By C. J. Amick (auth.), Reimund Rautmann (eds.)
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Extra resources for Approximation Methods for Navier-Stokes Problems: Proceedings of the Symposium Held by the International Union of Theoretical and Applied Mechanics (IUTAM) at the University of Paderborn, Germany, September 9 – 15, 1979
Sample text
In conclusion we should like to note tkat the extension of t~e above results to the case ~ ~ is of great importance. True, one may not wait for the theory to be developed but attack the problem on the stability of a flow past a circular cylinder numerically to clear up the question whether the periodic Karman flow is a bifurcation of the initial stationary one. The numerical procedure might be based on the above series. This method would be safe for Reynolds numbers not very much differing from ~C " @ 40 BIBLIOGRAPHY I.
C(R)'C~ , by Pro- is an analytical function of . 2) can have n ~ - t r i v i a l solutions only for discrete values 36 of ~ 6 A . Such values of O ~ will be called eigen- values. 2) has a non-trivial solution in ~/~ (~) O . I) and its decrease at infinity is defizLed by the fact that respective classes /,~(~) 7/" and p belong to the . I) ~ v o l v e the energy relation Separatimg the real and imaginary parts, we obtai~ where {$-= ~ % . It follows from these relations , eC' ~--- I " ~ ~ ' that A where /~ is some constant whose value is defined by the sta- tionary solution.
He~ce there exists -net for this image and so for the set ~1-~too there exists a finite proves the compactness of 3. 2). Theorem 3. I). The proof of this theorem is not simple at all so the reader is referred to paper [I8] . 4) has a non-trivial solution or if R --~ ~ o R ~ 0 . 2) is satisfied. The above propositions permit to solve rather simply the question about the bifurcation of the solution in the case when equation tions in ~/~ (] - - K ) TJ'= ~ has non-trivial solu- , of course under assumptions imposed by the theo- ry.