  In this article we introduce a novel coupled algorithm for massively parallel
direct numerical simulations of electrophoresis in microfluidic flows. This
multiphysics algorithm employs an Eulerian description of fluid and ions,
combined with a Lagrangian representation of moving charged particles. The
fixed grid facilitates efficient solvers and the employed lattice Boltzmann
method can efficiently handle complex geometries. Validation experiments with
more than $70\,000$ time steps are presented, together with scaling experiments
with over ${4\cdot10^{6}}$ particles and ${1.96\cdot10^{11}}$ grid cells for
both hydrodynamics and electric potential. We achieve excellent performance and
scaling on up to $65\,536$ cores of a current supercomputer.
