# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 4 Far-Field Solver: Level-Set Method  
We solve the transport Equations (15)-(17) with the level-set method, which allows us to follow the shapes of the bubble, see [13].  
We reformulate the transport Equations (15)-(17) in the level-set equations, which are given as  
\[\frac{\partial u_{i}}{\partial t}=-F_{0}|\
abla u_{i}|,\ ( \mathbf{x},t)\in\Omega\times[0,T], \tag{27}\] \[u_{i}(\mathbf{x},0)=u_{i,0}(\mathbf{x}),\] (28) \[u_{i}(\mathbf{x},t)=0.0,\ (\mathbf{x},t)\in\partial\Omega\times[0,T],\] (29) \[i=1,\dots,M, \tag{30}\]  
where \(\mathbf{v}\) is the constant velocity field with \(\mathbf{v}=F_{0}\mathbf{n}\), while \(\mathbf{n}=\
abla u_{i}/|\
abla u_{i}|\), see [13]. Furthermore, \(\Omega\) is the computational domain and \(T\) is the end time. The indicator functions are given as \(u_{i}(\mathbf{x},t)\) and the initialization \(u_{i}(\mathbf{x},0)\) is done based on the results of the near-field computations, see Section 5.  
These equations are well known as level-set equations and can be solved like convection-diffusion equations, see [13]. We concentrate on the benefits of the LS-methods, which are highly accurate and assume to deal with short transport-phases such that we could reduce the problem of the mass-conservation, see [18].  
In the following, we apply the explicit different discretization methods in space, while we apply the level-set equation with the explicit time-discretization and apply upwind methods for the advection and outer normal direction term only in the \(x\)-direction, the same is also done with the \(y\)-direction.  
We discretize the Level-set equation with the explicit time-discretization and apply upwind methods for the advection and outer normal direction term in the \(x\)- and \(y\)-direction and have the following terms:  
\[D_{x}^{-}u_{i,j,k} = \frac{u_{i,j,k}-u_{i,j-1,k}}{\Delta x}, \tag{31}\] \[D_{x}^{+}u_{i,j,k} = \frac{u_{i,j+1,k}-u_{i,j,k}}{\Delta x},\] (32) \[D_{y}^{-}u_{i,j,k} = \frac{u_{i,j,k}-u_{i,j,k-1}}{\Delta y},\] (33) \[D_{y}^{+}u_{i,j,k} = \frac{u_{i,j,k+1}-u_{i,j,k}}{\Delta y},\] (34) \[|D_{x}^{+}u_{i,j,k}|=\left((\max(D_{x}^{-}u_{i,j,k},0))^{2}+(\min( D_{x}^{+}u_{i,j,k},0))^{2}\right.\] (35) \[\left.+(\max(D_{y}^{-}u_{i,j,k},0))^{2}+(\min(D_{y}^{+}u_{i,j,k}, 0))^{2}\right)^{1/2}. \tag{36}\]  
The explicit discretization is given as:  
\[u_{i,j,k}^{n+1} = u_{i,j,k}^{n}+\Delta t\ F_{0}|D_{x}^{+}u_{i,j,k}^{n}|, \tag{37}\] \[i=1,\ldots,M,\ j=1,\ldots,M_{x},\ k=1,\ldots,M_{y},\ n=0,\ldots,N,\] (38) \[u_{i,j,k}^{0} = u_{i,j,k,0},\ \text{initial condition}, \tag{39}\]  
where we have \(F_{0}\geq 0\) based on the positive velocity \(v_{x},v_{y}\geq 0\) of the transport equations. Furthermore, \(M\) is the number of bubbles, \(M_{x}\) and \(M_{y}\) are the number of spatial grid-points in \(x\) and \(y\) direction. \(N\) is the number of time-steps.  
For our case of the pure velocity in the \(y\)-directions, means we have \(v_{x}=0,\ \ v_{y}\geq 0\), the discretization is simplified to:  
\[u_{i,j,k}^{n+1}=u_{i,j,k}^{n}+\Delta t\ F_{0}|\bar{D}_{x}^{+}u_{ i,j,k}^{n}|, \tag{40}\] \[|\bar{D}_{x}^{+}u_{i,j,k}|=\left((\max(D_{x}^{-}u_{i,j,k},0))^{2 }+(\min(D_{x}^{+}u_{i,j,k},0))^{2}\right)^{1/2},\] (41) \[i=1,\ldots,M,\ j=1,\ldots,M_{x},\ k=1,\ldots,M_{y},\ n=0,\ldots,N,\] (42) \[u_{i,j,k}^{0}=u_{i,j,k,0},\ \text{initial condition}, \tag{43}\]  
where \(F_{0}=v_{y}\) is the pure velocity in \(y\) direction.  
An alternative approach of the shape transport can be done with the VOF method. This method is based on a free-surface modeling technique, while the method is tracking and locating the free surface, see also [17]. These methods are mass-conserved but more delicate to implement and more computationally intensive; therefore, we deal with the idea to reduce computational time with fast LS-methods and update the far-near-field coupling as much as possible to reduce the error of mass-conservation.