Electroosmotic Flow In A Charged Microand Nano-Channel And Double Diffusive Convection In A Cavity. : The thesis presents a study on various aspects of electroosmotic flow (EOF) within a charged micro- and nano-channel. We have developed a two-dimensional numerical model to analyze the flow characteristics, electrical potential and concentration distributions of ions. The mathematical model is based on the coupling of Nernst-Plank equation, Poisson equation and Navier Stokes equations. In this study, the Debye-H¨uckel approximation is not invoked. The present model has the flexibility that it can handle the multivalent ions in electroosmotic flow. We also made a study on double-diffusive convection within a lid-driven cavity based on a three-dimensional analysis. In Chapter-2, we studied the steady EOF through a circular channel whose height varies between 5nm to 100nm. The unsteady EOF within a rectangular channel due to an externally applied alternating electric field is presented in Chapter-3. In both these studies, we considered the effect of overlapping double layers. A detailed analysis of the EOF is presented.