# Reduction of Order: Analytical Solution of Film Formation in the Electrostatic Rotary Bell Sprayer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

_{φ}and corresponding derivatives in the NSEs. Lastly, another method is based on applying Newton’s Law to a differential fluid parcel [7]. This alternative is similar to the NSEs, albeit less formal in the procedure.

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

U | dimensionless radial velocity |

W | dimensionless tangential velocity |

i | $\sqrt{-1}$ |

r | position vector |

s | dimensionless film coordinate |

s^{+} | dimensionless film thickness |

v | velocity vector |

v_{r} | radial fluid velocity |

v_{φ} | tangential fluid velocity |

v’_{φ} | tangential fluid velocity in rotating coordinates |

z | complex number |

Ψ | Complex solution to reduced order equation |

Φ | Complex solution to reduced order equation |

β | cone half angle |

ω | angular velocity |

Ω | angular velocity |

## References

- Saito, K.; Toda, K.; Salzar, A. (Eds.) Automotive Painting Technology; Springer: New York, NY, USA, 2013. [Google Scholar]
- Darwish Ahmad, A.; Abubaker, A.; Salaimeh, A.; Akafuah, N. Schlieren visualization of shaping air during operation of an electrostatic rotary bell sprayer: Impact of shaping air on droplet atomization and transport. Coatings
**2018**, 8, 1–13. [Google Scholar] [CrossRef] - Nikolaev, V.S.; Vachagin, K.D.; Baryshev, Y.N. Film flow of viscous liquids over surfaces of rapidly rotating conical disks. Int. Chem. Eng.
**1967**, 7, 595–598. [Google Scholar] - Rauscher, J.W.; Kelly, R.E.; Cole, J.D. An asymptotic solution for the laminar flow of a thin film on a rotating disk. J. Appl. Mech.
**1973**, 40, 43–47. [Google Scholar] [CrossRef] - Greenspan, H.P. The Theory of Rotating Fluids; University Printing House: Cambridge, UK, 1968. [Google Scholar]
- Bruin, S. Velocity distributions in a liquid film flowing over a rotating conical surface. Chem. Eng. Sci.
**1969**, 24, 1647–1654. [Google Scholar] [CrossRef] - Hinze, J.O.; Milborn, H.J. The atomization of liquids by means of a rotating cup. J. Appl. Mech.
**1950**, 17, 147–153. [Google Scholar] - Makarytchev, S.V.; Xue, E.; Langrish, T.A.G.; Prince, R.G.H. On modelling fluid flow over a rotating conical surface. Chem. Eng. Sci.
**1997**, 52, 1055–1057. [Google Scholar] [CrossRef] - Brown, J.W.; Churchill, R.V. Complex Variables and Applications; McGraw Hill Education: New York, NY, USA, 2014. [Google Scholar]
- Emslie, A.G.; Bonner, F.T.; Peck, L.G. Flow of a viscous liquid on a rotating disk. J. Appl. Phys.
**1958**, 29. [Google Scholar] [CrossRef] - Saito, K.; Williams, F.A. Scale modeling in the age of high speed computers. Prog. Scale Model.
**2014**, 2. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Doerre, M.; Akafuah, N.K.
Reduction of Order: Analytical Solution of Film Formation in the Electrostatic Rotary Bell Sprayer. *Symmetry* **2019**, *11*, 937.
https://doi.org/10.3390/sym11070937

**AMA Style**

Doerre M, Akafuah NK.
Reduction of Order: Analytical Solution of Film Formation in the Electrostatic Rotary Bell Sprayer. *Symmetry*. 2019; 11(7):937.
https://doi.org/10.3390/sym11070937

**Chicago/Turabian Style**

Doerre, Mark, and Nelson K. Akafuah.
2019. "Reduction of Order: Analytical Solution of Film Formation in the Electrostatic Rotary Bell Sprayer" *Symmetry* 11, no. 7: 937.
https://doi.org/10.3390/sym11070937