The oscillatory instability of a spatially homogeneous state in large aspect ratio systems of fluid dynamics

We derive the generalized Ginzburg-Landau equation for the case of an oscillatory instability of a spatially homogeneous state in systems whose geometry is characterized by two entirely different length scales. This evolution equation is applied to describe the spatio-temporal behaviour of the onset of convection in binary fluid mixtures in large aspect ratio systems. We obtain time periodic traveling wave motions, quasiperiodic fluid motions with two and more frequencies modulating the intensities of the traveling waves as well as chaotic temporal behaviour.

[1]  Harry L. Swinney,et al.  Flow regimes in a circular Couette system with independently rotating cylinders , 1986, Journal of Fluid Mechanics.

[2]  E. M. Lifshitz,et al.  Course in Theoretical Physics , 2013 .

[3]  P. Kolodner,et al.  The transition to finite-amplitude traveling-wave convection in binary fluid mixtures , 1987 .

[4]  P. Hohenberg,et al.  Codimension-2 bifurcations for convection in binary fluid mixtures , 1984 .

[5]  H. Haken,et al.  Transient patterns of the convection instability: A model-calculation , 1984 .

[6]  H. Snyder.  Stability of Rotating Couette Flow. I. Asymmetric Waveforms , 1968 .

[7]  J. K. Platten,et al.  Convection in Liquids , 1983 .

[8]  F. Takens,et al.  Occurrence of strange AxiomA attractors near quasi periodic flows onTm,m≧3 , 1978 .

[9]  藤村 薫.  J. K. Platten and J. C. Legros: Convection in Liquids, Springer-Verlag, Berlin and New York, 1984, xiv+680ページ, 25×17.5cm, DM180. , 1985 .

[10]  Gil,et al.  Topological defects of wave patterns. , 1987, Physical review letters.

[11]  Walden,et al.  Traveling waves and chaos in convection in binary fluid mixtures. , 1985, Physical review letters.

[12]  E. Jakeman,et al.  Soret-driven thermosolutal convection , 1971, Journal of Fluid Mechanics.

[13]  E. Hopf.  A mathematical example displaying features of turbulence , 1948 .

[14]  F. Takens,et al.  On the nature of turbulence , 1971 .

[15]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

[16]  Knobloch,et al.  Oscillatory convection in binary mixtures. , 1986, Physical review. A, General physics.

[17]  M. Golubitsky,et al.  Hopf Bifurcation in the presence of symmetry , 1984 .

[18]  A. Newell,et al.  Evolution of the order parameter in situations with broken rotational symmetry , 1986 .

[19]  Donald A. Nield,et al.  The thermohaline Rayleigh-Jeffreys problem , 1967, Journal of Fluid Mechanics.

[20]  Rehberg,et al.  Experimental observation of a codimension-two bifurcation in a binary fluid mixture. , 1985, Physical review letters.

[21]  Edward A. Spiegel,et al.  Intermittency through modulational instability , 1983 .

[22]  R. Heinrichs,et al.  Traveling waves and spatial variation in the convection of a binary mixture , 1987 .

[23]  D. Gutkowicz–Krusin,et al.  Rayleigh–Bénard instability in nonreactive binary fluids. I. Theory , 1979 .

[24]  Cross,et al.  Traveling and standing waves in binary-fluid convection in finite geometries. , 1986, Physical Review Letters.

[25]  Knobloch,et al.  Traveling waves and chaos in thermosolutal convection. , 1987, Physical review. A, General physics.