成了本问题的封闭方程组,对其直接进行求解较为困难。接下来我们将采用泰勒展开和润滑近似方法, 对上述方程进行化简。

1.2 润滑近似

根据润滑近似理论^[20],并参考Eggers等^[18]的做法,这里将液滴下落的轴向速度$u_z^\ast \left( {r^\ast ,z^\ast ,t^\ast } \right)$和压力$p^\ast \left( {r^\ast ,z^\ast ,t^\ast } \right)$沿液滴半径$r^\ast $做泰勒展开,可得

(11) $\begin{eqnarray} \label{eq11} u_z^\ast \left( {r^\ast ,z^\ast ,t^\ast } \right)=v_0^\ast \left( {z^\ast ,t^\ast } \right)+v_2^\ast \left( {z^\ast ,t^\ast } \right)r^{\ast 2}+\cdot \cdot \cdot \end{eqnarray}$

(12) $\begin{eqnarray} p^\ast \left( {r^\ast ,z^\ast ,t^\ast } \right)=p_0^\ast \left( {z^\ast ,t^\ast } \right)+p_2^\ast \left( {z^\ast ,t^\ast } \right)r^{\ast 2}+\cdot \cdot \cdot \end{eqnarray}$

由于假定流动轴对称,因此式(11)和式(12)中关于$r^\ast $的一次项均为零。将式(11)代入质量守恒方程(1)中,化简后可以得到

(13) $\begin{eqnarray} \label{eq13} u_r^\ast \left( {r^\ast ,z^\ast ,t^\ast } \right)=-\frac{1}{2}\frac{\partial v_0^\ast }{\partial z^\ast }r^\ast -\frac{1}{4}\frac{\partial v_2^\ast }{\partial z^\ast }r^{\ast 3}-\cdot \cdot \cdot \end{eqnarray}$

将式(12)和式(13)代入沿液滴半径$r^\ast $方向的动量守恒方程(2)中,化简并保留至$O\left( {r^\ast } \right)$阶精度,可以得到$\partial p^\ast /\partial r^\ast =0$。因此,液滴沿半径方向压强均匀分布,此即润滑近似。类似的,将式(12)和式(13)代入沿毛细管道$z^\ast $轴方向的动量守恒方程(3)中并忽略高阶小量,可得

(14) $\begin{eqnarray} \label{eq14} &&\frac{\partial v_0^\ast }{\partial t^\ast }+v_0^\ast \frac{\partial v_0^\ast }{\partial z^\ast }=-\frac{1}{\rho ^\ast }\frac{\partial p_0^\ast }{\partial z^\ast }+\\&&\qquad\frac{\mu ^\ast }{\rho ^\ast }\left( {4v_2^\ast +\frac{\partial ^2v_0^\ast }{\partial z^{\ast 2}}} \right)+g^\ast \end{eqnarray}$

同理,将式(11)$\sim$式(13)代入液滴自由面的法向和切向应力平衡条件方程式(6)和式(7)中,忽略高阶小量,可得

(15) $\begin{eqnarray} \label{eq15} &&p_0^\ast +\mu ^\ast \frac{\partial v_0^\ast }{\partial z^\ast }=\gamma ^\ast \kappa ^\ast \end{eqnarray}$

(16) $\begin{eqnarray} -\mu ^\ast \left( {3\frac{\partial v_0^\ast }{\partial z^\ast }\frac{\partial h^\ast }{\partial z^\ast }+\frac{1}{2}\frac{\partial ^2v_0^\ast }{\partial z^{\ast 2}}h^\ast -2v_2^\ast h^\ast } \right)=\frac{\partial \gamma ^\ast }{\partial z^\ast }\qquad \end{eqnarray}$

将方程式(15)和式(16)代入方程式(14)中,消去$p_0^\ast $和$v_2^\ast $,可得

(17) $\begin{eqnarray} \label{eq17} &&\frac{\partial v_0^\ast }{\partial t^\ast }+v_0^\ast \frac{\partial v_0^\ast }{\partial z^\ast }=-\frac{1}{\rho ^\ast }\frac{\partial \left( {\gamma ^\ast \kappa ^\ast } \right)}{\partial z^\ast }+\frac{2}{\rho ^\ast h^\ast }\frac{\partial \gamma ^\ast }{\partial z^\ast }+\\&&\qquad3\frac{\mu ^\ast }{\rho ^\ast }\frac{1}{h^{\ast 2}}\frac{\partial }{\partial z^\ast }\left( {h^{\ast 2}\frac{\partial v_0^\ast }{\partial z^\ast }} \right)+g^\ast \end{eqnarray}$

将式(11)$\sim$式(13)代入液滴自由面的运动学边界条件方程(5)中,忽略高阶小量可得

(18) $\begin{eqnarray} \label{eq18} \frac{\partial h^\ast }{\partial t^\ast }=-v_0^\ast \frac{\partial h^\ast }{\partial z^\ast }-\frac{1}{2}\frac{\partial v_0^\ast }{\partial z^\ast }h^\ast \end{eqnarray}$

同样,对表面活化剂的对流和扩散输运方程式(10)亦做如上处理,可得

(19) $\begin{eqnarray} \label{eq19} \frac{\partial }{\partial t^\ast }\left( {\varGamma ^\ast h^\ast } \right)+\frac{\partial }{\partial z^\ast }\left( {v_0^\ast \varGamma ^\ast h^\ast } \right)=D_s^\ast \frac{\partial }{\partial z^\ast }\left( {h^\ast \frac{\partial \varGamma ^\ast }{\partial z^\ast }} \right)\ \ \end{eqnarray}$

至此,我们将重力作用下,毛细管出口表面活化剂包覆轴对称液滴的自由下落问题简化为由方程式(17)$\sim$式(19)描述的一维动力学系统。该系统的待求变量为液滴对称轴线上的速度$v_0^\ast \left( {z^\ast ,t^\ast } \right)$,液滴自由面位置$h^\ast \left( {z^\ast ,t^\ast } \right)$和表面活化剂浓度$\varGamma ^\ast \left( {z^\ast ,t^\ast } \right)$。方程(17)中液滴自由面的局部平均曲率$\kappa ^\ast $和表面张力系数$\gamma ^\ast $分别可由式(8)和式(9)给出。这里需要指出,本文中式(8)为液滴自由面曲率的精确表达式, 它能更好地描述自由面的非线性演化行为。

1.3 无量纲化

接下来,对方程(17)$\sim$方程(19)组成的动力学系统进行无量纲化。选取毛细管道的内半径$R^\ast $为特征长度,$\left( {\rho ^\ast R^{\ast 3}/\gamma _0^\ast } \right)^{1/2}$和$\gamma _0^\ast /R^\ast $分别为特征时间和特征压强,以$\varGamma _{\rm m}^\ast $和$\gamma _0^\ast $分别作为表面活化剂特征浓度和特征表面张力系数,得到方程式(17)和式(18)的无量纲形式

(20) $\begin{eqnarray} \label{eq20} &&\frac{\partial h}{\partial t}+v_0 \frac{\partial h}{\partial z}=-\frac{1}{2}\frac{\partial v_0 }{\partial z}h \end{eqnarray}$

(21) $\begin{eqnarray} \frac{\partial v_0 }{\partial t}+v_0 \frac{\partial v_0 }{\partial z}=\varphi +\psi + 3Oh\frac{1}{h^2}\frac{\partial }{\partial z}\left( {h^2\frac{\partial v_0 }{\partial z}} \right)+Bo \end{eqnarray}$

其中的无量纲参数$Oh={\mu ^\ast } / {\left( {\rho ^\ast R^\ast \gamma _0^\ast } \right)^{1/2}}$为Ohnesorge数,代表了黏性应力$F_\mu $、惯性力$F_{\rm i} $和表面张力$F_{\rm s} $这三种力的量级比例关系：$F_\mu /\sqrt {F_{\rm i} F_{\rm s} } $;$Bo={\rho ^\ast g^\ast R^{\ast 2}} / {\gamma _0^\ast }$为Bond数,代表了流场中重力和表面张力的量级之比。方程(21)中的参数$\varphi $代表了表面张力引起的附加压强随液滴自由面曲率变化($z$轴方向变化)而引起的压强梯度效应;参数$\psi $反映了液滴自由面上,活化剂浓度梯度引起的Marangoni效应。$\varphi $和$\psi $的具体表达式为

(22) $\begin{eqnarray} \label{eq22} \varphi =-\frac{\partial (\gamma \kappa )}{\partial z},\ \ \psi =\frac{2}{h}\frac{\partial \gamma }{\partial z} \end{eqnarray}$

根据式(8)和式(9),可知液滴自由面曲率和无量纲表面张力系数的表达式分别为

(23) $\begin{eqnarray} \kappa =\frac{1}{h}\frac{1}{\left( {1+h_z^2 } \right)^{1/2}}-\frac{h_{zz} }{\left( {1+h_z^2 } \right)^{3/2}} \end{eqnarray}$

(24) $\begin{eqnarray} \gamma =1+\beta \ln (1-\varGamma ) \end{eqnarray}$

式中,$\beta ={\varGamma _{\rm m}^\ast R_{\rm g}^\ast T^\ast } / {\gamma _0^\ast }$是表面活化剂的活性常数。$\varGamma $为液滴自由面无量纲表面活化剂浓度,相应的表面活化剂输运方程(19)的无量纲形式为

(25) $\begin{eqnarray} \label{eq24} &&\frac{\partial \varGamma }{\partial t}+v_0 \frac{\partial \varGamma }{\partial z}=-\frac{1}{2}\frac{\partial v_0 }{\partial z}\varGamma +\\&&\qquad\frac{1}{Pe}\left( {\frac{\partial ^2\varGamma }{\partial z^2}+\frac{1}{h}\frac{\partial h}{\partial z}\frac{\partial \varGamma }{\partial z}} \right) \end{eqnarray}$

式中,$Pe=v^\ast R^\ast /D_s^\ast =R^{\ast2}/\left( {D_s^\ast T^\ast }\right)$是Peclet数, 代表了表面活化剂的对流和扩散速率的量级之比。其中,$v^\ast=R^\ast /T^\ast $是特征速度,$T^\ast $是特征时间$\left( {\rho ^\ast R^{\ast3}/\gamma _0^\ast } \right)^{1/2}$。从式(20)$\sim$式(21)和式(24)中, 可得$v_0$,$h$和$\varGamma$沿轴向坐标$z$的分布及其随时间$t$的演化过程。进一步,根据如下方程可以得到液滴内压强以及速度分布

(26) $\begin{eqnarray} &&p=\gamma \kappa -Oh\frac{\partial v_0 }{\partial z} \end{eqnarray}$

(27) $\begin{eqnarray} u_r =-\frac{1}{2}\frac{\partial v_0 }{\partial z}r-\frac{1}{4}\frac{\partial v_2 }{\partial z}r^3,\ \ u_z =v_0 +v_2 r^2\quad \end{eqnarray}$

其中

(28) $\begin{eqnarray} \label{eq27} v_2 =\frac{1}{2h}\left( {3\frac{\partial h}{\partial z}\frac{\partial v_0 }{\partial z}+\frac{1}{Oh}\frac{\partial \gamma }{\partial z}} \right)+\frac{1}{4}\frac{\partial ^2v_0 }{\partial z^2} \end{eqnarray}$

至此,通过求解式(20)$\sim$式(21)和式(24)组成的一维动力学系统, 即可获得毛细管出口表面活化剂包覆液滴在重力作用下自由下落问题的解。

2 边界条件与数值方法

接下来,对本文中采用的数值方法及其验证进行介绍。首先,假定初始时刻,毛细管出口位置液滴头部的形状为半球形, 液滴自由面上表面活化剂均匀分布,即

(29) $\begin{eqnarray} h=\sqrt {1-z^2} ,\ \ v=0,\ \ \varGamma =\varGamma _0 ,\ \ z\in [0,1] \end{eqnarray}$

令毛细管中液体的体积流率$Q=\pi U_{\rm f} $为常数{,} $U_{\rm f}$为毛细管出口位置的平均速度。在毛细管出口,液滴自由面上的表面活化剂浓度为$\varGamma \left( {z=0,t} \right)=\varGamma _0 $。将液滴的最低点(液滴头顶部位)到毛细管口的距离记为液滴长度$L\left( t \right)$,如图2所示。在这里为了简化问题,假设毛细管出口截面液滴下落速度是均匀的,相应的边界条件可表示为

(30) $\begin{eqnarray} \label{eq28} \left. {{\begin{array}{*{20}c} {h=1,} & {v_0 =U_{\rm f} ,\ \ \varGamma =\varGamma _0 \ \ \left( {z=0} \right)} \\ {h=0,} & {v_0 ={L}'(t)\ \ \left( {z=L(t)} \right)} \\ \end{array} }} \right\} \end{eqnarray}$

显然,液滴长度$L\left( t \right)$随液滴下落逐渐变化,是一个未知量。根据质量守恒定律,$L\left( t \right)$可以根据以下关系式确定^[25]

(31) $\begin{eqnarray} \label{eq29} \int_0^{L(t)} \pi h^2\mbox{d}z=V_0 +\pi U_{\rm f} t \end{eqnarray}$

式中,$V_0 $定义为液滴无量纲化初始体积。由于$L\left( t \right)$随时间不断变化,这里引入以下变换

(32) $\begin{eqnarray} \label{eq30} \xi =z/L(t),\ \ \tau =t \end{eqnarray}$

可将计算域$\left[ {0,L\left( t \right)} \right]$投影到$\left[ {0,1}\right]$,相应的导数变换关系如下

(33) $\begin{eqnarray} \label{eq31} \frac{\partial }{\partial z}=\frac{1}{L(t)}\frac{\partial }{\partial \xi },\ \ \frac{\partial }{\partial t}=\frac{\partial }{\partial \tau }-\xi \frac{L^\prime (t)}{L(t)}\frac{\partial }{\partial \xi } \end{eqnarray}$

根据上述边界条件,式(20)$\sim$式(21)和式(24)组成的一维动力学系统可以定解。在本文中采用差分方法对

一维动力学系统进行数值求解, 在空间方向采用二阶中心差分格式、在时间方向上采用四阶Runge--Kutta法。空间离散的网格宽度$\Delta \xi =$ $1/1200$,时间步长选择$\Delta \tau =5\times 10^{-7}$。为了对模型和计算方法的正确性进行验证, 首先对毛细管内无表面活化剂的甘油和水混合液滴的下落、变形及断裂过程进行了模拟,其中液滴表面无活化剂, 毛细管道半径$R^*$ = 1.375 mm,体积流量$Q^*$ = 15 mL/min, 结果如图2所示。数值模拟所得液滴外形与实验结果^[31]基本吻合。这验证了本文所给出的液滴外形演化的数学模型以及相应数值求解方法的可靠性。

图2

图2 水和20%甘油混合液滴颈部断裂前自由面形状的理论和实验结果^[31]对比图

3 结果分析

本小节中,参考文献^[25]中的数据设置了相关无量纲参数,并在$\beta \in \left[ {0,1.0} \right]$,$\varGamma _0 \in \left[ {0,0.7} \right]$的范围内取值,对包覆液滴下落过程进行了模拟。图3给出了不同时刻表面活化剂包覆的液滴从毛细管形成和下落演化的过程, 其中流动参数为$Oh = 0.2941$,$Bo = 0.5500$,$U_{\rm f} = 0.3040$,$Pe = 100$,$\varGamma _0= 0.1$,$\beta = 0.3$。在液滴形成($t = 0$)和下落的初始阶段($t = 10.00$),随着液体从毛细管口不断流出, 液滴的体积逐渐增大,并在表面张力作用下逐渐形成类半球形头部。当液滴体积继续增大并超过稳定液滴最大临界体积时^[22], 液滴的中段半径略小于毛细管道出口半径形成颈部($t= 15.00$),并在表面张力作用下开始快速收缩。液滴在继续下落的过程中,头部逐渐演化为近似球形。在液滴颈部即将发生断裂前($t = 16.40$),由于液滴头部下落拉伸的作用,液滴颈部呈现为一段细长的圆柱形液桥,分别与毛细管下方锥形流体区域和液滴头部相连接。

图3

图3 液滴自由面形状演化过程图

图4给出了液滴颈部断裂时刻,不同表面活化剂活性常数$\beta $对液滴下落高度以及液滴演化形状的影响规律,其中流动参数为：$Oh = 0.2941$,$Bo = 0.5500$,$U_{\rm f} = 0.3040$,$Pe = 100$,$\varGamma _0 = 0.3$。当$\beta =0$时,液滴自由面无表面活化剂存在,此时液滴自由面上表面张力系数为常数,即$\gamma =1$ (见式(24))。从图4中可以明显看出,随着表面活化剂活性常数$\beta$逐渐增大,液滴颈部断裂时刻液滴下落的长度(极限长度) $L_{\rm b}$呈单调增加的趋势。与此同时, 液滴的外形也从近似球形逐渐变为卵形。这主要是由于液滴自由面上表面活化剂非均匀分布引起的Marangoni力所致^[27]。

图4

图4 表面活化剂活性常数$\beta {\rm g}$对液滴下落和自由面形状演化规律的影响

图5给出液滴自由面不同活化剂初始浓度$\varGamma _0 $条件下,液滴颈部断裂时间$t_{\rm b} $、液滴颈部断裂时刻液滴下落的长度(极限长度)$L_{\rm b}$随表面活化剂活性常数$\beta $的变化曲线,其中流动参数：{\it Oh} = 0.2941,{\it Bo} = 0.5500,$U_{\rm f}$ = 0.3040,{\it Pe} = 100。从图中可以非常明显地看出,液滴形成过程中,液滴颈部断裂时间$t_{\rm b} $随着表面活化剂活性常数$\beta $的增大而减小。换言之,表面活化剂能够促进毛细管内液体下落过程中液滴的形成过程, 并且这一效应随活化剂初始浓度的增大而愈发显著。当活化剂初始浓度相对较小时($\varGamma _0 \leqslant 0.4$),从图5中可以发现,液滴颈部断裂时间$t_{\rm b} $以及液滴下落的长度(极限长度) $L_{\rm b} $与表面活化剂活性常数$\beta $之间近似呈线性关系。

图5

图5 液滴自由面上不同活化剂初始浓度$\varGamma _0 $的条件下,液滴断裂时间$t_{\rm b}$、极限长度$L_{\rm b}$随表面活化剂活性常数$\beta$的变化规律

图6给出了不同表面活化剂活性常数$\beta$取值情况下,液滴颈部断裂时间$t_{\rm b} $、液滴颈部断裂时刻液滴下落的长度(极限长度) $L_{\rm b} $随表面活化剂初始浓度$\varGamma _0 $的变化曲线,其中流动参数：{\it Oh} = 0.2941,{\it Bo} = 0.5500,$U_{\rm f}$ = 0.3040,{\it Pe} = 100 。从图中可以看出,随着初始浓度$\varGamma _0 $的增大,$t_{\rm b} $逐渐减小而$L_{\rm b} $逐渐增大。综合上述结果,我们可以得出如下结论,与无表面活化剂的液滴形成和下落演化过程相比, 表面活化剂一方面促进液滴形成(缩小颈部断裂时间),另一方面可以增大液滴下落的长度(极限长度)。与此同时, 表面活化剂的存在使得形成的液滴趋向于卵球形。相比于表面活化剂的活性常数, 液滴自由面上表面活化剂的初始浓度对于液滴形状的影响更为显著。

图6

图6 不同表面活化剂活性常数$\beta$条件下,液滴断裂时间$t_{\rm b} $、极限长度$L_{b}$随液滴自由面上表面活化剂初始浓度$\varGamma _0 $的变化规律

从前面分析可知,方程(21)中的参数$\varphi $反映了液滴自由面曲率沿$z$轴方向变化所引起的表面张力附加压强梯度效应,参数$\psi $反映液滴自由面上活化剂浓度梯度引起的Marangoni效应。图7给出了两种活化剂初始浓度以及活性常数条件下,液滴断裂前下落的长度$L$基本一致时方程(21)中参数$\varphi $和$\psi $、液滴自由面表面张力系数$\gamma $、活化剂浓度$\varGamma $以及液滴自由面位置沿轴向坐标$z$的分布。其中实线代表初始活化剂浓度和活性常数分别为$\varGamma _0$ = 0.1, $\beta =0.1$,下落时间$t$ = 15.00的结果;虚线代表初始活化剂浓度和活性常数分别为$\varGamma _0 $ = 0.3, $\beta =0.5$,下落时间$t$ = 13.60的结果。其余流动参数为：{\it Oh} = 0.2941,{\it Bo} = 0.5500,$U_{\rm f}$ = 0.3040,{\it Pe} = 100。从图中可以看出, 表面活化剂的浓度沿轴向坐标的变化趋势与自由面径向位置变化趋势基本一致。换言之,在对流和扩散输运的作用下, 在液滴半径较小(较细)的部位,表面活化剂的浓度相对较低,而在液滴半径较大(较粗)的部位, 表面活化剂的浓度相对较高。根据液滴表面无量纲表面张力系数与表面活化剂浓度之间的关系式(23), 可知在液滴半径较细的区域,局部表面张力较大,在液滴半径较粗的位置,局部表面张力较弱。因此, 在液滴自由面上存在从液滴半径较粗位置指向半径较细位置的Marangoni牵引力。进一步,根据式(22)可知,若$\psi >0$将促进流体沿$z$轴正向流动;反之,$\psi <0$将促使流体沿$z$轴负向流动。从图7中可以发现,在毛细管出口附近$\psi >0$,即Marangoni牵引力促进了流体向下运动,使毛细管出口形成的锥形液体区域被进一步拉伸;相反,在液滴颈部区域$\psi <0$,Marangoni牵引力使得液体有向液滴颈部汇聚的趋势;然而,在液滴的头部区域又出现$\psi >0$,此时Marangoni牵引力具有促使液滴头部沿轴向拉伸的效应。因而, 可使液滴头部外形从近似球形演化为卵形。对比图7中实线和虚线结果,可以发现,当液滴自由面上表面活化剂初始浓度$\varGamma _0 $和活化剂活性常数$\beta $增大时,由于Marangoni效应增强,$\psi $的变化更为明显,因此液滴下落过程中毛细管出口形成的锥形液体区域被迅速拉长。换言之,Marangoni效应较强时, 液滴自由下落过程中,锥形区域拉伸相同长度所需时间相对缩短。另一方面,Marangoni效应较强时,在液滴自由下落过程中, 颈部区域$\psi $和$\varphi $明显减小,这使得流体具有向颈部区域流动的趋势,从而导致颈部半径的收缩速度相对减缓。综合上述两方面的影响, 若在自由下落液滴表面包覆表面活化剂,在Marangoni效应的影响下,液滴下落速度会增加、液滴下落的极限长度会增长, 与此同时液滴头部更趋近于卵形。

图7

图7 液滴下落的长度基本一致时,方程(21)中与表面张力相关的两项$\varphi $和$\psi$、液滴自由面上表面张力系数$\gamma $、活化剂浓度$\varGamma$和液滴自由面形状

4 结论

本文研究了毛细管出口表面活化剂包覆液滴形成与下落过程的动力学特性。在轴对称流动假设下, 通过采用泰勒展开并根据润滑近似理论,推导了液滴自由面以及自由面上表面活化剂浓度分布随时间演化的一维动力学模型, 并对其进行了数值求解。结果显示,液滴自由面上的表面活化剂浓度分布不均匀所引起的Marangoni效应, 可以促进液滴形成和下落过程。在促进液滴颈部断裂的同时,还可以增大液滴下落过程的极限长度。相比于表面活化剂的活性常数, 液滴自由面上表面活化剂的初始浓度对液滴外形演化过程的影响更为明显。

(责任编辑: 胡漫)

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