# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 5 Coupling Near-Field and Far-Field
### Bubble Formation: Oscillation of the Air-Bubbles in the Electrical-Field (Decoupled Version)  
In the following, we simulate a bubble filled with air in an electrical field, see [10]. We apply the decoupled computation of the near- and far-field, as discussed in Section 5.1.  
We have the following setting of the influenced bubble, see Figure 21.  
Figure 19: Transport of 10 bubbles with the level-set function at the initialization \(t=0.1\).  
Figure 20: Transport of 10 bubbles with the level-set function at time \(t=50\).  
We apply the following near-field equation based on an extension of the pressure-term with an E-field.  
The near-field equations with the E-field are given as:  
\[\frac{dr}{ds} =\cos(\theta), \tag{67}\] \[\frac{dz}{ds} =\sin(\theta),\] (68) \[\frac{d\theta}{ds} +\frac{\sin(\theta)}{r}=\frac{\Delta p}{\sigma}+\frac{p_{E}}{ \sigma}, \tag{69}\]  
where \(s\) is the arc length along the curve and \(\theta\) the angle of elevation for its slope and \(\sigma\) is the mono-layer surface tension, and the approximated pressure difference is given as \(\Delta p=\frac{2}{R_{0}}\), see Section 2.1. \(p_{E}=\frac{9}{8}|\mathbf{E}_{0}|^{2}\ \sin(\theta)\) is the electrical stress related to an external electrical field \(\mathbf{E}_{0}\).  
We apply the following parameters:  
* Computation of the near-field bubbles (a representing bubble is computed)
* The electrical field parameters are given as: \(\sigma=0.1,\,\rho=0.1,\ g=9.81,\ |\mathbf{E}_{0}|^{2}=0.1\).
* Input-parameters of the near-field bubbles computation are given in Appendix A.2.
* Output-parameters of the near-field bubble computation (formation) are given in the Appendix A.2.
* Output-parameters of the near-field bubble computation (in the E-field) are given in Appendix A.2.
* Ellipse: \((x-x_{bubble_{i}})^{2}+((y-y_{bubble_{i}})*a_{bubble}/b_{bubble})^{2}-a_{ bubble}^{2}\) where \((x_{bubble_{i}},y_{bubble_{i}})\) is the origin of the \(i\)-th bubble.
* Computation of the far-field bubbles (level-set initialization):  
Figure 21: Bubble formation influenced by an electrical field.  
Parameterization of the level-set initial-function, such as two bubbles:  
\[\phi_{0}(x,y)=\left\{\begin{array}{c}(x-x_{bubble_{1}})^{2}+((y-y_{bubble_{1}}) ^{a_{bubble_{2}}})^{a_{bubble_{1}}})^{2}-a_{bubble_{1}}^{2},\\ a_{x}\leq x\leq 50,\ a_{y}\leq x\leq b_{y},\\ (x-x_{bubble_{2}})^{2}+((y-y_{bubble_{2}})^{a_{bubble_{2}}})^{2}-a_{bubble_{2}}^ {2},\\ 50\leq x\leq b_{x},\ a_{y}\leq x\leq b_{y},\end{array}\right. \tag{70}\]  
where \((x_{bubble_{1}},y_{bubble_{1}})=(20,50)\), \((x_{bubble_{2}},y_{bubble_{2}})=(80,50)\) with the coordinates of the grid \((a_{x},a_{y})=(0,0)\) and \(b_{x},b_{y})=(100,200)\).  
The numerical results of the formation of the bubbles with and without the E-field are given in Figure 22.  
The numerical results of the near-far-field coupled bubble transport code are given in Figures 23 and 24.  
The bubble modifications are given by the E-field, while it changes the formation of the bubble, see the Figure 24. The level-set method is flexible and is modified by multi-level-set domains. Therefore, it allows us to deal with multiple level-set functions of different concentrations and initial conditions, such that we could transport multiple bubbles with E-fields. Such a flexibilization is necessary to deal with the laboratory experiments, see [29].  
Figure 23: Transport of 10 bubbles in the E-field with the level-set function at the initialization \(t=0.5\).  
Figure 22: (**left**) Formation of 10 bubbles without an E-field. (**right**) Formation of 10 bubbles with an E-field.  
Figure 27: Transport of 10 bubbles with the level-set function: (**top**) \(t=7.5\) with E-field switched off and (**bottom**) \(t=15.5\) with E-field switched on.  
Figure 26: Algorithm for updating a bubble formation with an oscillating E-field in a transport regime based on level-set methods.