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)

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 languageEnglish
Pages (from-to)248-287
Number of pages40
JournalJournal of Mathematical Analysis and Applications
Volume436
Issue number1
DOIs
Publication statusPublished - 2016 Apr 1

Fingerprint

Uniform Estimates
Inclined
Viscous Fluid
Incompressible Fluid
Thin Films
Navier Stokes equations
Thin films
Navier-Stokes Equations
Fluids
Approximation
Perturbation Expansion
Surface Tension
Justification
Aspect Ratio
Surface tension
Aspect ratio
Error Estimates
Gravity
Gravitation
Motion

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.

In: Journal of Mathematical Analysis and Applications, Vol. 436, No. 1, 01.04.2016, p. 248-287.

Research output: Contribution to journalArticle

@article{23ab299e0b714085989fe1a37f9ee535,
title = "Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime",
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.",
keywords = "Free boundary problem, Inclined plane, Incompressible Navier-Stokes equations, Thin film approximation",
author = "Hiroki Ueno and Akinori Shiraishi and Tatsuo Iguchi",
year = "2016",
month = "4",
day = "1",
doi = "10.1016/j.jmaa.2015.11.053",
language = "English",
volume = "436",
pages = "248--287",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "1",

}

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 -