CAREER: Fast Algorithms for Particulate Flows

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The objective of this Faculty Early Career Development (CAREER) project is to build stable, high-accuracy, and optimal algorithms for direct numerical simulations of particulate flows. Dense suspensions of deformable particles in viscous fluids are ubiquitous in natural and engineering systems. Examples include drop, bubble, vesicle, swimmer, and blood cell suspensions. Unlike simple Newtonian fluids, the laws describing their flow behavior are not well established, owing to the complex interplay between the deformable micro-structure and the macro-scale flow. For instance, interactions between soft particles modify their trajectories and cause shear-induced diffusion, and interaction with confining walls controls the spatial organization of the suspensions. Besides experiments, direct numerical simulations are often the only means for gaining insights into the non-equilibrium behavior of such complex fluids. Hence, there is a need for robust and optimal algorithms that are scalable. Integrated with the research effort, this project will undertake educational, mentoring, and outreach activities including a new interdisciplinary graduate-level course on computational methods for complex fluids and an interactive education module illustrating the non-intuitive phenomena observed in complex fluid systems that is accessible to high school students. The specific aims of this project include (i) highly accurate algorithms to compute the nearly singular integrals that arise within the context of boundary integral methods when particles approach very close to each other when subjected to flow, (ii) fast, high-order, and adaptive algorithms to simulate multiphase flows through arbitrary periodic geometries, and (iii) stable time-marching and reparameterization schemes for the coupled systems of stiff, nonlinear, time-dependent differential and integro-differential equations governing the evolution of particles and some material concentration on their surfaces (e.g., surfactants or multi-phase lipids). The proposed computational infrastructure will lead to better predictive capabilities for blood flow through complex geometries, margination of platelets and targeted carriers, and new design tools for microfluidic devices. They can be applied to a large class of particulate flow problems. Besides fluid mechanics, the proposed numerical methods and the computational infrastructure can be applied to solve partial differential equations that arise in various other disciplines. The research outcomes will have an impact on a broad spectrum of disciplines in sciences and engineering.