The present thesis deals with microfluidic systems under the influence of electric fields. The purpose of this research is to identify key behaviours which are highly relevant for applications in lab-on-chip devices such as pH control, patterning, mixing and cell trapping. In the first part of the thesis we consider the electrokinetics of charged, porous membranes and present a mathematical model for the ionic transport under the effects of a horizontal electric field. First, we investigate the behaviour of a system that consists of one anion membrane with two reservoirs and produce numerical solutions with the aim to gain a better understanding of the mechanisms that lead to overlimiting current. We then analyse the features of a system where a bipolar membrane is held in an electrolyte bath with water and a salt. We use our model to confirm findings from experiments such as the hysteretic behaviour of the IV curve and the water splitting phenomenon.
In the second part of the thesis we examine the behaviour of interfaces between two fluids that are sandwiched between two electrodes. We find that introducing a constant flow rate into the system leads to time modulated travelling waves. In the case of flat channel walls these look like moving strips which are reminiscent of the patterns found in the no flow case. Adding corrugations on one or both electrodes leads to a rich variety of dynamics. We develop a Floquet stability analysis which takes into account the fact that the base state of the system is nonuniform. This is a very useful tool for identifying the different types of behaviours which arise as we change the applied voltage and the overall flow rate. We examine the streamlines of the fluid to explore the advantages of the different regimes: just by changing the applied voltage it is possible to transition from an environment which is favourable to efficient mixing to one which could enhance cell trapping.