# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## Appendix A
### Parameters of Section 6.5  
* Input-parameters of the near-field bubbles computation:
* Bubble 1: \(r_{1}(0)=0.01,\ z_{1}(0)=0,\ r_{1}(L)=0,\ L=2,\ \frac{\Delta p_{1}}{\sigma_{1}}=0.2\),  
Figure 28: Transport of 10 bubbles with the level-set function: (**top**) at \(t=38.0\) with E-field switched off and (**bottom**) \(t=45.5\) with E-field switched on. * [1] [PERSON]; [PERSON]; [PERSON] Droplets formation and merging in two-phase flow microfluidics. _Int. J. Mol. Sci._**2011**, _12_, 2572-2597. [CrossRef] [PubMed]
* [2] [PERSON]; [PERSON]; [PERSON]; [PERSON] Gas-liquid two-phase chemical reaction model of reactive plasma inside a bubble for water treatment. In Proceeding of the 22 nd International Symposium on Plasma Chemistry, Antwerp, Belgium, 5-10 July 2015. * [3] [PERSON]; [PERSON] The formation of bubbles and drops. _Adv. Chem. Eng._**1970**, \(8\), 255-368. * [4] [PERSON]; [PERSON]; [PERSON] Bubble formation and dynamics in gas-liquid-solid fluidization--A review. _Chem. Eng. Sci._**2007**, _62_, 2-27. [CrossRef]
* [5] [PERSON] _Electrorheological Fluids: Modeling and Mathematical Theory_; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2008; Volume 1748. * [6] [PERSON]; [PERSON]; [PERSON] _Chemical Reactor Design and Operation_, 2 nd ed. ; John Wiley and Sons: Chichester, UK; New York, NY, USA, 1984. * [7] [PERSON]; [PERSON] Non-thermal plasmas in and in contact with liquids. _J. Phys. D Appl. Phys._**2009**, _42_, 053001. [CrossRef]
* [8] [PERSON]; [PERSON]; [PERSON] Two-phase bubble columns: A comprehensive review. _ChemEngineering_**2018**, \(2\), 13. [CrossRef]
* [9] [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] Thermo-electro-hydrodynamic convection under microgravity: A review. _Fluid Dyn. Res._**2016**, _48_, 061413. [CrossRef]
* [10] [PERSON]; [PERSON] Nonlinear oscillations of gas bubbles submerged in water: Implications for plasma breakdown. _J. Phys. D Appl. Phys._**2012**, _45_, 415203. [CrossRef]
* [11] [PERSON]; [PERSON]; [PERSON] The coalescence speed of a pendent and a sessile drop. _J. Fluid Mech._**2005**, _527_, 85-114. [CrossRef]
* [12] [PERSON]; [PERSON] Porous Media Based Modeling of PE-CVD Apparatus: Electrical fields and Deposition Geometries. In _Special Topics and Reviews in Porous Media;_ Begell House Inc.: Redding, CA, USA, 2010; Volume 1, pp. 215-229. * [13] [PERSON] _Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science_; Cambridge University Press: Cambridge, UK, 1996. * [14] [PERSON]; [PERSON] _Computer Simulation Using Particles_; CRC Press: Boca Raton, FL, USA, 1985. * [15] [PERSON] DEMOCRITUS: An adaptive particle in cell (PIC) code for object-plasma interactions. _J. Comput. Phys._**2011**, _230_, 4679-4695. [CrossRef]
* [16] [PERSON]; [PERSON]. The formation of a bubble from a submerged orifice. _Eur. J. Mech. B Fluids_**2015**, _53_ (Suppl. C), 24-36. [CrossRef]* [17] [PERSON]; [PERSON] Volume of fluid (VOF) method for the dynamics of free boundaries. _J. Comput. Phys._**1981**, _39_, 201-225. [CrossRef]
* [18] [PERSON]; [PERSON]; [PERSON] Coupled level-set and volume-of-fluid method for two-phase flow calculations. _Numer. Heat Transf. Part B Fundam._**2017**, _71_, 173-185. [CrossRef]
* [19] [PERSON]; [PERSON] _Multi-Dimensional ENO Schemes for General Geometries_; USA, NASA Contractor Report 187637; ICASE Report No. 91-76; NASA Langley Research Center: Hamption, VA, USA, 1991; Available online: [[https://ntrs.nasa.gov/search.jsp?R=19920001131](https://ntrs.nasa.gov/search.jsp?R=19920001131)]([https://ntrs.nasa.gov/search.jsp?R=19920001131](https://ntrs.nasa.gov/search.jsp?R=19920001131)) (accessed on 4 November 2019). * [20] [PERSON]; [PERSON] Weighted essentially non-oscillatory schemes on triangular meshes. _J. Comput. Phys._**1999**, _150_, 97-127. [CrossRef]
* [21] [PERSON]; [PERSON]; [PERSON] Streamer dynamics in gases containing dust particles. _Plasma Sources Sci. Technol._**2006**, _15_, 591-602.