Speaker
Alex Barnett,Flatiron Institute
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.