Hilbert Space Methods in Hydrodynamics with Applications to Couette Flow

Z. Naturforsch. 48 a, 679-691 (1993); received February 17, 1993 This paper is dedicated to the memory of our friends and colleagues Christoph K. Goertz, Dwight R. Nicholson, Linhua Shan, and Robert A. Smith, who were tragically slain on November 1, 1991 A new base for three-dimensional divergence-free vector fields in Hilbert space (Galerkin method) is proposed. The base fields have vanishing boundary conditions for their curl and are useful to solve the incompressible Navier-Stokes equation. The base vector potentials are obtained as the eigensolutions of the squared Laplace operator. We first derive the operator in a simply connected domain and then study Couette flow in the small gap approximation. The method yields a rapidly converging critical Taylor number and in lowest approximation a three mode model for the Taylor vortices, similar to the Lorenz model. It represents the first bifurcation of the flow very well.