Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Stability of Poiseuille flow in pipes, annuli, and channels


Authors: D. D. Joseph and S. Carmi
Journal: Quart. Appl. Math. 26 (1969), 575-599
DOI: https://doi.org/10.1090/qam/99836
MathSciNet review: QAM99836
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Abstract | References | Additional Information

Abstract: The value of $ R = 180$ which has been given by Orr [1] as a limit for sure stability of Hagen-Poiseuille flow is incorrect. A lower value, $ R = 82.88$, can be associated with an eigenfunction possessing a first mode azimuthal variation $ \left( {N = 1} \right)$ and no streamwise variation. This eigenfunction is obtained as an exact solution of the appropriate Euler equation. A yet lower value, $ R = 81.49$, is associated with a spiral mode with $ N = 1$ and wave number $ \alpha \approx 1$. Corresponding results are obtained for Poiseuille flow between cylinders. For all but the very smallest radius ratios the smallest eigenvalue of Euler's equation is assumed for the purely azimuthal disturbance. The mode shape for the pipe flow is consistent with the experimental situation as it is now understood [2], though the stability limit is much smaller than the experimental value $ \left( {R \approx 2100} \right)$. For the annulus, the variation of the energy limit with the radius ratio (from pipe flow to channel flow) is consistent with the experimentally observed stability limit. All the results of the energy analysis hold equally if the pipe is in rigid rotation about its axis. If the rotation is ``fast'' linear and energy results nearly coincide.


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  • [1] W. McF. Orr, The stability or instability of steady motions of a liquid, Part II: A viscous liquid, Proc. Roy. Irish. Acad. Sect. (A) 27, 69-138 (1907)
  • [2] J. Fox, M. Lessen and W. Bhat, Experimental investigation of the stability of Hagen-Poiseuille flow, Phys. Fluids. 11, 1 (1968)
  • [3] O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. Trans. Roy. Soc. London Ser. A 186, 123-164 (1895)
  • [4] J. Serrin, On the stability of viscous fluid motions, Arch. Rational Mech. Anal. (1) 3, 1-13 (1959) MR 0105250
  • [5] D. D. Joseph, Nonlinear stability of the Boussinesg equations by the method of energy, Arch. Rational Mech. Anal. (3) 22, 163-184 (1966) MR 0192725
  • [6] F. Reisz, and B. Sz.-Nagy, Lecons d'analyse fonctionnelle, Chap. VI, Akad. Kiado, Budapest, 1925
  • [7] W. Velte, Über ein Stabilitatäkriterium der Hydrodynamik, Arch. Rational Mech. Anal. 9, 9-20 (1962) MR 0155501
  • [8] Von Klaus Kirchgässner, Die Instabilität der Stromüng zwischen zwer rotierenden Zylindern gegenüber Taylor-Wirbeln für beliebige Spaltbreiten, Z. Angew. Math. Phys. 12, 14-30 (1961) MR 0140239
  • [9] R. Jentzsch, Über Integralgleichungen mit positivem Kern, J. Math. 141, 235 (1912) MR 1580852
  • [10] F. Busse, Bounds on turbulent transport of mass and momentum, Z. Angew. Math. Phys.
  • [11] L. Y. Hung, Stability of Couette flow by the method of energy, M.S. thesis, Department of Aeronautics and Engineering Mechanics, University of Minnesota (1968)
  • [12] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966 MR 0203473
  • [13] H. Salwen and E. Grosch, Stability of Poiseuille flow in a circular pipe, Bull. Amer. Phys. Soc. (Abstract) 13, 814 (1968)
  • [14] M. Lessen, S. Sadler and T. Liu, Stability of pipe Poiseuille flow, Phys. Fluids. 11, 1404-1409 (1968)
  • [15] R. J. Leite, An experimental investigation of the stability of Poiseuille flow, J. Fluid Mech. 5, 81 (1959)
  • [16] T. J. Pedley, On the instability of viscous flow in a rapidly rotating pipe, J. Fluid Mech. (in press)
  • [17] R. Hanks, The laminar-turbulent transition for flow in pipe, concentric annuli, and parallel plates, A. I. Ch. E. J. 9 (1963), 45-48
  • [18] J. Mott and D. Joseph, Stability of parallel flow between concentric annuli, Phys. Fluids (1968), (to appear)
  • [19] D. D. Joseph and L. N. Tao, Transverse velocity components in fully-developed flows, J. Appl. Math. Mech. 30, 147-148 (1963)
  • [20] E. L. Ince, Ordinary differential equations, Dover, New York, 1956, p. 405 MR 0010757
  • [21] C. C. Lin, On the stability of two-dimensional parallel flow, Quart. Appl. Math. 3, 277 (1966) MR 0014894


Additional Information

DOI: https://doi.org/10.1090/qam/99836
Article copyright: © Copyright 1969 American Mathematical Society

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