Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Justification of the linear long-wave approximation to viscous fluid flow down an inclined plane

Authors: S. M. Sun and M. C. Shen
Journal: Quart. Appl. Math. 52 (1994), 759-775
MSC: Primary 76D05; Secondary 35Q30, 76D33
DOI: https://doi.org/10.1090/qam/1306049
MathSciNet review: MR1306049
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Abstract: The objective of this paper is to justify the linear long-wave approximation used in the derivation of approximate equations for long waves on the free surface of a two-dimensional viscous fluid flow down an inclined plane. To the first order of a small parameter, the approximate equation is a heat equation, which becomes ill-posed if a Reynolds number $ R$ is greater than some critical value $ {R_c}$. To overcome this difficulty we consider a higher-order approximate equation, which is well-posed even if $ R > {R_c}$, and show that the solution of the higher-order equation is an approximation to the solution of the linearized Navier-Stokes equations. The justification is based upon a set of long-wave initial conditions, and the error bounds can also be expressed in terms of pointwise estimates.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/qam/1306049
Article copyright: © Copyright 1994 American Mathematical Society

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