Longitudinal and transverse electrohydrodynamic flows through a plane channel, of which the walls are micropatterned with a periodic array of stripes, are considered. One unit of wall pattern consists of a slipping stripe and a non-slipping stripe, each with a distinct zeta potential. The problems are solved by a semi-analytical method, where the basic solutions satisfying the electrohydrodynamic equations are expressed by eigenfunction expansions, and the coefficients are determined numerically by point collocation satisfying the mixed stick-slip boundary conditions. In the regime of linear response, the Onsager relations for the fluid and current fluxes are deduced as linear functions of the hydrodynamic and electric forcings. The phenomenological coefficients are explicitly expressed as functions of the channel height, the Debye parameter, the slipping area fraction of the wall, the intrinsic slip length, and the zeta potentials. Attention is paid to some particular kinds of patterns, with a view to revisit and to generalize the theoretical limits made in previous studies on electrokinetic flow over an inhomogeneously slipping surface. One should be cautious when applying the theoretical limits. We show that when a surface is not 100% uniformly slipping but has a small fraction of area being covered by no-slip slots, the electro-osmotic enhancement can be appreciably reduced. We also show that when the electric double layer is only moderately thin, slipping-uncharged regions on a surface will have finite inhibition effect on the electro-osmotic flow. V

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Comparisons between slip lengths predicted by a liquid-gas coupled model and that by an idealized zero-gas-shear model are presented in this paper. The problem under consideration is pressure-driven flow of a liquid through a plane channel bounded by two superhydrophobic walls which are patterned with longitudinal or transverse gas-filled grooves. Effective slip arises from lubrication on the liquid-gas interface and intrinsic slippage on the solid phase of the wall. In the mathematical models, the velocities are analytically expressed in terms of eigenfunction series expansions, where the unknown coefficients are determined by the matching of velocities and shear stresses on the liquid-gas interface. Results are generated to show the effects due to small but finite gas viscosity on the effective slip lengths as functions of the channel height, the depth of grooves, the gas area fraction of the wall, and intrinsic slippage of the solid phase. Conditions under which even a gas/liquid viscosity ratio as small as 0.01 may have appreciable effects on the slip lengths are discussed.

We have discovered novel super-diffusion, arrested spreading, and shuttling of diffusiophoretic colloids, driven by simultaneous diffusion of an ionic solute.

A multiple-scale perturbation theory is developed to analyze the advection-diffusion transport of a passive solute through a parallel-plate channel. The fluid velocity comprises a steady and a time-oscillatory component, which may vary spatially in the transverse and streamwise directions, and temporally on the fast transverse diffusion timescale. A long-time asymptotic equation governing the evolution of the transverse averaged solute concentration is derived, complemented with Taylor dispersion coefficients and advection speed corrections that are functions of the streamwise coordinate. We demonstrate the theory with a two-dimensional flow in a channel comprising alternating shear-free and no-slip regions. For a steady flow, the dispersion coefficient changes from zero to a finite value when the flow transitions from plug-like in the shear-free section to parabolic in the no-slip region. For an oscillatory flow, the dispersion coefficient due to an oscillatory flow can be negative and two orders of magnitude larger than that due to a steady flow of the same amplitude. This motivates us to quantify the relative magnitude of the steady and oscillatory flow such that there is an overall positive dispersion coefficient necessary for an averaged (macrotransport) equation. We further substitute the transport coefficients into the averaged equation to compute the evolution of the concentration profile, which agrees well with that obtained by solving the full two-dimensional advection-diffusion equation. In a steady flow, we find that while the shear-free section suppresses band broadening, the following no-slip section may lead to a wider band compared with the dispersion driven by the same pressure gradient in an otherwise homogeneously no-slip channel. In an unsteady flow, we demonstrate that a naive implementation of the macrotransport theory with a (localized) negative dispersion coefficient will result in an aphysical finite time singularity (or “blow-up solution”), in contrast to the well-behaved solution of the full advection-diffusion equation.