Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

Hiroki Ueno, Akinori Shiraishi, Tatsuo Iguchi

Research output: Contribution to journalArticle

2 Citations (Scopus)


We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, it is hard to treat the Navier-Stokes equations directly, so that a thin film approximation is often used. It is an approximation obtained by the perturbation expansion with respect to the aspect ratio δ of the film under the thin film regime δ. ≪ 1. Our purpose is to give a mathematically rigorous justification of the thin film approximation by establishing an error estimate between the solution of the Navier-Stokes equations and those of approximate equations. To this end, in this paper we derive a uniform estimate for the solution of the Navier-Stokes equations with respect to δ under appropriate assumptions.

Original languageEnglish
Pages (from-to)248-287
Number of pages40
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 2016 Apr 1



  • Free boundary problem
  • Inclined plane
  • Incompressible Navier-Stokes equations
  • Thin film approximation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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