In this thesis, we investigate three fluid dynamic problems involving various physical mechanisms which exhibit interfacial instability. These problems have wide ranging industrial, scientific and engineering applications.;In the first problem, we investigate the linear stability of the unbounded Couette flow of two fluids separated by a plane interface. The exact dispersion relation is solved asymptotically and numerically to analyze the effects of the four stability parameters of the flow; the ratio of the viscosities, the ratio of the density, the surface tension and gravity. While our results confirm most of the earlier reported theories involving shear flows of fluids of equal densities, they also resolve the reported discrepancies between the numerical and the asymptotic solutions. For the general case of fluids with different densities, new asymptotic expressions for the growth rates of the flow are obtained and numerical calculations of marginal states are carried out in order to examine the effects of the stability parameters on the flow. The numerical results confirm the remarkable accuracy of our asymptotic expressions.;In the second problem, the electrohydrodynamic extension of the first problem is presented. Here, the plane interface is stressed by applying external electric fields normal to the interface. A linear stability analysis similar to that employed in the first problem is used to investigate the effects of six additional stability parameters on the stability of the flow; the ratio of the permittivities, the two conductivities, the two initial electric fields and the velocity of the upper fluid in the unperturbed motion. Various limiting cases having practical applications are investigated. We examine the effects of electrical shear stresses and initial streaming of the fluids on the onset of static instability. We also examine finite electric charge relaxation effects.;Finally, we investigate the dynamic behaviour of viscous droplets in the presence of applied electric fields in zero gravity conditions. Here, the full nonlinear equations of motion are solved numerically by adapting the NASA-VOF2D algorithm. The numerical computations carried out for axisymmetric droplets in zero gravity successfully simulate microgravity experiments conducted on KC-135 NASA aircraft flights. Further experimental and modelling modifications are discussed
