Asymptotic and particle methods in nonlinear transport phenomena: Membrane separations and drop dynamics

This thesis applies asymptotic methods in the context of membrane separations in order to resolve the hydrodynamics at the entrance region in membrane channels. The particle method was used to study the deformation of liquid drops in Stokes flows and the prevaling numerical method was modified to improve the computational efficiency.;Asymptotic analysis was used to study a numerical flow singularity in the entrance region of membrane channels which has hitherto been glossed over in numerical treatments of desalination and membrane separations. The singularity arises due to the discontinuity in flow boundary conditions at the junction between an impermeable and porous wall. This fine structure is important to resolve due to concentration polarization and the resulting limits on throughput which are sensitive to the pressure field.;The numerical solution was obtained for the problem and then used to determine the inner solution which was an asymptotic series with the leading order radial dependence for the pressure being r -1/2. The inner solution was matched to the outer solution by phrasing both solutions in terms of the same angular basis functions and obtaining the radial dependence using Fourier integration. Thus a full analytical equation was obtained that described the singularity in both the inner and outer regions.;The particle method was used to model the hydrodynamics of liquid drops at low Reynolds numbers. The particle method computes hydrodynamic interactions between particles representing small volumes of liquid drops in order to determine the velocity of each particle so that the positions can be updated by the Lagrangian approach. In this thesis, the particle method was modified from a purely Lagrangian approach to a hybrid method, which used cubically distributed cohesive force and Stokeslet interaction kernels, to improve the computational efficiency of the simulations. The particle simulations were validated using test problems to model sedimentation and interfacial tension-driven flows.