The Navier-Stokes equations for steady viscous incompressible axisymmetric flow can be reduced to a set of ordinary differential equations if the flows are assumed to satisfy a certain set of similarity relations. These flows which are called self-similar are studied in this thesis from a mathematical viewpoint and their connection with flows which are not completely self-similar is explored.
The work is divided into three parts. In the first part we consider self-similar flows in domains which are of infinite extent. When the swirl velocity is identically zero, the governing equations reduce to a third order equation with three different subcases, of which only one has been treated before. All solutions of this equation are studied and classified according to their asymptotic behaviour. This study suggests some new results concerning the boundary conditions that can be imposed on self-similar flow with swirl which are of interest in connection with the self-similar flow near a rotating disk. These results are then proved rigorously. The general self-similar flow near a rotating disk is discussed, and it is shown t h a t a solution exists for perturbation of solid body rotation.
The study of self-similar flows in a domain bounded by two infinite coaxial disks rotating in two parallel planes a finite distance apart forms the second part. The main emphasis is on flow at high values of the Reynolds number (low values of the Ekman number), and approximate solutions are obtained by the method of matched asymptotic expansions. This analysis suggests that there exist several types of flow which have not been studied in detail before. An approximate solution is also obtained both for near solid body rotation and for low Reynolds number flow. It is shown that for sufficiently small values of the Reynolds number a solution always exists.
In most rotating flows of physical interest, the velocity components must satisfy some boundary conditions at a finite value of the radial coordinate, so those flows cannot be completely self-similar. In the third part we consider, as an example of such physical flow problems, the flow between two finite rotating disks contained in a rotating cylinder. The solution for Stokes flow is obtained in a form different from that calculated by earlier workers, and it is shown that the flow is self-similar in part of the flow domain. When the angular velocity of one of the disks is slightly different from that of the other disk and the cylinder, the Navier-Stokes equations can be linearized. Approximate solutions, which are self-similar except in the neighbourhood of the cylinder, are calculated for high Reynolds number. A detailed theoretical picture of the flow pattern which suggests an interesting intensive circulation in the neighbourhood of the junction of the cylinder with the slower disk is obtained from these solutions.