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An analysis of transient behavior in the onset of convection

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Part of the Lecture Notes in Mathematics book series (LNM,volume 594)

Abstract

A layer of viscous fluid lying between two heated horizontal, parallel plates is considered. When a critical adverse temperature gradient is established across the layer, heat is no longer transferred by conduction alone, and a convective motion is established. We model this physical phenomenon with the Boussinesq approximation to the Navier-Stokes equations, and using this mathematical model, we study the evolution of a single periodic disturbance for values of the adverse temperature gradients near the critical value. A method of matched asymptotic expansions is used to construct the time dependent solutions of the system. In the course of this analysis, the Landau equation is rigorously derived and a domain of stability of the convective state is determined. Another result of this analysis is the rigorous justification of a perturbation method which is quite similar to that introduced by Stuart and Watson and other investigators.

Keywords

  • Rayleigh Number
  • Boussinesq Approximation
  • Landau Equation
  • Newton Polygon
  • Perturbation Scheme

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.

This research was partially supported by the Faculty Research Award Program of CUNY under Grant No. 10618 (N.G.) and by the Air Force Office of Scientific Research under Grant No. AFOSR-71-2107 (F.C.H.).

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Bibliography

  • Chandrasekhar, S., (1961) Hydrodynamic and Hydromagnetic Stability, Oxford University Press.

    Google Scholar 

  • Fujita, H., and Kato, T., (1964) On the Navier-Stokes Initial Value Problem I, Arch. Rational Mech. Anal, Vol. 16 #4, pp. 269–315.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hoppensteadt, F., and Gordon, N., (1975) Asymptotic Solutions of Nonlinear Partial Differential Equations, Comm. Pure Appl. Math. (In press).

    Google Scholar 

  • Iooss, G., (1971) Nonlinear stability theory of laminar flows in the case of exchange of stabilities, Xth Biennial Fluid Dynamics Symposium Rynia (Poland) T. P. No. 999, Office National d'Etudes et de Recher. Aerospatiales.

    Google Scholar 

  • Iudovich, V. I., (1967) Free Convection and Bifurcation, J. Appl. Math. Mech. Vol. 31 #1, pp. 101–111.

    Google Scholar 

  • Kirchgassner, K., and Sorger, P., (1969) Branching Analysis for the Taylor Problem, Quart. J. Mech. Appl. Math., Vol. 22 #2, pp. 183–210.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Kirchgassner, K., and Kielhoffer, H., (1973) Stability and Bifurcation in Fluid Mechanics, Rocky Mt. J. of Math., Vol. 3 #2, pp. 275–318.

    CrossRef  Google Scholar 

  • Kogelman, S., and Keller, J. B., (1971) Transient Behavior of Unstable Nonlinear Systems with Application to the Bénard and Taylor Problems, SIAM J. Appl. Math., Vol. 20 #4, pp. 619–637.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Krishnamurti, R., (1968) Finite Amplitude Convection with Changing Mean Temperature, J. Fluid Mech., Vol. 33 #3, pp. 445–464.

    CrossRef  MATH  Google Scholar 

  • Ladyshenskaya, O. A., (1965) Functional Analytische Untersuchungen der Navier-Stokesschen Gleichungen, Akademie-Verlag, Berlin.

    Google Scholar 

  • Matkowsky, B. J., (1970) Nonlinear Dynamic Stability: A Formal Theory, SIAM J. Appl. Math., Vol. 18 #4, pp. 872–883.

    CrossRef  MATH  Google Scholar 

  • Rabinowitz, P. H., (1968) Existence and Nonuniqueness of Rectangular Solutions of the Bénard Problem, Arch. Rational Mech. Anal., Vol. 29, pp. 32–57.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Segel, L. A., (1965) Nonlinear Hydrodynamic Stability Theory and Its Applications to Thermal Convection and Curved Flows, pp. 165–198, Non-equilibrium Thermodynamics Variational Techniques and Stability, ed. Donally, R. I., Herman, P., and Prigogine, I., Univ. of Chicago Press.

    Google Scholar 

  • Sobolevskii, P. E., (1966) Equations of Parabolic Type in Banach Space, A.M.S. Trnsl. ser. 2, Vol 49, pp. 1–62.

    Google Scholar 

  • Stuart, J. T., (1960), On the Nonlinear Mechanisms of Wave Disturbances in Stable and Unstable Parallel Flow, J. Fluid Mech. 9, 352–370.

    CrossRef  Google Scholar 

  • Watson, J., (1960), On the Nonlinear Mechanisms of Wave Disburbances in Stable and Unstable Parallel Flow J. Fluid Mech. 9, 371–389.

    CrossRef  MathSciNet  Google Scholar 

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© 1977 Springer-Verlag

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Gordon, N., Hoppensteadt, F.C. (1977). An analysis of transient behavior in the onset of convection. In: Brauner, CM., Gay, B., Mathieu, J. (eds) Singular Perturbations and Boundary Layer Theory. Lecture Notes in Mathematics, vol 594. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086089

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  • DOI: https://doi.org/10.1007/BFb0086089

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08258-3

  • Online ISBN: 978-3-540-37340-7

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