Advertisement

On the Place of Energy Methods in a Global Theory of Hydrodynamic Stability

  • D. D. Joseph
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Abstract

The point of departure for the global theory to be described is the system of the nonlinear Boussinesq equations (1, 2) governing the disturbance of some given motion. For simplicity, let (U, T, T) be a basic steady velocity, temperature and concentration (say, salt) field which satisfies the Boussinesq equations in a bounded domain (or in a period cell) and such that these fields take on prescribed values on ∂V (or periodic values on appropriate parts of ∂V ). Let (u,0 6, y) be disturbances of the given motion which are induced at time zero as initial conditions. Subsequently [30]
$$\left( {u,\,\theta ,\,\gamma } \right)/\partial V = 0,\,\,\,\,\,div\,u = 0/V,$$
(1a,b)
$$\frac{{du}}{{dt}} + R\left( {u\cdot\nabla } \right)U + \left( {u\cdot\nabla } \right)u = - \nabla P - \left( {R\theta - \ell \gamma } \right)n + \Delta u,$$
(2a)
$${P_T}\frac{{d\theta }}{{dt}} + {P_T}u \cdot \nabla \theta + Ru \cdot {\eta _T} = \Delta \theta ,$$
(2b)
and
$${P_T}\frac{{d\gamma }}{{dt}} + {P_\Gamma }u \cdot \nabla \gamma + \ell u \cdot {\eta _T} = \Delta \gamma ,$$
(2c)
where
$$\frac{d}{{dt}} = \frac{\partial }{{\partial t}} + RU \cdot \nabla .$$

Keywords

Global Stability Couette Flow Energy Method Fluid Layer Poiseuille Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Busse, F.: Dissertation, Munich 1962. Cf. also The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 4, 225 (1968).Google Scholar
  2. 2.
    Carothers, S.: Portland experiments on the flow of oil in tubes. Proc. Roy. Soc. 87, 154 (1912).CrossRefGoogle Scholar
  3. 3.
    Davis, S.: Surface tension driven convection by the method of energy. J. Fluid Mech. 39, 347–359 (1969).MATHCrossRefGoogle Scholar
  4. 4.
    Davies, S. J., White, C. M.: An experimental study of the flow of water in pipe of rectangular section. Proc. Roy. Soc. 119, 92 (1928).MATHCrossRefGoogle Scholar
  5. 5.
    Fife, P., Joseph, D.: Existence of convective solutions of the generalized Benard problem which are analytic in their norm. Arch. Rat. Mech. Anal. 33, 116 (1969).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fox, J., Lessen, M., Bhat, W.: Experimental investigation of the stability of Hagen-Poiseuille flow. Phys. Fluids 11, 1 (1968).CrossRefGoogle Scholar
  7. 7.
    Grindley, J., Gibson, A.: On the frictional resistances to the flow of air through a pipe. Proc. Roy. Soc. 80, 114 (1907/1908).MATHCrossRefGoogle Scholar
  8. 8.
    Hanks, R.: The laminar-turbulent transition for flow in a pipe, concentric annuli, and parallel plates. A.I.Ch.E.J. 9, 45 (1963).Google Scholar
  9. 9.
    Harrison, W.: On the stability of the steady motion of viscous liquid con-tained between two rotating circular cylinders. Trans. Camb. Phil. Soc. 22, 455 (1921).Google Scholar
  10. 10.
    Hopf, E.: On nonlinear partial differential equations. Lecture series of the Symposium on partial differential equations. University of California 1955, pp. 7–11.Google Scholar
  11. 11.
    Joseph, D.: On the stability of the Boussinesq equations. Arch. Rat. Mech. Anal. 20, 59 (1965).MATHCrossRefGoogle Scholar
  12. 12.
    Joseph, D.: Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Rat. Mech. Anal. 22, 163 (1966).MATHCrossRefGoogle Scholar
  13. 13.
    Joseph, D.: Uniqueness criteria for the conduction-diffusion of the Boussinesq equations. Arch. Rat. Mech. Anal. 35, 169–177 (1969).MATHCrossRefGoogle Scholar
  14. 14.
    Joseph, D., Shir, C. C.: Subcritical convective instability. Parti: Fluid layers heated from below and internally. J. Fluid Mech. 26, 753–768 (1966).MATHCrossRefGoogle Scholar
  15. 15.
    Joseph, D., Carmi, S.: Part 2: Spherical shells. J. Fluid Mech. 26, 769 (1966).MATHCrossRefGoogle Scholar
  16. 16.
    Joseph, D., Carmi, S.: Stability of Poiseuille flow in pipes, annuli and channels. Quart. Appl. Math. 26, 575–599 (1969).MATHGoogle Scholar
  17. 17.
    Joseph, D., Goldstein, R., Graham, D.: Subcritical instability and exchange of stability in a horizontal fluid layer. Phys. Fluids 11, 903 (1968).MATHCrossRefGoogle Scholar
  18. 18.
    Krishnamurti, R.: Finite amplitude convection with changing mean tem-perature. Part I: Theory. J. Fluid Mech. 88, 445 (1968).CrossRefGoogle Scholar
  19. 19.
    Lindgren, R.: The transition process and other phenomena in viscous flow. Arkiv for Fysik 12, 1–169 (1957).Google Scholar
  20. 20.
    Lindgren, R.: Liquid flow in tubes. Parti: The transition process under highly disturbed entrance flow conditions. Arkiv for Fysik 15, 97 (1959).Google Scholar
  21. 21.
    Mott, J., Joseph, D.: Stability of parallel flow between concentric cylinders. Phys. Fluids 11, 2065 (1968).MATHCrossRefGoogle Scholar
  22. 22.
    Naumann, A.: Experimentelle Untersuchungen über die Entstehung der turbulenten Rohrströmung. Forsch. Geb. Ingw. 2, 85 (1931).CrossRefGoogle Scholar
  23. 23.
    Orr, W. MCF.: The stability or instability of steady motions of a liquid. Part II: A viscous liquid. Proc. Roy. Irish Acad. A 27, 69 (1907).Google Scholar
  24. 24.
    Payne, L., Weinberger, H.: An exact stability bound for Navier-Stokes flow in a sphere. Nonlinear problems, ed. R. E. Langer, University of Wisconsin Press 1963.Google Scholar
  25. 25.
    Pedley, T.: On the stability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 85, 97 (1969).CrossRefGoogle Scholar
  26. 26.
    Reynolds, O.: On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. Roy. Soc. London A 186, 123 (1895).MATHCrossRefGoogle Scholar
  27. 27.
    Sani, R.: On finite amplitude roll cell disturbances in a fluid layer subjected to heat and mass transfer. A.I.Ch.E.J. 11, 971 (1965).MathSciNetGoogle Scholar
  28. 28.
    Serrin, J.: On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1 (1959).MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Shir, C. C.: Ph. D. thesis, University of Minnesota, 1967.Google Scholar
  30. 30.
    Shir, C. C., Joseph, D.: Convective instability in a temperature and concen-tration field. Arch. Rat. Mech. Anal. 30, 38 (1968).MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Sorger, P.: Über ein Variationsproblem aus der nichtlinearen Stabilitäts-theorie zäher inkompressibler Strömungen. Z. angew. Math. Phys. 17, 201 (1966).MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Thomas, T. Y.: On the uniform convergence of the solutions of the Navier- Stokes equations. Proc. Nat. Acad. Sei. U.S.A. 29, 243 (1943).MATHCrossRefGoogle Scholar
  33. 33.
    Türner, J., Stommel, A.: A new case of convection in the presence of combined vertical salinity and temperature gradient. Proc. Nat. Acad. Sei. U.S.A. 52, 49–53 (1964).CrossRefGoogle Scholar
  34. 34.
    Velte, W.: Über ein Stabilitätskriterium der Hydrodynamik. Arch. Rat. Mech. Anal. 9, 9 (1962).MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Veronis, G.: On finite amplitude instability in thermohaline convection. J. Marine Res. 23, 1 (1965).Google Scholar
  36. 36.
    Veronis, G.: Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315 (1968).MATHCrossRefGoogle Scholar
  37. 37.
    Westbrook, D.: The stability of convective flow in a porous medium. Phys. Fluids 12, 1547–1551 (1969).CrossRefGoogle Scholar
  38. 38.
    Joseph, D. D., Munson, B.: Global stability of spiral flow. J. Fluid Mech. (in press).Google Scholar
  39. 39.
    Joseph, D. D.: Global stability of the conduction-diffusion solution. Arch. Rat. Mech. Anal. 36, 285–292 (1970).MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag, Berlin/Heidelberg 1971

Authors and Affiliations

  • D. D. Joseph
    • 1
  1. 1.Imperial College of Science and TechnologyLondonUK

Personalised recommendations