Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Stability and bifurcation in a modulated Burgers system

Authors: W. E. Olmstead and S. H. Davis
Journal: Quart. Appl. Math. 39 (1982), 467-477
MSC: Primary 35B32; Secondary 35Q20, 58E07, 76E99
DOI: https://doi.org/10.1090/qam/644101
MathSciNet review: 644101
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Abstract: The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude $ A\left( \tau \right)$ of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate $ \theta \left( \tau \right)$. Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in $ \theta \left( \tau \right)$. For each of these oscillatory-type modulations, it is found that $ {A^2}\left( \tau \right)$ has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

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  • [1] J. M. Burgers, Mathematical example illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, 1-53 (1939)
  • [2] J. M. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in applied mechanics (R. von Mises and T. von Kármán, editors) 1, Academic Press, New York, 171-199 (1948) MR 0027195
  • [3] G. B. Whitham, Linear and nonlinear waves, John Wiley & Sons, 1974 MR 0483954
  • [4] W. Eckhaus, Studies in nonlinear stability theory, Springer Tracts in Natural Philosophy, Vol. 6, Springer-Verlag, Berlin, 1965 MR 0196298
  • [5] C. Golia and J. M. Abel, Path integral synthesis of Lyapunov functionals for partial differential equations, Int. J. Nonlinear Mech. 10, 333-334 (1975) MR 0430489
  • [6] C. O. Horgan and W. E. Olmstead, Stability and uniqueness for a turbulence model of Burgers, Quart. Appl. Math. 36, 121-127 (1978) MR 0495602
  • [7] S. H. Davis, The stability of time-periodic flows, Ann. Rev. Fluid Mech. 8, 57-74 (1976)
  • [8] S. H. Davis, Finite amplitude instability of time-dependent flows, J. Fluid Mech. 45, 33-48 (1971)
  • [9] R. C. DiPrima and J. T. Stuart, Nonlocal effects in the stability of flow between eccentric rotating cylinders, J. Fluid Mech. 54, 393-416 (1972)
  • [10] P. Hall, The stability of unsteady cylinder flows, J. Fluid Mech. 67, 29-64 (1975)
  • [11] L. A. Rubenfeld, A model bifurcation problem exhibiting the effects of slow passage through critical, SIAM J. Appl. Math. 37, 302-306 (1979) MR 543949
  • [12] B. J. Matkowsky, A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., 76, 620-625 (1970). MR 0257544
  • [13] W. C. Reynolds and M. C. Potter, Finite-amplitude instability of parallel shear flows, J. Fluid Mech. 27, 465-492 (1967)
  • [14] A. Davey, The growth of Taylor vortices in flow between rotating cylinders, J. Fluid Mech. 14, 336-368 (1962) MR 0145788
  • [15] L. A. Segal and J. T. Stuart, On the question of the preferred mode in cellular thermal convection, J. Fluid Mech. 13, 289-306 (1962) MR 0140237

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DOI: https://doi.org/10.1090/qam/644101
Article copyright: © Copyright 1982 American Mathematical Society

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