### Abstract

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 language | English |
---|---|

Pages (from-to) | 248-287 |

Number of pages | 40 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 436 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Apr 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime.** / Ueno, Hiroki; Shiraishi, Akinori; Iguchi, Tatsuo.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 436, no. 1, pp. 248-287. https://doi.org/10.1016/j.jmaa.2015.11.053

}

TY - JOUR

T1 - Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

AU - Ueno, Hiroki

AU - Shiraishi, Akinori

AU - Iguchi, Tatsuo

PY - 2016/4/1

Y1 - 2016/4/1

N2 - 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.

AB - 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.

KW - Free boundary problem

KW - Inclined plane

KW - Incompressible Navier-Stokes equations

KW - Thin film approximation

UR - http://www.scopus.com/inward/record.url?scp=84953351554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84953351554&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2015.11.053

DO - 10.1016/j.jmaa.2015.11.053

M3 - Article

AN - SCOPUS:84953351554

VL - 436

SP - 248

EP - 287

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -