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The Rayleigh problem for a wavy wall

Published online by Cambridge University Press:  11 April 2006

P. N. Shankar  and
U. N. Sinha
P. N. Shankar
Affiliation:
Aerodynamics Division, National Aeronautical Laboratory, Bangalore 560017, India
U. N. Sinha
Affiliation:
Aerodynamics Division, National Aeronautical Laboratory, Bangalore 560017, India
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Abstract

The problem of the flow generated in a viscous fluid by the impulsive motion of a wavy wall is treated as a perturbation about the known solution for a straight wall. It is shown that, while a unified treatment for high and low Reynolds numbers is possible in principle, the two limiting cases have to be treated separately in order to get results in closed form. It is also shown that a straightforward perturbation expansion in Reynolds number is not possible because the known solution is of exponential order in that parameter. At low Reynolds numbers the waviness of the wall quickly ceases to be of importance as the liquid is dragged along by the wall. At high Reynolds numbers on the other hand, the effects of viscosity are shown to be confined to a narrow layer close to the wall and the known potential sohtion emerges in time. The latter solution is a good illustration of the interaction between a viscous fluid field and its related inviscid field.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 77 , Issue 2 , 24 September 1976 , pp. 243 - 256
Copyright
© 1976 Cambridge University Press

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References

Ladyzhenskaya, O. A. 1959 Solution ‘in the large’ of the non-stationary boundary value problem for the Navier-Stokes system in two space variables. Comm. Pure Appl. Math. 12, 655.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flows, revised 2nd ed. Gordon & Breach.