  In this paper, we develop a general mathematical framework for perfect and
approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which
is governed by a coupled PDE system via the field-effect electro-osmosis. We
first establish the representation formula of the solution of the coupled
system using the layer potential techniques. Based on Fourier series, the
perfect hydrodynamic cloaking and shielding conditions are derived for the
control region with the cross-sectional shape being annulus or confocal
ellipses. Then we further propose an optimization scheme for the design of
approximate cloaks and shields within general geometries. The well-posedness of
the optimization problem is proved. In particular, the condition that can
ensure the occurrence of approximate cloaks and shields for general geometries
are also established. Our theoretical findings are validated and supplemented
by a variety of numerical results. The results in this paper also provide a
mathematical foundation for more complex hydrodynamic cloaking and shielding.
