  In this paper, we develop a general mathematical framework for enhanced
hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes,
which is obtained by simultaneously perturbing the inner and outer boundaries
of the perfect cloaking structure. We first derive the asymptotic expansions of
perturbed fields and obtain a first-order coupled system. We then establish the
representation formula of the solution to the first-order coupled system using
the layer potential techniques. Based on the asymptotic analysis, the enhanced
hydrodynamic near-cloaking conditions are derived for the control region with
general cross-sectional shape. The conditions reveal the inner relationship
between the shapes of the object and the control region. Especially, for the
shape of a deformed annulus or confocal ellipses cylinder, the cloaking
conditions and relationship of shapes are quantified more accurately. 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.
