TY - JOUR

T1 - A mathematical justification of a thin film approximation for the flow down an inclined plane

AU - Ueno, Hiroki

AU - Iguchi, Tatsuo

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/12/1

Y1 - 2016/12/1

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, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous examples of the approximate equations are the Burgers equation, Kuramoto–Sivashinsky equation, KdV–Burgers equation, KdV–Kuramoto–Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier–Stokes equations and those of approximate equations.

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, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous examples of the approximate equations are the Burgers equation, Kuramoto–Sivashinsky equation, KdV–Burgers equation, KdV–Kuramoto–Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier–Stokes equations and those of approximate equations.

KW - Burgers equation

KW - Free boundary problem

KW - Inclined plane

KW - Navier–Stokes equations

KW - Thin film approximation

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U2 - 10.1016/j.jmaa.2016.06.064

DO - 10.1016/j.jmaa.2016.06.064

M3 - Article

AN - SCOPUS:84993968393

VL - 444

SP - 804

EP - 824

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -