  The electro-osmotic flow through a channel between two undulated surfaces
induced by an external electric field is investigated. The gap of the channel
is very small and comparable to the thickness of the electrical double layers.
A lattice Boltzmann simulation is carried out on the model consisting of the
Poisson equation for electrical potential, the Nernst--Planck equation for ion
concentration, and the Navier--Stokes {\color{black}equations} for flows of the
electrolyte solution. An analytical model that predicts the flow rate is also
derived under the assumption that the channel width is very small compared with
the characteristic length of the variation along the channel. The analytical
results are compared with the numerical results obtained by using the lattice
Boltzmann method. In the case of a constant surface charge density along the
channel, the variation of the channel width reduces the electro-osmotic flow,
and the flow rate is smaller than that of a straight channel. In the case of a
surface charge density distributed in homogeneously, one-way flow occurs even
under the restriction of a zero net surface charge along the channel.
