**Project leader:** Professor **Thomas Witelski****Project manager: **Yuqing Dai**Team members: **Dominic Jeong, Juliet Jiang, Ada Zhang

Understanding particle contamination behavior and clogging in membrane pores at a microscopic scale provides applicable information in a multitude of industrial fields and practices. Our team analyzes particle-laden flow in a simplified, symmetric 2D cross-section of a pore tube with a spatially and time-dependent model that uses two approaches. The first half of our project models particle concentration and pore radius evolving with two coupled PDEs with moving surfaces due to evaporation and deposition and otherwise reflecting boundaries. Reynolds Transport Theorem and Conservation of Mass provide the main PDEs in the model. Using nondimensionalization, we create a standardized model to observe pore radius evolution and concentration evolution over time. For further simplification, we utilize perturbation expansions and asymptotic reduction to scale our problem down to 1D pores with a slender aspect ratio limit. Numerical methods such as the Forward Euler time-stepping, computational scaling, and finite differences are used to acquire computational solutions. We then provided systematic analyses on all model parameters and model properties, such as evaporation and deposition rates, volume scaling of particles on the moving wall, saturated and initial concentrations, and the shape of the pore. We observe the monotonicity of concentration at the base, the accumulation of mass over time, and the stability of parameters as well.

The second half of our project approaches the same 2D model with stochastic processes. With the use of the Fokker Planck equation, we interpret concentration as a distribution of spatial random variables describing the Brownian motion of individual particles. We also derive a system of Euler Maruyama step equations for the stochastic motion of particle trajectories over time. We implement our own boundary conditions that mirror those of the PDE model, such as reflective and partially absorbing boundaries at a moving boundary from an evaporating fluid. We construct a probabilistic model that parallels the results of the PDE approach and is more flexible for modifications such as wall geometry and more complex boundary conditions. Lastly, we observe similar dynamics in a parabolic droplet with a fixed contact radius and a fully or partially absorbing floor. We compare concentration gradients and boundary flux for both PDE and SDE approaches, introducing a quasi-static approximation method for the moving surface. With similar scaling techniques and computational methods, our results again provide comparative results between PDE and SDE models and new insight into deposition behavior over time.