Nonlinear Convection in Binary Mixtures
The experimental observation of travelling wave convection1 has stimulated much of the recent work, both experimental and theoretical, on convection in binary fluids. Two systems have been extensively studied experimentally: 3He-4He mixtures above the λ-point2, and water-ethanol mixtures3,4. Of these the former cannot be visualized, and the dynamics has to be inferred from point measurements. The experiments reveal the existence of both time-independent patterns5, and a variety of time-dependent travelling waves 3,4. The latter are small amplitude states that come into existence near the Hopf bifurcation from the pure conduction state. This bifurcation occurs in binary fluid mixtures characterized by a sufficiently negative separation ratio S. This ratio provides a measure of the stabilizing concentration gradient set up in response to a destabilizing temperature gradient by the (negative) Soret effect. With increasing Rayleigh number the conduction state loses stability to growing oscillations provided the restoring force due to the concentration gradient is sufficiently strong to overcome viscous dissipation. The instability occurs because heat diffuses faster than concentration, setting up a phase difference between the concentration and temperature fields, which persists into the nonlinear regime, regardless of whether the instability evolves into a travelling or standing pattern6.
KeywordsRayleigh Number Hopf Bifurcation Bifurcation Diagram Global Bifurcation Oscillatory Convection
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- 3.V. Steinberg, E. Moses and J. Fineberg: “Spatio-temporal complexity at the onset of convection in a binary fluid.” Nuclear Phys. B (Proc. Suppl.) 2: 109 (1987)Google Scholar
- 15.E. Knobloch and D. R. Moore: “A minimal model of binary fluid convection.” Preprint (1989)Google Scholar
- 21.E. Knobloch and D. R. Moore: “Particle drifts associated with travelling wave convection in binary fluid mixtures.” Bull. Amer. Phys. Soc. 33: 370 (1988)Google Scholar
- 23.E. Knobloch and J. B. Weiss: Mass transport by wave motion, in “The Internal Solar Angular Velocity,” B. R. Durney and S. Sofia, eds., Reidel, Dordrecht (1987)Google Scholar
- 24.D. R. Moore and E. Knobloch: “Nonlinear convection in binary mixtures.” Bull. Amer. Phys. Soc. 33: 2285 (1988) and preprint (1989)Google Scholar
- 30.J. W. Swift: “Bifurcation and symmetry in convection.” Ph.D. Thesis, University of California, Berkeley (1984)Google Scholar
- 31.M. Silber: “Bifurcations with D 4 symmetry and spatial pattern selection.” Ph.D. Thesis, University of California, Berkeley (1989)Google Scholar
- 35.E. Knobloch: Nonlinear binary fluid convection at positive separation ratios, in “Cooperative Dynamics in Complex Systems,” H. Takayama, ed., Springer-Verlag, Berlin (1989)Google Scholar
- 36.G. Dangelmayr and E. Knobloch: On the Hopf bifurcation with broken O(2) symmetry, in “The Physics of Structure Formation: Theory and Simulation,” W. Güttinger and G. Dangelmayr, eds., Springer-Verlag, Berlin (1987); Dynamics of slowly varying wave trains in finite geometry, this volumeGoogle Scholar