Analytic expressions for the stochastic amplitude equation for taylor-couette flow

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 53, Issue 1, 1996, Pages 577-585

  Treiber, Martin ^a  

^a UNIVERSITY OF BAYREUTH   (Germany)

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EID: 4243348797     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.53.577     Document Type: Article

References (34)

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11. Strictly speaking, this criterion is valid only for stationary bifurcations citecross hohenberg. Presumably it can be applied also for traveling patterns if the direction of the propagation is perpendicular to the direction of the width.
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23. M. Treiber, in Pattern Formation in Liquid Crystals, edited by L. Kramer and A. Buka (Springer, New York, 1995).
24. For ε approx εNL propto Q2/3, Eq. ( refagl) would lead to amplitude fluctuations where the nonlinear terms proportional to -A2 A become of the same order of magnitude as the linear terms;
25. see P. C. Hohenberg and J. B. Swift, Phys. Rev. A 46, 4773 (1992).
26. In the TC system, εNL is of the order of 10-7.
27. The result of SBH can be directly obtained by taking as basic equations for ( refQ) and ( refO) the curl of the NS equations [instead of ( refbasiceqtaylor)], giving three equations for vr,vφ, and vz with the fluctuating forces ξi = [ bbox nabla times( bbox nabla tilde underline bboxσ)]i, where i=r,φ, or z.
28. This is plausible because for this ratio the Landau expressions citelandau hyd for fluctuations of the stress tensor in compressible fluids, used in citedeissler TC, give vanishing fluctuations of the diagonal components. In incompressible fluids, these pressure fluctuations cannot lead to fluctuations of the observable quantities.
29. P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1982), p. 98.
30. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
31. For η=0.738, the exact second order expressions correspond to ω0 defined as in Eq. ( refapprox), but with δ=0.203 instead of 3(1-η)/4=0.197. The first order approximation corresponds to δ=0.
32. Nonzero through flow has very little influence on these parameters. Within the range where axisymmetric modes become first unstable, the influence on Tc and kc is smaller than 0.7 citebabcock TCexp. Similar orders of magnitude apply for τ0 and ξ0sup 2.
33. Note, however, that in citebabcock TCtheo the amplitude is related to vz instead of vr and the scalar product is defined differently as well.
34. M. Treiber (unpublished).