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

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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.