HILBERT-SPACE METHODS IN HYDRODYNAMICS WITH APPLICATIONS TO COUETTE-FLOW

A new base for three-dimensional divergence-free vector fields in Hilb ert space (Galerkin method) is proposed. The base fields have vanishin g boundary conditions for their curl and are useful to solve the incom pressible Navier-Stokes equation. The base vector potentials are obtai ned as the eigensolutions of the squared Laplace operator. We first de rive the operator in a simply connected domain and then study Couette flow in the small gap approximation. The method yields a rapidly conve rging critical Taylor number and in lowest approximation a three mode model for the Taylor vortices, similar to the Lorenz model. It represe nts the first bifurcation of the flow very well.