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Pairwise interactions in inertially-driven one-dimensional microfluidic crystals
In microfluidic devices, inertia drives particles to focus on a finite number
of inertial focusing streamlines. Particles on the same streamline interact to
form one-dimensional microfluidic crystals (or "particle trains"). Here we
develop an asymptotic theory to describe the pairwise interactions underlying
the formation of a 1D crystal. Surprisingly, we show that particles assemble
into stable equilibria, analogous to the motion of a damped spring. Although
previously it has been assumed that particle spacings scale with particle
diameters, we show that the equilibrium spacing of particles depends on the
distance between the inertial focusing streamline and the nearest channel wall,
and therefore can be controlled by tuning the particle radius.Comment: 15 pages, 9 figure
Vortex sheets and diffeomorphism groupoids
In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics
in which the motion of an inviscid incompressible fluid is described as the
geodesic flow of the right-invariant -metric on the group of
volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic,
group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex
sheets. It turns out that the corresponding dynamics is related to a certain
groupoid of pairs of volume-preserving diffeomorphisms with common interface.
We also develop a general framework for Euler-Arnold equations for geodesics on
groupoids equipped with one-sided invariant metrics.Comment: Final version accepted to Advances in Mathematics; 46 pages, 6
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