DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 513-522

Hisashi Okamoto

Title: A Study of Bifurcation of Kolmogorov Flows with an Emphasis on the Singular Limit

We consider a family of stationary Navier-Stokes flows in 2D flat tori. The flow is driven by an outer force which is of the form $(\sin y, 0)$. Varying the Reynolds number and the aspect ratio of the torus, we numerically compute bifurcating solutions by a path-continuation method. Folds and cusps are obtained in the range where the Reynolds number is $< 100 $. Some solutions are computed up until the Reynolds number becomes 10,000. Asymptotic properties as the Reynolds number tends to infinity are discussed. Also given is an analysis as the aspect ratio of the torus tends to zero.

1991 Mathematics Subject Classification: Primary 76D30; Secondary 76C05, 35Q30, 35Q35.

Keywords and Phrases: Kolmogorov flows in 2D tori, incompressible fluid, bifurcation, singular perturbation, internal layer, inviscid limit.

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