Several applications such as liquid-liquid extraction in micro-fluidic devices are concerned with the flow of two immiscible liquid phases. The stratified flow regime in these micro-channels has the inherent advantage that separation at the exit is efficient. In pressure driven flows in microchannels the flow profile is laminar and parabolic. This induces axial dispersion in the system which is known to reduce the efficiency of the process. In micro-channels electro-osmosis has been employed to overcome this drawback, leading to a flow profile that approaches the classical plug flow behaviour.In this work fundamental features of the velocity profiles in stratified flows are analysed. First the flow between two rectangular infinite flat plates is discussed. The influence of physical properties, in particular viscosity of the two liquids, on the velocity profiles is determined symbolically. The flow-profiles are classified in the parameter space of physical properties (viscosity ratio) and operating conditions (flow-rate ratio). The mass transfer in these systems is compared with that found when the two liquids follow an ideal plug flow behaviour. A symbolic solution based on the method of separation of variables is proposed for concentration when the flow profile is a plug flow. It is found that the plug flow profile gives a poorer performance than the laminar profile. This is attributed to the lower velocity prevailing at the interface in the plug flow regime. To understand this more clearly the effect of changing the height of the interface was also analysed and it was found that under some conditions the ideal plug flow can give a better performance than the laminar reactor. The results of this work will aid the experimentalist establish guidelines to determine if introducing electro-osmosis in a Hagen-Poiseuille flow will be beneficial or detrimental in liquid-liquid extraction.