Oscillations of a fluid in a channel. (English) Zbl 1038.76001

Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (Diss.). i, 68 p. (2003).
A free boundary problem describing the motion of viscous incompressible fluid in three-dimensional rectangular channel is investigated. The author analyses the qualitative behaviour of the flow (oscillations of periodic solutions) by using the bifurcation theory with symmetry. The first chapter deals with the existence of solutions for nonstationary linear and nonlinear problems, and with the spectral behavior of linearized problem \(\partial_t x +Lx=0\) in an appropriate Hilbert space \(X^r\). The second chapter contains the main result, Hopf bifurcation theorem with \(Z_k\)-symmetry for the investigated Navier-Stokes system. Here the results of the article [B. Schweizer, SIAM J. Math. Anal. 28, No. 5, 1135–1157 (1997; Zbl 0889.35075)] are used.
The group O(2) of symmetries is determined by the shape of the domain and by boundary conditions. Using the eigenfunctions of Laplace operator in the rectangle, the author finds an \(L\)-invariant decomposition of spaces \(X^r=\bigoplus X^{r}_{n,k}, n\in N, k\in Z\). The isotropy subgroup of the position of boundary function in \(X^{r}_{n,k}\) is isomorphic to the cyclic group \(Z_k\). In such spaces \(X^{r}_{n,k}\) the eigenvalues of \(L\) are investigated, and the detailed picture of their position depending on gravity and surface tension is obtained. At last, the equivariant version of Hopf bifurcation is proved together with the existence of a branch of \(Z_k\)-symmetric and periodically oscillating solutions of Navier-Stokes system.


76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76E99 Hydrodynamic stability


Zbl 0889.35075
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