Reduction of order: Analytical solution of film formation in the electrostatic rotary bell sprayer

Mark Doerre, Nelson K. Akafuah

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


This brief paper explains the slight differences in governing equations for a fluid film in a spinning cone, and the mechanism that reduces the order of a solution. Spinning cones with a centrally supplied fluid that spreads over its inner surface as a thin film have been the subject of interest for many years. Though often cast as a mathematical analysis, understanding this process is important, especially in the application of automotive painting. The analysis consists of a system of equations obtained from the Navier-Stokes equations along with simple boundary conditions that describe radial and tangential momentum conservation. Solutions to this system of equations are shown using several techniques. The connection between these techniques is slightly subtle. However, the conditions that enable reduction of order are clear once they are exposed. Directional velocity profiles in the film can be a combination of four roots in the complex plane. This system of roots also contains two diagonal axes of symmetry that are offset by 90 degrees. Alternatively, if the radial and tangential velocity profiles are expressed as a single complex function, a reduced order solution that is a combination of one set of diagonal set of roots can be found.

Original languageEnglish
Article number937
Issue number7
StatePublished - Jul 1 2019

Bibliographical note

Funding Information:
Acknowledgments: This work was partially supported by development funds of the Institute of Research for Technology Development (IR4TD) College of Engineering, University of Kentucky.

Publisher Copyright:
© 2019 by the authors.


  • Centrifugal force
  • Coriolis force
  • Navier-stokes
  • Reduction of order
  • Spinning cone

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)


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