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

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• 作者：Hiroki Ueno ; ^{hueno@math.keio.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Tatsuo Iguchi ^{iguchi@math.keio.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Navier&ndash ; Stokes equations ; Free boundary problem ; Thin film approximation ; Inclined plane ; Burgers equation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：444
• 期：1
• 页码：804-824
• 全文大小：472 K

文摘

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.