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?)

  • [1] T. B. Benjamin, Wave formulation in laminar flow down an inclined plane, J. Fluid Mech. 2, 554-574 (1957)
  • [2] C. S. Yih, Stability of liquid flow down an inclined plane, Phys. Fluids 6, 321-334 (1963)
  • [3] C. C. Mei, Nonlinear gravity waves in a thin sheet of viscous fluid, J. Math. Phys. 45, 266-288 (1966)
  • [4] D. J. Benney, Long waves on liquid films, J. Math. Phys. 45, 150-155 (1966)
  • [5] D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990
  • [6] S. M. Shih and M. C. Shen, Uniform asymptotic approximation for viscous fluid flow down an inclined plane, SIAM J. Math. Anal. 6, 560-582 (1975)
  • [7] A. Carasso and M. C. Shen, On viscous fluid flow down an inclined plane and the development of roll waves, SIAM J. Appl. Math. 33, 399-426 (1977)
  • [8] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969
  • [9] S. M. Shih, Contributions to the theory of surface waves on a viscous fluid, Ph.D. Thesis, University of Wisconsin-Madison, 1973
  • [10] M. C. Shen, S. M. Sun, and R. E. Meyer, Surface waves on viscous magnetic fluid flow down an inclined plane, Phys. Fluids A 3, 439-445 (1991)

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

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