Slow oscillatory Stokes flow

Author:
S. H. Smith

Journal:
Quart. Appl. Math. **55** (1997), 1-22

MSC:
Primary 76D07

DOI:
https://doi.org/10.1090/qam/1433748

MathSciNet review:
MR1433748

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Abstract: Two transient problems in slow viscous flow are considered where the corresponding steady-state behaviour leads to the paradoxical results of Stokes and Jeffery. First, the oscillatory flow past a circular cylinder is investigated when the frequency tends to zero, where an outer domain of size is required to ensure that the velocity conditions at infinity are satisfied. The flow close to the cylinder is quasi-steady except for a time of length about the time of separation; most of the action takes place in the outer domain where the dominant transient behaviour extends over a time that is of a complete cycle.

**[1]**G. G. Stokes,*On the effect of the internal friction of fluids on the motion of a pendulum*, Trans. Cambridge Philos. Soc.**9**, 9-106 (1851)**[2]**C. W. Oseen,*Über die Stokessche Formel und über eine verwandt Aufgabe in der Hydrodynamik*, Ark. Mat. Astr. Fys.**6**, no. 29 (1910)**[3]**S. Kaplun and P. A. Lagerstrom,*Asymptotic expansion of Navier-Stokes solutions for small Reynolds numbers*, J. Math. Mech.**6**, 585-593 (1957) MR**0091693****[4]**I. Proudman and J. R. A. Pearson,*Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder*, J. Fluid Mech.**2**, 237-262 (1957) MR**0086545****[5]**G. B. Jeffery,*The rotation of two circular cylinders in a viscous fluid*, Proc. Roy. Soc. (A)**101**, 169-174 (1922)**[6]**S. Tomotika and T. Aoi,*The steady flow of viscous fluid past a sphere and circular cylinder at small Reynolds numbers*, Quart. J. Mech. Appl. Math.**3**, 140-161 (1950) MR**0036110****[7]**J. M. Dorrepaal, M. E. O'Neill, and K. B. Ranger,*Two-dimensional Stokes flows with cylinders and line singularities*, Mathematika**31**, 65-75 (1984) MR**762178****[8]**C. Pozrikidis,*A study of linearized oscillatory flow past particles by the boundary-integral method*, J. Fluid Mech.**202**, 17-41 (1989) MR**1000879****[9]**S. Kim and S. J. Karrila,*Microhydrodynamics*, Butterworth-Heinemann, Boston, 1991**[10]**A. B. Basset,*A treatise on hydrodynamics*, Vol. 2, Deighton Bell, Cambridge, 1888**[11]**M. Abramowitz and I. A. Stegun,*Handbook of Mathematical Functions with Formulas*,*Graphs, and Mathematical Tables*, Dover Publications, New York, 1965 MR**1225604****[12]**G. Wannier,*A contribution to the hydrodynamics of lubrication*, Quart. Appl. Math.**8**, 1-32 (1950) MR**0037146****[13]**K. B. Ranger,*Eddies in two dimensional Stokes flow*, Internat. J. Engrg. Sci.**18**, 181-190 (1980)

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Additional Information

DOI:
https://doi.org/10.1090/qam/1433748

Article copyright:
© Copyright 1997
American Mathematical Society