# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 2 Mathematical Model
### Near-Field Model  
The near-field model is discussed with respect to the formation of a drop or bubble, see [11; 16].  
The basic modeling idea is based on the so-called Young-Laplace equation, see [28]. We assumed to have quasi-static cases, while we had static pressure, and the surface tension forces were also effective elements, see [29].  
The Young-Laplace-equation is derived by the balance of the liquid-gas interface, see [29], where we have:  
\[\Delta p+p_{ex}=\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\sigma, \tag{4}\]  
where \(\Delta p\) is the pressure-difference between the gas and liquid and \(p_{ex}\) is an extra pressure; for example, in our case the electrical stress \(p_{ex}=p_{e}\). Furthermore, \(r_{1}\) and \(r_{2}\) are the radii of the curvature, see Figure 5, and \(\sigma\) is the surface tension coefficient between the liquid and gas.  
The parameters of the model and the geometry of the bubble are presented in Figure 5.  
The parameterization of the free surface of the bubble in terms of the arclength \(s\) can be modelled as a system of ordinary differential equation with a boundary value problems and is given as, see also [11; 29]:  
\[\frac{dr}{ds} =\cos(\theta), \tag{5}\] \[\frac{dz}{ds} =\sin(\theta),\] (6) \[\frac{d\theta}{ds} =\frac{\Delta p(z)}{\sigma}+\frac{p_{e}}{\sigma}-\frac{\sin(\theta )}{r}, \tag{7}\]where the pressure difference is given as:  
\[\frac{\Delta p(z)}{\sigma}=\frac{2}{r_{0}}-\frac{gz}{\sigma}(\rho_{l}-\rho_{g}), \tag{8}\]  
where we have \(r_{0}\) the radius of the curvature at the apex, \(\sigma\) is the surface-tension coefficient, \(\rho_{l}\) is the liquid density, \(\rho_{g}\) is the gas density, \(g\) is the gravity and \(p_{e}\) the electrical stress.  
In the following, we assume a small radius \(r_{0}\leq 1\) and assume small vertical extensions; that is, \(z\leq 1\). Then, we have a dominant \(\frac{2}{R_{0}}\) term in the pressure difference, see also [29]:  
\[\frac{\Delta p(z)}{\sigma}=\frac{2}{r_{0}}-\frac{gz}{\sigma}(\rho_{l}-\rho_{g} )\approx\frac{2}{r_{0}}. \tag{9}\]  
Further, we have the boundary conditions, which are given as:  
\[r =a,\;z=0,\;\text{at}\;s=0, \tag{10}\] \[r =0,\;\frac{dz}{ds}=\frac{dz}{dr}=0,\;\text{at}\;s=L, \tag{11}\]  
where \(L\) is the arc length of the bubble which is a-priori unknown and is numerically computed via the boundary value problem. Furthermore, we apply \(a\to 0\), while \(a\) is the initial radius of the bubble at \(s=0\).  
The ODE system (5)-(7) is a so-called boundary value problem, which can be solved with a multiple shooting methods, see also a discussion in Section 3 and the literature [30; 31; 32]. We apply a boundary-value solver of the MATLAB software-package, which is given as programmed in the function _bvp4c_. Based on the boundary values, we can solve the possible curvature of the bubble. We solve the axial-symmetric bubble, while it is sufficient to solve the half shape of the bubble. Then, we can measure the different diameters of the bubble ellipse and such results are interpolated on a grid, which is used in the far-field solver methods, see Section 5.