| Abstract [eng] |
In the dissertation one of the Rivlin-Erikson differential type fluids model – the second grade fluids flow problem is considered. The problem is studied in three different unbounded domains: • the two-dimensional channel, • the three-dimensional axially symmetric pipe, • the three-dimensional pipe with an arbitrary cross section. For the two-dimensional channel and the three-dimensional axially symmetric pipe we assume that the initial data and the external force have only the last component and are independent of the coordinate x_n: u_0(x,t)=(0, …, u_{n0}(x’,t)), f(x,t)=(0, …, f_n(x’,t)). We look for an unidirectional (having just the last component) solution u(x,t) =(0, …, u_n(x’,t)), which satisfies the flux condition. Such solution we call Poiseuille type solution. In the first two cases the existence of a unique unidirectional Poiseuille type solution is proved and the relation between the flux of the velocity field and the pressure drop (the gradient of the pressure) is found. The analogous results were obtained for the time periodic problem in the two-dimensional channel. It is shown that in the three-dimensional pipe with an arbitrary cross section the unidirectional solution does not exists even if data are unidirectional. However, for sufficiently small data in this case exists a unique solution having all three components u(x’,t)=( u_1, u_2, u_3). To analyze the problem we use Galerkin method with the special bases. |
