1 Introduction
We are motivated to simulate bubble transport and their formation in liquid, based on Efields and different flowfields. Such formation and transport of gasbubbles in liquid are important for the controlled production in additive manufacturing (AM), convective cooling, electrostatic precipitation, plasma assisted combustion, see [7], [9], [14] and [8].
For simulating such processes, we are motivated to extend an underlying hydrodynamics model, which is based on the Burgers’ equation with an electrodynamical equation and additional a twophase equation, which models the bubble transport and modifications, see [2].
We consider the single models and coupled them into a multiphysics model based on the different modeled processes, i.e., flowfield model, Efield model and phase field model. Here, the numerical solvers are important, while we deal with different scaledependent models, see [3] and [4].
We present splitting methods, which separate each modeling equation and solve them with the optimal solver for each equations, see [6].
2 Mathematical Model
The modeling is based on an electrohydrodynamics model coupled with phase field models.
We have the following coupled modeling equations:

Electrical field equation

Phasefield equation

Fluidflow equation
2.1 Modeling: Electrical field equations
We deal with the electrohydordynamics. We have given the Poisson equation as:
(1) 
is the permittivity of vacuum, the dielectric constant, the potential and the free charge density.
The free charge conservation is given as:
(2) 
where is the electrical conductivity, the velocity field and is the electrical field. While the second term on the left hand side presents the convection of the free charges and the righthand side term presents the transport by electromigration.
2.2 Modeling: Phase field equations
We deal with the volumefraction , which satisfies the advection equation:
(3) 
where is the velocity field and is the bubble/drop phase, while is the bulk fluid phase.
Remark 1
The full phase field equation is given by the CahnHillard equation, while the parameter is the phase parameter.
2.3 Modeling: Fluidflow equation
We deal with the fluidflow equation, which satisfy the NavierStokes equation:
(4)  
(5) 
where is the velocity field and is the pressure, is the dynamic viscosity, is the fluid density, is the surface tension, is the electrostatic force, is the gravitational acceleration.
The forces are given as:
(6)  
(7) 
where is the surface tension, is the interface curvature with and
is the Maxwell stress tensor.
3 Solvers and splitting approaches
We deal with the modeling and solverideas, as presented in Fig. 1.
3.1 Reinitialization of the LevelSet method
Based on the problem of the conservation of the volume of the levelset method, we deal with the improvement of the reinitialisation, see [Radams2019].
We have the following algorithm of the reinitialization:

We initialize with and . we apply and we have the indicator function of the density and the volume of the bubble with .

We compute with the next levelset step with timestep . We have the old concentration at time with and we obtain the new concentration at .

We compute the volume of the bubble in the new timepoint .

We apply the correction, which is based on the factor:
(8) 
We apply the reinitialization of the concentration, which is given as:
(9) 
We set and apply , if we are done, else we go to Step .
The idea of the algorithm is given in the Figure 2.
3.2 Solver methods for the Burgers’ and NavierStokes equation
In the following, we discuss the discretization of the Burger’s and NS equation.
3.2.1 Discretization of the Burgers’ equation
We apply the following splitting approach to the 2D Burgers’ equation, which is given as:
(10)  
(11) 
with initial and boundary conditions.
We rewrite into the operatornotation as:
(12) 
where is the nonlinear term (flowterm), is the linear viscosityterm and is the righthand side electrical field term.
We apply the following splitting approach:

AStep (Burgers’ Step) with explicit timediscretization:
We apply the finite difference or finite volume scheme for the viscous Burgers’ equation and deal with the following discretized differential equations:
We apply the following splitting method with the timestep size :
(13) where we discretize the spatial operator with upwinding methods.

BStep (Diffusion Step) with implicit timediscretization:
We apply the finite difference or finite volume scheme for the Diffusion equation and deal with the following discretized differential equations:
We apply the following splitting method with the timestep size :
(15) (16) where is the semidiscretized operator of the diffusion operator.

CStep (RHSStep)
We apply the RHSoperator
(17) where we apply the rhs with . We obtain the solution of the Burgers’ equation . Then, we could go on to the first step.
3.2.2 Discretization of the NS equation
We apply the following splitting approach to the NavierStokes equation.
We deal with the 2D NS equation, which is given as:
(18)  
(19)  
(20) 
with initial and boundary conditions.
We rewrite into the operatornotation as:
(21)  
(22) 
where is the nonlinear term (flowterm), is the linear viscosityterm and is the pressureterm.
We apply the following splitting approach:

AStep (Burgers’ Step):
We apply the finite difference or finite volume scheme for the viscous Burgers’ equation and deal with the following semidiscretized differential equations:We apply the following splitting method with the timestep size :
(23) where is the semidiscretized operator of the nonlinear convection operator of the NS equation and is the semidiscretized operator of the diffusion operator of the NS equation.

BStep (CorrectionStep)
We apply the pressure term in the NS equation, which is solved as following:
We apply an implicit formulation as:
(25) where we apply the divergence of the solution and we have . Then, we reformulate with an additional derivation into the following Poisson equation:
(26) Therefore we solve:
(27) (28) (29) (30) where we obtain the solution of the NS equation . Then, we could go on to the first step.
4 Numerical Experiments
In the next subsections, we deal with the different testexperiments based on the coupled equations.
We apply the following test and SimulationSoftware, that allows a flexibilisation of the software with respect to the EHDmodel:
4.1 Test example 1: Slow and fast velocity fields
In the following, we deal with the modelingequation:
The modeling equations of the gasbubble 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 with , while is the number of bubblesources.
We define the values of the function as:
(31) 
Furthermore, we assume the absence of phase changes and then, we deal with the transport equation of the gasphase as:
(32)  
(33)  
(34) 
where is the velocity of the twophase flow equation, and are the indicator functions of the gasphases and are the initial conditions.
Assumption 4.1
We deal with the following assumption to reduce the computational amount of the farfield 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 gasbubbles are sufficient, see [13].

The concentration of the bubbles are given as , while is the initial concentration of the bubble.
We study 2 different examples:

Example 1: Two layer problem:
We have the following velocity field for the bubbletransport, see Figure 4. For the low velocityfield we have for the high velocityfield, we have , as shown in Fig. 4.
The results are computed with a MATLABcode and are given in see Figure 5.
Figure 5: Transport of 2 bubbles in a velocity field with different layers (low velocityfield with and high velocity field with ), the levelset function (left figures) and the contour functions (right figures) are given at at time , and (from top to bottom). 
Example 2: Multilayer problem:
In the following, we deal with five different layers, while we could also extend the number of layers. We have the following velocity field for the bubbletransport, see Figure 6. For the lowest velocityfield we have , for the low velocityfield, we have and for the high velocityfield, we have .
The results are computed with the MATLABcode and are given in see Figure 7.
Figure 7: Transport of 2 bubbles in a velocity field with 5 different layers (low , medium and high velocity field with ), the levelset function (left figures) and the contour functions (right figures) are given at at time , and (from top to bottom).
4.2 Test example 2: Decoupled EHDmodel with Burgers’ equation
We apply the following decoupled model, which is applied with :
(35)  
(36) 
where is the 2Dvelocity field and is the pressure, is the dynamic viscosity, is the fluid density, is the electrostatic force. We apply and and .
The forces are given as:
(37) 
where we have ,
and
.
The velocity field is applied in the phase field model, which is given as:
(38)  
(39)  
(40) 
where is the velocity of the twophase flow equation, and are the indicator functions of the gasphases and are the initial conditions. The indicator functions is given as:
(41) 
with , while is the number of bubblesources.
Further, we have the CFLcondition of the Burgers’ Equation and the levelset equation for the timeinterval is given as:

CFLCondition for the Burgers’ equation:
(42) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep and is the diffusion parameter.

CFLCondition for the Levelset equation:
(43) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep .

EHDmodel without Efield, we have :
The velocity field is computed by the Burgers’ equation in advance at the beginning, see the Figures 8 (here without the Efield). Further, we apply the computed velocity field based on the decoupled Burgers’ equation (without the Efield) into the phase field model, which is computed with modified levelset method, see the Figures 8.
Figure 8: Upper figures: Velocityfield of the Burgers’ equation without Efield in contour visualization (upper left figure) and velocityfield of the Burgers’ equation without Efield in vectorialvisualization (upper right figure). Lower Figures: Transport of 2 bubbles in a velocity field with decoupled velocity field computed by the Burgers’ equation without an Efield, the levelset function is given in the lower left figure and the contour functions is given in the lower right figure at time . 
EHDmodel with Efield:, we have
The velocity field is computed by the Burgers’ equation and is computed in previous at the beginning, while we also apply the maximum amplitude of the Efield, see the Figures 9 (here with the Efield). Further, we apply the computed velocity field based on the decoupled Burgers’ equation (with the Efield) into the phase field model, which is computed with modified levelset method, see the Figures 9.
Figure 9: Upper figures: Decoupled velocityfield of the Burgers’ equation with weak Efield, which is computed in a maximum of the amplitudes (contour visualization, left figure), right figure: Velocityfield of the Burgers’ equation with weak Efield (vectorialvisualization, right figure). Lower Figures: Transport of 2 bubbles in a velocity field with decoupled velocity field computed by the Burgers’ equation with an Efield, the levelset function is given in the lower left figure and the contour functions is given in the lower right figure at time .
4.3 Test example 3: Coupled EHDmodel with Burgers’ equation
We apply the following coupled model, which is applied with and we assume, that the velocity of the Burgers’ equation are in the scales of the Levelset equation, means we also have to apply Burgers’ equation simultaneously to the Levelset equation, means we have .
Here, we have a fast process that influences the bubbles, while the velocity field influences the speed of the bubbles in the field, we have to deal with the repeatedly update, see Figure 10.
We have the Burgers’ equation, which is given as:
(44)  
(45) 
where is the 2Dvelocity field and is the pressure, is the dynamic viscosity, is the fluid density, is the electrostatic force. We apply and and .
The forces are given as:
(46) 
where we have ,
and
we assume only positive Efields.
Further, we have the CFLcondition of the Burgers’ Equation and the levelset equation for the timeinterval is given as:

CFLCondition for the Burgers’ equation:
(47) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep and is the diffusion parameter.

CFLCondition for the Burgers’ equation:
(48) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep and is the diffusion parameter.
The velocity field is applied in the phase field model, which is given as:
(49)  
(50)  
(51) 
where is the velocity of the twophase flow equation, and are the indicator functions of the gasphases and are the initial conditions. The indicator functions is given as:
(52) 
with , while is the number of bubblesources.
Further, we have the CFLcondition of the Burgers’ Equation and the levelset equation for the timeinterval is given as:

CFLCondition for the Burgers’ equation:
(53) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep and is the diffusion parameter.

CFLCondition for the Levelset equation:
(54) where is the timestep, is the spatial step and is the absolute value of the velocityfield in timestep .

EHDmodel without Efield, we have :
The velocity field is computed by the Burgers’ equation, see the Figures 11 (here without the Efield). Further, we apply the computed velocity field based on the Burgers’ equation (without the Efield) into the phase field model, which is computed with modified levelset method, see the Figures 11.
Figure 11: Upper figures: Coupled velocityfield of the Burgers’ equation without Efield in contour visualization (upper left figure) and velocityfield of the Burgers’ equation without Efield in vectorialvisualization (upper right figure). Lower Figures: Transport of 2 bubbles in a velocity field with coupled velocity field computed by the Burgers’ equation without an Efield, the levelset function is given in the left lower figure and the contour functions is given in the lower right figure at time . 
EHDmodel with Efield:, we have
The velocity field is computed by the Burgers’ equation, see the Figures 12 (here with the Efield). Further, we apply the computed velocity field based on the Burgers’ equation (with the Efield) into the phase field model, which is computed with modified levelset method, see the Figures 12.
Figure 12: Upper figures: Coupled velocityfield of the Burgers’ equation with weak Efield, which is computed by each timestep, in contour visualization (upper left figure) and velocityfield of the Burgers’ equation with weak Efield in vectorialvisualization (upper right figure). Lower Figures: Transport of 2 bubbles in a velocity field with coupled velocityfield computed by the Burgers’ equation with an Efield, the levelset function is given in the lower left figure and the contour functions is given in the lower right figure at time .
5 Conclusion
In this paper, we discussed coupled models, which are based on electrohydrodynamics equations and transport models, which are based on phasefield equations. The models are coupled via splitting approaches and we could present decoupled and coupled versions of the electrohydrodynamical model with the phase field model. We obtained numerical results, while we have to be careful with the different scales of the bubbleoscillations in the Efield and the flowscale if the hydrodynamical equation. We solved such problems with time and spacestep controls. First numerical results are presented with the coupled EHD and phasefield models, here we could see the influence of the different and changing velocities, due to the EField on the bubbles.
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