| This thesis emphasizes the interplay of asymptotic and numerical methods in solving the PDE's of transport phenomena for two applications: (i) validation and design of a novel membrane separation process; and (ii) rheological modeling of dilute polymer solutions. The first application involves a novel membrane fractionation mechanism called anti-polarization dialysis (APD) and is formulated in two space dimensions plus time.; An atomistic variant of smoothed particle hydrodynamics (ASPH) was developed to solve the two-dimensional problem. ASPH is also applied to the second main problem of this thesis: a one-dimensional dumbbell model for the transient stress build-up in a dilute solution of finitely extensible polymers during sudden start-up of an elongation flow.; For three elastic dumbbell models (linear locked with fixed friction, linear-locked with variable friction, FENE with variable friction), the numerical solutions intimate the asymptotic structure of the problem. In all cases there is a central core of probability density (outer solution), and a boundary layer (inner solution). This physical insight enables rigorous asymptotic analysis for the linear-locked dumbbell with fixed friction and the FENE dumbbell with variable friction. The analysis combines singular perturbations with the method of multiple scales. The analytical results in this thesis provide marked improvements on the transient stresses, as checked against the numerical solutions. |
