  We present an analytical study, validated by numerical simulations, of
electroosmotic flow in a Hele-Shaw cell with non-uniform surface charge
patterning. Applying the lubrication approximation and assuming thin electric
double layer, we obtain a pair of uncoupled Poisson equations which relate the
pressure and the stream function, respectively, to gradients in the zeta
potential distribution parallel and perpendicular to the applied electric
field. We solve the governing equations for the fundamental case of a disk with
uniform zeta potential and show that the flow-field in the outer region takes
the form of a pure dipole. We illustrate the ability to generate complex
flow-fields around smooth convex regions by superposition of such disks with
uniform zeta potential and a uniform pressure driven flow. This method may be
useful for future on-chip devices, allowing flow control without the need for
mechanical components.
