الفهرس 
GENERAL INTRODUCTIONOnly 14 pages are availabe for public view
 |  | from | 135 |  |  |
| | | from | 135 | | |
Abstract
The aim of the thesis is to study the Navier-Stokes equations for incompressible viscous fluid flow. These equations that describe the motion of a Newtonian fluid with appropriate boundary conditions do not have a general analytic solution. Only a few special cases can be solved analytically. The solution of the Navier-Stokes problem is sought numerically. In this thesis we consider a class of such methods which are based on the work of [3], [11], [12], [13], [14], [18], [25], [38] and [50]. Mathematical difficulties of Navier-Stokes problem solution especially for small value of viscosity required special methods. One of them was introduced by Chorin [11]. He was first to propose the vortex blobs method. This method allows to by passing mentioned troubles. Also, it is useful especially when the vorticity is created in small regions of the domain. Theoretically the vortex motion forms the continuous family of Ito’s stochastic processes governed by Ito’s stochastic differential equations. Of course, this continues family must be cut off to finite one. The main goal of the thesis is to study the flow field for an axisymmetric jet viscous flow in a large tank by using this method. The main idea of that is to approximate the vorticity by a large set of small vortex creatures being transported in velocity field and performing random walks according to Wiener process. The velocity field evolves in time due to change of vorticity distribution.
In order to satisfy the boundary conditions, new vortex blobs are continuously created on the boundary. Of course, the velocity field contains proper potential components, which it is allows to formulae boundary integral equation for the vorticity created at the moment. Thus, the Neumann boundary problem should be solved. Generally, this method is approximate the solution to Helmholtz’s equation and it can be regarded as a discretization of this equation in a special form. The vortex blobs method brings some advantages as the result of elimination of pressure and is believed to belong to a new class of powerful numerical techniques for simulation of fluid motion.
In an axisymmetric viscous flow, we studied the solution of potential problems that reduced the boundary problems to Hankel transforms. Also, the construction of axisymmetric vorticity carrier is considered. The solutions of these problems seem to be proper, especially when we choose the shape of simple vorticity carrier. The boundary integral equations for vorticity creation are given and solved. Finally, the formulation of the stochastic equation which it is describing the motion of vorticity carriers is considered. The thesis contains four chapters, three appendix and list of publications.