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.
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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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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