Eduard Marusic-Paloka (University of Zagreb, Croatia).
We study the effects of the perturbation of the domain on the flow of viscous fluid in two cases.
In first case we study the flow through a pipe which is curved and we are interested in effects caused by flexion and torsion of the pipe. The goal is to understand how does such flow differ from the Poiseuille flow through a straight pipe. The method of asymptotic expansions is applied, with respect to the ratio between pipes thickness and length, which is assumed to be small. Terms in the expansion are explicitly computed.
In the second part we study the viscous flow through domain Omega_epsilon obtained from domain Omega by a small perturbation of its boundary. We are interested in influence of this perturbation on the solution. In real life boundaries of fluid domains can be very complicated (e.g in meteorology or hydrology) and, to perform numerical computations we frequently planish the boundary. By doing so, we create an error. How big is it? Is it localized near the boundary? To obtain our goal a method of particular asymptotic analysis is proposed and an asymptotic expansion for the solution is derived. Its terms are studied and, some of them, are explicitly computed. As expected, the effect has the same magnitude as the magnitude of the boundary perturbation, but it is not localized near the boundary. Thus it is not of the boundary layer type. An effective boundary condition on the planished boundary, taking into account its rugosities, is derived.