%0 Generic
%A Bodea, Simina
%D 2003
%F heidok:3753
%K Dynamischer KontaktwinkelNavier-Stokes equations , Free Boundary Problem , Dynamic Contact Angle , Existence and Regularity Theory , Hopf Bifurcation
%R 10.11588/heidok.00003753
%T Oscillations of a Fluid in a Channel
%U https://archiv.ub.uni-heidelberg.de/volltextserver/3753/
%X In this work we investigate the motion of a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel.  The upper surface changes with the motion of the fluid, so we deal with a  free boundary problem. We consider small perturbations of a uniform flow with  a flat free surface. We include the effect of the surface tension; the external  forces are gravity, and the wind force which acts on the free boundary.  The motion of the fluid in the channel is governed by the Navier-Stokes  equations. We consider the system to be periodic in the direction of the length of  the channel. Technically, we identify the inflow boundary with the  outflow boundary of the channel and then we consider the second spatial  variable belonging to the circle $S^1$. In order to obtain a well-posed model,  we have to prescribe the value of the dynamic contact angle between the walls  and the free boundary and we choose it to be $\frac{\pi}{2}$.  As boundary conditions, we consider that the walls are impenetrable together  with a perfect slip condition, and a no slip condition for the bottom. The main aim of this paper is to analyse the qualitative behavior of the flow  (oscillations of periodic solutions) using tools of bifurcation theory. In order to do this we need fundamental facts of existence and regularity  of solutions, spectral analysis of the linear system connected with the free  boundary value problem taking into account the underlying symmetries, and  techniques of equivariant Hopf bifurcation theorem.