# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 2 Mathematical Model
### Far-Field Approach  
For the far-field model, we base our work on macroscopic transport, see [18; 33]. We assume that the flow is laminar and incompressible. Furthermore, we concentrate on the two-dimensional tube-problem, which is given in Figure 2.  
Figure 5: Bubble geometries: (**left**) 3D bubble with the \(R_{1}\) the radius of the curvature in a horizontal section and \(R_{2}\) the radius of the curvature in the vertical section, (**right**) set-up of the bubble shape in 2D, where \(\phi\) is the contact angle, \(r\) is the horizontal length, \(z\) is the vertical length and \(s\) is the arclength along the free surface of the bubble. The conservation equation for the two fluids are given by the Navier-Stokes equation, which is also called the two-phase flow equation, see also [16]:  
\[\
abla\cdot\mathbf{v}=0,\ (x,y,t)\in\Omega\times[0,T], \tag{12}\] \[\rho(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\
abla \mathbf{v})=-\
abla p+\
abla\cdot(\mu(\
abla\mathbf{v}+\
abla\mathbf{v}^{t}))+ \rho_{l}\mathbf{g}+\mathbf{F}_{\sigma},\ (x,y,t)\in\Omega\times[0,T], \tag{13}\]  
where \(\mathbf{v}\) is the velocity vector, \(\rho_{l}\) is the fluid density of the liquid, \(\sigma\) is the surface tension of the liquid, \(\mu\) is the fluid viscosity of the liquid and \(\rho\) is the fluid density of the liquid. \(\mathbf{g}\) is the gravitational acceleration and \(\mathbf{F}_{\sigma}\) is the surface tension force. Furthermore, \(\Omega\in\mathbf{R}^{2}\) is the underlying reactor-domain and \(T\in\mathbf{R}^{+}\) is the end-time of the simulations. The detailed coordinates of the bubble reactor is given in Figure 6.  
The modeling equations of the gas bubble in the liquid is given by a transport equation, which is based on the idea of the volume of fluid and level set methods. We assume the indicator functions \(u_{i}(\mathbf{x},t)\) with \(i=1,\ldots,M\), while \(M\) is the number of bubble sources. We define the values of the function as:  
\[\left\{ \begin{array}{cl}u_{i}(\mathbf{x},t)=1,&\text{gas-phase},\\ u_{i}(\mathbf{x},t)=0,&\text{liquid-phase},\\ 0<u_{i}(\mathbf{x},t)<1,&\text{interface}.\end{array} \right. \tag{14}\]  
Furthermore, we assume the absence of phase changes and then, we deal with the transport equation of the gas-phase as:  
\[\frac{\partial u_{i}}{\partial t}+\
abla(\mathbf{v}\ u_{i})=0, \ (x,y,t)\in\Omega\times[0,T],\ i=1,\ldots,M, \tag{15}\] \[u_{i}(\mathbf{x},0)=u_{i,0}(\mathbf{x}),\] (16) \[i=1,\ldots,M, \tag{17}\]  
Figure 6: The coordinates of the model-reactor with the concentrations of the bubbles \(c_{1},\ldots,c_{M}\), the velocity \(v\), the gravitational acceleration \(g\), the pressure \(p\) and the surface tension force \(F_{\sigma}\). where \(\mathbf{v}\) is the velocity of the two-phase flow equation, and \(u_{i}\) are the indicator functions of the gas-phases and \(u_{i,0}(\mathbf{x})\) are the initial indicator conditions. In the following Assumption 1, we simplify the flow model in the bubbling reactor, such that we could concentrate on the transport-equations as far-field modeling equations. We deal with the following assumption to reduce the computational amount of the far-field model:  
* We can assume a constant velocity in the reactor. * We assume that the water and the gas are not interacting or change the constant velocity. * We have such small velocities, such that the transport equations of the gas bubbles are sufficient, see [18]. * For such a simplification, we deal with \(\mathbf{v}=\mathbf{v}_{0}=const\) in the transport Equations (15)-(17). * The concentration of the bubbles are given as \(c_{i}(\mathbf{x},t)=c_{i,0}(\mathbf{x})u_{i}(\mathbf{x},t)\), while \(c_{i,0}(\mathbf{x})\) is the initial concentration of the bubble. Based on this assumption, we have the following benefits and drawbacks of the far-field modeling approach:  
* The model is simple and fast to compute. * The model also allows us to discuss a dynamical shape. * The shape of the bubble is not preserved, while we assume a static shape.