Asymptotic conditions at infinity for the Stokes problem in a system of pipes
In the theoretical hydromechanics special attention is given to the investigation of problems in unbounded domains, in particular, in domains with semi-infinite outlets to infinity. The boundary value problems in domains with cylindrical outlets (pipes) belong to this class. This is self-understood, there are no infinite volumes of liquids in the nature and, hence, these problems should be considered only as certain model ones. At the same time, exactly such problems are used by engineers while solving the practical problems. Performing the computer simulation of the flow in long thin pipes, engineers, at first, replace them by semi-infinite cylinders and basing on the intuition, experiments and the engineering know how, suggest some sensible conditions in each of semi-cylinders at infinity. After this, the semi-infinite cylinders must be cut again in order to allow the numerical simulation of the flow. In this connection the proper formulation of asymptotic boundary conditions at infinity gain the decisive significance for the adequate description of real situations. In the talk I discuss mathematical tools which allow to realise the correct setting of asymptotic conditions at infinity in a system of pipes and show how to apply the general theory for concrete physically sensible problems, e.g. for open drains, for pipes occupied by pumps or check valves and for connected pipes with check valve inside. The results were obtained jointly with S. A. Nazarov.
Konstantin Pileckas, Vilnius University, Lithuania
February 11, 2000