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
Full-text PDF Free Access
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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.
T. B. Benjamin, Wave formulation in laminar flow down an inclined plane, J. Fluid Mech. 2, 554–574 (1957)
C. S. Yih, Stability of liquid flow down an inclined plane, Phys. Fluids 6, 321–334 (1963)
C. C. Mei, Nonlinear gravity waves in a thin sheet of viscous fluid, J. Math. Phys. 45, 266–288 (1966)
D. J. Benney, Long waves on liquid films, J. Math. Phys. 45, 150–155 (1966)
D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990
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)
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)
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969
S. M. Shih, Contributions to the theory of surface waves on a viscous fluid, Ph.D. Thesis, University of Wisconsin-Madison, 1973
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)
T. B. Benjamin, Wave formulation in laminar flow down an inclined plane, J. Fluid Mech. 2, 554–574 (1957)
C. S. Yih, Stability of liquid flow down an inclined plane, Phys. Fluids 6, 321–334 (1963)
C. C. Mei, Nonlinear gravity waves in a thin sheet of viscous fluid, J. Math. Phys. 45, 266–288 (1966)
D. J. Benney, Long waves on liquid films, J. Math. Phys. 45, 150–155 (1966)
D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990
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)
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)
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969
S. M. Shih, Contributions to the theory of surface waves on a viscous fluid, Ph.D. Thesis, University of Wisconsin-Madison, 1973
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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Article copyright:
© Copyright 1994
American Mathematical Society