Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A unified solution of several classical hydrodynamic stability problems


Author: Isom H. Herron
Journal: Quart. Appl. Math. 76 (2018), 1-17
MSC (2010): Primary 76E05; Secondary 47B25, 47N50
DOI: https://doi.org/10.1090/qam/1489
Published electronically: October 31, 2017
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Abstract: The longstanding problems of the linear stability of plane Couette flow and circular pipe flow (to axisymmetric disturbances) are solved by operator theory. It is shown simply that both are stable for all Reynolds numbers and wave numbers. The proof is based on the von Neumann extension of a semi-bounded symmetric operator and the notion of a square root of an unbounded positive definite selfadjoint operator. The use of the latter operator representation is new for this type of hydrodynamic stability problem. It is made clear how the method will apply in other problems with a similar structure such as the planar stability of Couette flow between rotating coaxial cylinders and parabolic Poiseuille flow.


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  • [1] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Monographs and Studies in Mathematics, vol. 9, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. Translated from the third Russian edition by E. R. Dawson; Translation edited by W. N. Everitt. MR 615736
  • [2] Halima N. Ali and Isom H. Herron, The two-dimensional stability of a viscous fluid between rotating cylinders, J. Math. Anal. Appl. 203 (1996), no. 2, 481-489. MR 1410935, https://doi.org/10.1006/jmaa.1996.0392
  • [3] Pascal Chossat and Gérard Iooss, The Couette-Taylor problem, Applied Mathematical Sciences, vol. 102, Springer-Verlag, New York, 1994. MR 1263654
  • [4] S. H. Davis, On the principle of exchange of stabilities, Proc. Roy. Soc. Ser. A 310 (1969), 341-358. MR 0278615
  • [5] R. C. Di Prima and G. J. Habetler, A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability, Arch. Rational Mech. Anal. 34 (1969), 218-227. MR 0266499, https://doi.org/10.1007/BF00281139
  • [6] 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
  • [7] Isom H. Herron, Observations on the role of vorticity in the stability theory of wall bounded flows, Stud. Appl. Math. 85 (1991), no. 3, 269-286. MR 1124375, https://doi.org/10.1002/sapm1991853269
  • [8] I. H. Herron, The linear stability of circular pipe flow to axisymmetric distrubances, Stab. & Appl. Anal. Cont. Med. 2 (1992), 293-303.
  • [9] Isom H. Herron, Hydrodynamic stability, differential operators and spectral theory, African Americans in mathematics (Piscataway, NJ, 1996) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 34, Amer. Math. Soc., Providence, RI, 1997, pp. 57-67. MR 1482256
  • [10] J. N. Hunt, Incompressible fluid dynamics, John Wiley & Sons, Inc. American Elsevier Publishing Co., Inc., New York, New York, 1964. MR 0207282
  • [11] Daniel D. Joseph, Stability of fluid motions. I, Springer-Verlag, Berlin-New York, 1976. Springer Tracts in Natural Philosophy, Vol. 27. MR 0449147
  • [12] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • [13] C. C. Lin, The theory of hydrodynamic stability, Cambridge, at the University Press, 1955. MR 0077331
  • [14] Warren S. Loud, Some examples of generalized Green's functions and generalized Green's matrices, SIAM Rev. 12 (1970), 194-210. MR 0259223, https://doi.org/10.1137/1012042
  • [15] M. S. Maserumule, On Eigenvalue Problems of Poiseuille Flows in a Circular Pipe, Ph.D. Dissertation, Rensselaer Polytechnic Institute, Troy, NY, 2000.
  • [16] Á. Meseguer and L. N. Trefethen, Linearized pipe flow to Reynolds number $ 10^7$, J. Comput. Phys. 186 (2003), no. 1, 178-197. MR 1967366, https://doi.org/10.1016/S0021-9991(03)00029-9
  • [17] J. v. Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1930), no. 1, 49-131 (German). MR 1512569, https://doi.org/10.1007/BF01782338
  • [18] P. L. O'Sullivan and K. S. Breuer, Transient growth in circular pipe flow I Linear disturbances, Phys Fluids 6 (1994), 3652-3664.
  • [19] C. L. Pekeris, Stability of the laminar flow through a straight pipe of circular cross-section to infinitesimal disturbances which are symmetrical about the axis of the pipe, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 285-295. MR 0025855
  • [20] Vít Průša, Sufficient conditions for monotone linear stability of steady and oscillatory Hagen-Poiseuille flow, SIAM J. Appl. Math. 67 (2006/07), no. 2, 354-363. MR 2285867, https://doi.org/10.1137/060652506
  • [21] V. A. Romanov, Stability of plane-parallel Couette flow, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 62-73 (Russian). MR 0326191
  • [22] D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. 46 (1988), no. 4, 751-773. MR 973388, https://doi.org/10.1090/qam/973388
  • [23] Peter J. Schmid and Dan S. Henningson, Stability and transition in shear flows, Applied Mathematical Sciences, vol. 142, Springer-Verlag, New York, 2001. MR 1801992
  • [24] H. B. Squire, On the stability of three dimensional disturbances of viscous flow between parallel walls, Proc. Roy. Soc. A 142 (1933), 621-638.
  • [25] H. L. Swinney and J. P. Gollub (eds.), Hydrodynamic instabilities and the transition to turbulence, 2nd ed., Topics in Applied Physics, vol. 45, Springer-Verlag, Berlin, 1985. MR 796811
  • [26] J. L. Synge, Hydrodynamical Stability, Semicent. Publ. Am. Math. Soc. 2 (1938), 227-269.
  • [27] Peng-Fei Yao and De-Xing Feng, Structure for nonnegative square roots of unbounded nonnegative selfadjoint operators, Quart. Appl. Math. 54 (1996), no. 3, 457-473. MR 1402405, https://doi.org/10.1090/qam/1402405

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Additional Information

Isom H. Herron
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180
Email: herroi@rpi.edu

DOI: https://doi.org/10.1090/qam/1489
Keywords: Hydrodynamics, stability, operator theory
Received by editor(s): September 2, 2015
Published electronically: October 31, 2017
Article copyright: © Copyright 2017 Brown University

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