Distinguished Seminar in Computational Science and Engineering
April 17, 2025, 12-1PM
45-432 in Building 45 and Zoom Webinar
Slender and close: accurate Stokes flows for rigid particles in challenging geometries
Alex Barnett
Group Leader, Numerical Analysis. Center for Computational Mathematics, Flatiron Institute, NY
Abstract:
The modeling of suspensions of rigid particles in a viscous incompressible fluid in the low-Reynolds-number limit is crucial to applications including sedimentation, rheology, microfluidic devices, active matter, and bacterial or cellular transport. Accurately modeling the relation between hydrodynamic forces and motions demands solving a Stokes boundary-value problem throughout the fluid domain at every time-step, yet the available numerical tools are far from satisfactory. This is especially true when objects become relatively close (“lubrication effects”). I will overview two new tools to address this with controlled accuracy using potential theory: 1) For the common case of spheres we show that interior fundamental solutions (MFS) augmented by simple image systems accurately handle separations down to a thousandth of the radius, and that large collections of spheres/ellipsoids can be tackled via block-diagonal least-squares preconditioning. 2) For slender fibers of circular cross-section we present a boundary integral (BIE) scheme with adaptive quadrature. Unlike widely-used slender body theory—which is non-convergent (merely asymptotic in the fiber radius) and incorrect when fibers approach—our scheme is convergent, and handles very close fibers with up to 10 accurate digits. We combine it with high-order time-stepping for sedimentation. For both tools we show high-order convergence, and well-conditioned iterative solution with close-to-linear cost scaling.
Joint work with Anna Broms, Anna-Karin Tornberg, and Dhairya Malhotra.
Bio:
Alex Barnett is an applied mathematician and numerical analyst. He is a Senior Research Scientist, and Group Leader for Numerical Analysis, at the Center for Computational Mathematics at the Flatiron Institute in New York City. After a Ph.D in physics from Harvard, he did postdoctoral work in radiology at Massachusetts General Hospital and as a Courant Instructor at New York University. He served on the mathematics faculty at Dartmouth College for 12 years, becoming a full professor, and creating several new courses on topics such as the math of music and sound. His research includes numerical partial differential equations (wave scattering, Stokes flow, and high-frequency eigenvalues), integral equations, fast algorithms, signal processing, statistics, imaging, inverse problems, quantum physics and biomathematics. He has authored or coauthored over 70 articles and two books, and developed popular scientific software libraries. His awards include several NSF grants, Dartmouth’s Karen E. Wetterhahn Memorial Award for Distinguished Creative or Scholarly Achievement, and 1st prize in the 1990 International Physics Olympiad.
Slender and close: accurate Stokes flows for rigid particles in challenging geometries
Alex Barnett
Flatiron Institute