Anna Broms: An accurate method of fundamental solutions for large and dense particle systems in Stokes flow

Abstract: The Stokes mobility and resistance problems for rigid particles in viscous fluids have broad applications in both natural processes and industrial contexts and are important in the study of transport or diffusion processes, rheology and nonlinear shear thickening. Hydrodynamic interactions between particles are challenging to determine, as they are simultaneously long-ranged and expensive to resolve for dense suspensions. The latter is caused by near-singular lubrication forces resulting from close-to-touching bodies in relative motion. With the aim of controlling the accuracy for dense suspensions with a computationally cheap method, we present a new technique for the two problems that combines the method of fundamental solutions (MFS) with the method of images. For rigid spheres, we propose to represent the flow using fundamental solutions (Stokeslets) located on interior spheres, augmented by lines of image sources adapted to each near-contact to resolve lubrication. Source strengths are found by a least-squares solve at contact-adapted boundary collocation nodes. In the talk, we also discuss efficient preconditioning. The resulting well-conditioned schemes are competing with state-of-the art solvers tailored for spherical particles in Stokes flow, yet are compatible also with other smooth particle shapes, and scale linearly in the number of objects. For instance, a problem of 10,000 ellipsoids is solved to 5-digit accuracy on a workstation in less than two hours.

Joint with A.-K. Tornberg, KTH, and A.H. Barnett, Flatiron Institute, NY.