Analytic expressions for the stochastic amplitude equation for Taylor-Couette flow

Martin Treiber

doi:10.1103/PhysRevE.53.577

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Analytic expressions for the stochastic amplitude equation for Taylor-Couette flow

Martin Treiber

Physikalisches Institut, Universität Bayreuth, Postfach 101251, D-95440 Bayreuth, Germany

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Phys. Rev. E 53, 577 – Published 1 January, 1996

DOI: https://doi.org/10.1103/PhysRevE.53.577

Abstract

Expressions for the thermal noise strength in stochastic amplitude equations of quasi-one-dimensional hydrodynamic systems near a pattern-forming instability are given in a generally applicable form. The expressions can be evaluated for any system where the macroscopic equations of motion and the entropy production in terms of generalized forces and fluxes are known and the linear stability analysis can be performed. We apply the results mainly to the Taylor-Couette system (with corotating cylinders) and derive an approximate analytic expression for the noise strength of the amplitude equation near the first threshold. An analytical expression is given also for the mean kinetic energy of the velocity fluctuations. The analytic formulas are easy to evaluate and are of second order in the gap width; their deviation from numerical results is less than 2.5% for a radius ratio of 0.738 and all (co) rotation rates of the outer cylinder. By comparing the mean energy of the fluctuations with the equipartition theorem, we separate equilibrium from nonequilibrium effects. © 1996 The American Physical Society.

References (32)

I. Rehberg, S. Rasenat, M. de la Torre Juarez, W. Schöpf, F. H. Hörner, G. Ahlers and H. Brand, Phys. Rev. Lett. 67, 596 (1991).
I. Rehberg, F. H. Hörner, L. Chiran, H. Richter and B. L. Winkler, Phys. Rev. A 44, 7885 (1991).
M. Wu, G. Ahlers and D. S. Cannell, Phys. Rev. Lett. 75, 1743 (1995).
G. Quentin and I. Rehberg, Phys. Rev. Lett. 74, 1578 (1995).
C. W. Meyer, G. Ahlers and D. S. Cannell, Phys. Rev. Lett. 59, 1577 (1987).
C. W. Meyer, G. Ahlers and D. S. Cannell, Phys. Rev. A 44, 2514 (1991).
W. Schöpf and I. Rehberg, J. Fluid Mech. 2717, 235 (1994).
K. L. Babcock, G. Ahlers and D. S. Cannell, Phys. Rev. E 50, 3670 (1994).
A. Tsameret and V. Steinberg, Phys. Rev. E 49, 1291 (1994).
A. Tsameret, E. Goldner and V. Steinberg, Phys. Rev. E 49, 1309 (1994).
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.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford, 1961).
A. Tsameret and V. Steinberg, Phys. Rev. E 49, 4077 (1994).
V. Zaitsev and M. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583 (1970) [Sov. Phys. JETP 32, 866 (1971)].
R. Graham, Phys. Rev. A 10, 1762 (1974).
P. C. Hohenberg and J. B. Swift, Phys. Rev. A 46, 4773 (1992).
W. Schöpf and W. Zimmermann, Phys. Rev. E 47, 1739 (1993).
M. Treiber and L. Kramer, Phys. Rev. E 49, 3184 (1994).
J. Swift, K. Babcock and P. Hohenberg, Physica A 204, 625 (1994).
R. J. Deissler, Phys. Rev. E 49, 31 (1994).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison Wesley, Reading, MA, 1959).
S. DeGroot and P. Mazur, Nonequilibrium Thermodynamics, 2nd ed. (Dover, New York, 1984).
M. Treiber, in Pattern Formation in Liquid Crystals, edited by L. Kramer and A. Buka (Springer, New York, 1995).
For |ε| approx $ε_{NL}$ propto $Q^{2 / 3}$ , Eq. ( refagl) would lead to amplitude fluctuations where the nonlinear terms proportional to -|A $|^{2}$ A become of the same order of magnitude as the linear terms; see P. C. Hohenberg and J. B. Swift, Phys. Rev. A 46, 4773 (1992). In the TC system, $ε_{NL}$ is of the order of $10^{- 7}$ .
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 $v_{r}$ , $v_{φ}$ , and $v_{z}$ with the fluctuating forces $ξ_{i}$ = [ bbox nabla times( bbox nabla tilde underline bboxσ}) $]_{i}$ , where i=r,φ, or z.
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.
P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1982), p. 98.
M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
For η=0.738, the exact second order expressions correspond to $ω_{0}$ 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.
Nonzero through flow has very little influence on these parameters. Within the range where axisymmetric modes become first unstable, the influence on $T_{c}$ and $k_{c}$ is smaller than 0.7 citebabcock TCexp. Similar orders of magnitude apply for $τ_{0}$ and $ξ_{0}$ sup 2.
Note, however, that in citebabcock TCtheo the amplitude is related to $v_{z}$ instead of $v_{r}$ and the scalar product is defined differently as well.
M. Treiber (unpublished).

[c1] I. Rehberg, S. Rasenat, M. de la Torre Juarez, W. Schöpf, F. H. Hörner, G. Ahlers and H. Brand, Phys. Rev. Lett. 67, 596 (1991).

[c2] I. Rehberg, F. H. Hörner, L. Chiran, H. Richter and B. L. Winkler, Phys. Rev. A 44, 7885 (1991).

[c3] M. Wu, G. Ahlers and D. S. Cannell, Phys. Rev. Lett. 75, 1743 (1995).

[c4] G. Quentin and I. Rehberg, Phys. Rev. Lett. 74, 1578 (1995).

[c5] C. W. Meyer, G. Ahlers and D. S. Cannell, Phys. Rev. Lett. 59, 1577 (1987).

[c6] C. W. Meyer, G. Ahlers and D. S. Cannell, Phys. Rev. A 44, 2514 (1991).

[c7] W. Schöpf and I. Rehberg, J. Fluid Mech. 2717, 235 (1994).

[c8] K. L. Babcock, G. Ahlers and D. S. Cannell, Phys. Rev. E 50, 3670 (1994).

[c9] A. Tsameret and V. Steinberg, Phys. Rev. E 49, 1291 (1994).

[c10] A. Tsameret, E. Goldner and V. Steinberg, Phys. Rev. E 49, 1309 (1994).

[c11] 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.

[c12] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford, 1961).

[c13] A. Tsameret and V. Steinberg, Phys. Rev. E 49, 4077 (1994).

[c14] V. Zaitsev and M. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583 (1970) [Sov. Phys. JETP 32, 866 (1971)].

[c15] R. Graham, Phys. Rev. A 10, 1762 (1974).

[c16] P. C. Hohenberg and J. B. Swift, Phys. Rev. A 46, 4773 (1992).

[c17] W. Schöpf and W. Zimmermann, Phys. Rev. E 47, 1739 (1993).

[c18] M. Treiber and L. Kramer, Phys. Rev. E 49, 3184 (1994).

[c19] J. Swift, K. Babcock and P. Hohenberg, Physica A 204, 625 (1994).

[c20] R. J. Deissler, Phys. Rev. E 49, 31 (1994).

[c21] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison Wesley, Reading, MA, 1959).

[c22] S. DeGroot and P. Mazur, Nonequilibrium Thermodynamics, 2nd ed. (Dover, New York, 1984).

[c23] M. Treiber, in Pattern Formation in Liquid Crystals, edited by L. Kramer and A. Buka (Springer, New York, 1995).

[c24] For |ε| approx $ε_{NL}$ propto $Q^{2 / 3}$ , Eq. ( refagl) would lead to amplitude fluctuations where the nonlinear terms proportional to -|A $|^{2}$ A become of the same order of magnitude as the linear terms; see P. C. Hohenberg and J. B. Swift, Phys. Rev. A 46, 4773 (1992). In the TC system, $ε_{NL}$ is of the order of $10^{- 7}$ .

[c25] 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 $v_{r}$ , $v_{φ}$ , and $v_{z}$ with the fluctuating forces $ξ_{i}$ = [ bbox nabla times( bbox nabla tilde underline bboxσ}) $]_{i}$ , where i=r,φ, or z.

[c26] 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.

[c27] P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1982), p. 98.

[c28] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).

[c29] For η=0.738, the exact second order expressions correspond to $ω_{0}$ 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.

[c30] Nonzero through flow has very little influence on these parameters. Within the range where axisymmetric modes become first unstable, the influence on $T_{c}$ and $k_{c}$ is smaller than 0.7 citebabcock TCexp. Similar orders of magnitude apply for $τ_{0}$ and $ξ_{0}$ sup 2.

[c31] Note, however, that in citebabcock TCtheo the amplitude is related to $v_{z}$ instead of $v_{r}$ and the scalar product is defined differently as well.

[c32] M. Treiber (unpublished).

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Analytic expressions for the stochastic amplitude equation for Taylor-Couette flow

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