Abstract
Starting from the Navier-Stokes equations we treat the onset of the wavy vortex flow of the Taylor problem in a fully nonlinear manner. To this end we first transform the original partial differential equations into a set of ordinary differential equations by means of a Galerkin method. By a specific choice of the basic functions the Galerkin coefficients acquire a very concise form. Since these functions fulfil the boundary conditions individually, there are no additional constraints on these coefficients, which usually stem from the boundary conditions. We then first calculate the Taylor vortex flow and perform a linear stability analysis of it. Finally we calculate the coefficients of the generalized Ginzburg-Landau equations in the vicinity of the second threshold and solve these equations.
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Marx, K., Haken, H. Numerical derivation of the generalized Ginzburg-Landau equations of wavy vortex flow. Z. Physik B - Condensed Matter 75, 393–411 (1989). https://doi.org/10.1007/BF01321827
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DOI: https://doi.org/10.1007/BF01321827