Numerical computations in problems of fluid flow instability

The thesis consists of two distinct parts. In the first part of the thesis the stability of an inviscid fluid between two concentric rotating cylinders to finite amplitude axisymmetric disturbances is investigated. The limiting form of the disturbance equations for small gap-to-radius ratio is derived. Computations are carried out for the case in which the cylinders rotate in the same direction. To determine the stability of the basic flow we used the energy method. Numerical results are given for distinct values of {dollar}mu{dollar} (where {dollar}mu{dollar} is the ratio of the angular velocity of the outer cylinder to the angular velocity of the inner cylinder). It is found that for those values of {dollar}mu{dollar} {dollar}>{dollar} 1 the basic flow is stable. Whereas, for those in the interval (0,1) the basic flow is unstable. Furthermore, during the course of this investigation we examine in detail the application of pseudospectral approximation techniques to the numerical computation of the non linear problem. Comparing the results obtained from two distinct integrals, we show that the aliasing interactions within the pseudospectral method lead to errors increasing in time. Thus to overcome the numerical instabilities which arise due to this error, some techniques which help to reduce the aliasing error are discussed. Numerical results obtained from these techniques are presented and they are found to be better than those obtained without them.; In the second part, the linear stability of a plane Couette flow of a viscoelastic fluid at low Reynolds numbers is considered. The results are obtained using the linear theory of hydrodynamic stability and assuming that the fluid rheological properties are adequately described by the corrotational Jeffery's model. For this problem, fixing the Reynolds number at 10 the effect of the Weissenberg number on the stability of the basic flow is examined. The eigenvalues of the linear equations which govern the perturbed state of the problem are solved using a compound matrix method and a spectral method. Our numerical results obtained from the compound matrix method show that, in general, the Weissenberg number has a distabilizing effect. In fact for large wave numbers the numerical result predicts instability for those values of {dollar}Wsb{lcub}e{rcub}{dollar} greater than one.