In this dissertation, we build mathematical tools for applications to the transport properties of human lung mucus. The first subject is the microscopic diffusive transport of micron-scale particles in viscoelastic fluid. Inspired by the technique of passive microrheology, we model the motion of Brownian beads in general viscoelastic fluids by the generalized Langevin equation (GLE) with a memory kernel (the diffusive transport modulus). The GLE is a stochastic differential equation, which admits a discrete formulation as an autoregressive (AR) process. We further use exponential series for the memory kernel in the GLE, in which case the GLE has an explicit formulation as a vector Ornstein-Uhlenbeck process. In this framework, we can develop fast and accurate direct algorithm for pathogen transport in viscoelastic fluids, and the Kalman filter and maximum likelihood method give a new method for inversion of the memory kernel from experimental position time series. The framework is illustrated with multimode Rouse and Zimm chain models. In the second topic, we revisit the classical oscillatory shear wave model of Ferry et al., and extend the theory for active microrheology of small volume samples of viscoelastic fluids. In Ferry's original setup, oscillatory motion of the bottom plate generates uni-directional shear waves propagating in the viscoelastic fluid. Our colleague David Hill built a device to handle small volume viscoelastic samples. We extend the Ferry analysis to include finite depth and wave reflection off the top plate. We further consider nonlinear viscoelastic constitutive laws. The last problem considered is the numerical simulation of viscoelastic fluid flow, which will eventually be used to predict bulk transport of mucus layers. We start with the analysis of the system of model equations and demonstrate the difficulty of a robust numerical scheme. We develop an extension of projection method, which involves a new treatment of stress evolution based on stress splitting in the numerical scheme and show the advantage over previous work
