Recent experimental studies, in conjunction with numerical simulations, have investigated the possibility of utilising microporous membranes in microfluidic devices to separate particles within a fluid. The specific design of this class of separation device involves pumping a particle-laden fluid back and forth through a membrane of axisymmetric, periodic micropores. Even though there is no net flow of the carrier fluid in such cases, introducing spatial asymmetry in the periodic pore profile has been shown to lead to directed particle transport; however, it is not currently understood how to control and optimise this transport. Deterministic solutions for the motion of individual particles are not available due to the presence of random thermal fluctuations in these microscale systems. Hence, a comprehensive statistical routine for predicting the magnitude and direction of the bias in suspended particle trajectories would be beneficial in the design of effective particle separation devices. Since modelling this two-phase flow problem is commonly divided into two parts: carrier fluid flow field determination and stochastic simulations of travelling suspended particles, this thesis aims to refine existing techniques and establish new methods for approximating solutions to these two sub-problems. [...]