Exchange of stabilities in Couette flow between cylinders with Navier-slip conditions

Herron, Isom; Suárez, Pablo

doi:10.1090/S0033-569X-2012-01274-4

Authors: Isom H. Herron and Pablo U. Suárez
Journal: Quart. Appl. Math. 70 (2012), 743-758
MSC (2010): Primary 76E07; Secondary 76U05, 76A02
DOI: https://doi.org/10.1090/S0033-569X-2012-01274-4
Published electronically: June 22, 2012
MathSciNet review: 3052088
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Viscous Couette flow is derived for flow between two infinitely long concentric rotating cylinders with Navier slip on both. Its axisymmetric linear stability is studied within a regime that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum, based on the rotation rates and radii of the cylinders. Stability conditions are analyzed, by methods based on those of Synge and Chandrasekhar. For sufficiently small slip length on the outer cylinder no instability occurs with arbitrary slip length on the inner cylinder. As a corollary, slip on the inner cylinder is shown to be stabilizing, with no slip on the outer cylinder. Two slip configurations are investigated numerically, first with slip only on the outer cylinder, then second with equal slip on both cylinders. It is found that instability does occur (for large outer slip length), and the principle of exchange of stabilities emerges. The instability disappears for sufficiently large slip length in the second case; Rayleigh’s criterion provides an explanation for these phenomena.

References

G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226

Coles, D. Transition in circular Couette flow J. Fluid Mech., 1965, 21, 385-425.

Conlisk, A. T. Essentials of Micro and Nanofluidics with Applications to the Biological and Chemical Sciences. Cambridge University Press, 2011.

P. G. Drazin and W. H. Reid, Hydrodynamic stability, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. With a foreword by John Miles. MR 2098531

Isom H. Herron and Halima N. Ali, The principle of exchange of stabilities for Couette flow, Quart. Appl. Math. 61 (2003), no. 2, 279–293. MR 1976370, DOI https://doi.org/10.1090/qam/1976370

Isom H. Herron and Michael R. Foster, Partial differential equations in fluid dynamics, Cambridge University Press, Cambridge, 2008. MR 2441490

J. N. Hunt, Incompressible fluid dynamics, John Wiley & Sons, Inc. American Elsevier Publishing Co., Inc., New York, New York, 1964. MR 0207282

Daniel D. Joseph, Stability of fluid motions. I, Springer-Verlag, Berlin-New York, 1976. Springer Tracts in Natural Philosophy, Vol. 27. MR 0449147

Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617

Miccal T. Matthews and James M. Hill, Flow around nanospheres and nanocylinders, Quart. J. Mech. Appl. Math. 59 (2006), no. 2, 191–210. MR 2219884, DOI https://doi.org/10.1093/qjmam/hbj003

Matthews, M. T. and Hill, J. M. Nanofluidics and the Navier boundary condition, 2008, Int. J. Nanotechnol. 5, 218-239.

Orszag, S. A. Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 1971, 50, 689-703.

Synge, J. L., On the stability of a viscous liquid between rotating coaxial cylinders, Proc. Roy. Soc. A, 1938, 167, 250-256.

Lloyd N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1776072

Vinogradova, O. I. Drainage of a thin liquid film confined between hydrophobic surfaces, Langmuir, 1995, 11, 2213-2220.

Watanabe, K. and Akino, T., Drag reduction in laminar flow between two vertical coaxial cylinders, J. Fluids Eng., 1999, 121, 541-547.

Watanabe, K., Takayama, T., Ogata, S. and Isozaki, S., Flow between two coaxial rotating cylinders with a high water-repellent wall, AIChE Journ., 2003, 49, 1956-1963.

J. A. C. Weideman and S. C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000), no. 4, 465–519. MR 1939962, DOI https://doi.org/10.1145/365723.365727