# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 1 Introduction  
We are motivated to model bubble formation and transport in dielectric liquids, which are applied in controlled production of gas/plasma bubbles in chemical, petrochemical, plasma or biomedical processes, see [1, 2, 3, 4]. The benefit of additional particles or charged-bubbles in the dielectric carrier fluids are that they can influence their fluid behavior or have additional reactions in the fluid, see [5] or [6]. Therefore, the carrier or background liquid can be influenced in the following directions:  
* The charged bubbles can be used as plasma bubbles and can be discharged, such applications can then be used to clean a polluted fluid, see [7]. * The charged bubbles can influence the flow of the underlying liquid, such applications can then be used in designing chemical reactors, see [6, 8]. In our model, we concentrate on the first applications, we model the influence of the bubbles in a dielectric carrier fluid, such as water. Our modeling approach is based on a decomposition into scale-dependent models:  
* Formation of bubbles (near-field model): the fast-fluctuating bubbles are formed with a near-field model and are influenced by the electric-stress related to the dielectric liquid, [9; 10; 11]. * Transport of bubbles (far-field model): the slow-fluctuating bubbles are transported with a far-field model, [12; 13]. Here, we can assume that the fast-fluctuations of the formations and the slow-fluctuations of the transport of the bubble in the liquid carrier fluid has a clear spectral gap, which can be overcome with interpolating (e.g., spline-functions) between the different models, see [14]. Therefore, we apply a so-called modified particle in cell (PIC) method, see [14] and interpolate between the fine-model (microscopic model) and the coarse-model (macroscopic model). The benefit of such a decomposition allows us to simulate the two separate models independently, while we only couple them with the interpolation functions, see [15]  
We also deal with numerical methods that take into account the different scale-dependent model. The formation process of bubbles are done with a microscopic model, the most standard solvers are done in the following:  
* Fast ordinary differential equation (ODE) solvers for so-called bubble shape models, which are based on Young-Laplace equations: We apply the Young-Laplace equations, which are a system of ODEs. The benefit is that the solvers are very fast, while the drawback is that we are restricted to quasi-static processes, see [11]. * Coupled partial differential equation (PDE) solvers for so-called free surface models, which are based on Navier-Stokes equations with kinematic boundary condition. Here, we have to solve a free surface problem with PDE solvers. The benefit is that we can apply the model to instationary processes, the drawback is that we need to solve a system of PDEs, which is more time consuming, see [16]. The macroscopic model is done with a transport model based on an advection equation, where the most standard solver approaches are:  
* Volume-of-fluid (VOF) methods: the VOF function presents the fraction of the volume in the grid cells, which is occupied by one of the two fluids. This is modeled with an advection equation. The benefit of the methods are mass conservation, the drawback of the methods are the problem to solve the sharp change in interface-region, see [17; 18]. * Level-set (LS) methods: the level set function presents a signed distance, which is positive on one fluid side and negative on the other fluid side, see [13]. It is also modeled with the same advection equation. The benefit of the methods is that we can solve if with fast high-speed flow methods, such as total variational diminishing (TVD) or essentially non-oscillatory (ENO) methods, see [19; 20]. The drawback is the lack of mass conservation. Combining the LS and VOF methods can also provide many benefits, although they are delicate to implement, see [18].