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1994 (1)
1993 (1)
1Author    Dirk-A BeckerRequires cookie*
 Title    A Simplified Algorithm for Finding Partially Invariant Solutions of Quasilinear Systems  
 Abstract    The method of partially invariant solutions of PDE systems was introduced by Ovsiannikov as a generalization of the classical similarity analysis. It offers a possibility to calculate exact solutions possessing a higher degree of freedom than similarity solutions. Ovsiannikov's algorithm, however, is somewhat hard to apply because one has to deal with three equation systems derived from the original PDE system. By means of the two-dimensional Euler equations, we show how the algorithm can be essentially simplified if classical similarity solutions are already known. Further, we prove a necessary criterion for the simplified algorithm to be senseful. 
  Reference    Z. Naturforsch. 49a, 458—464 (1994); received October 7 1993 
  Published    1994 
  Keywords    Hydrodynamics, Nonlinear systems, Group theory 
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 TEI-XML for    default:Reihe_A/49/ZNA-1994-49a-0458.pdf 
 Identifier    ZNA-1994-49a-0458 
 Volume    49 
2Author    Georg Knorr, Gerhard Ströhmer, ChristophK. Goertz, DwightR. Nicholson, Linhua Shan, RobertA. SmithRequires cookie*
 Title    Hilbert Space Methods in Hydrodynamics with Applications to Couette Flow  
 Abstract    , 1993 This paper is dedicated to the memory of our friends and colleagues, 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. 
  Reference    Z. Naturforsch. 48a, 679—691 (1993); received February 17 
  Published    1993 
  Keywords    Hydrodynamics, Eigenvector field, Self-adjoint operator, Couette flow, Taylor vortices 
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 TEI-XML for    default:Reihe_A/48/ZNA-1993-48a-0679.pdf 
 Identifier    ZNA-1993-48a-0679 
 Volume    48