ALBERT

Description: The classic hydrodynamic Hele-Shaw problem is revisited in the context of evaluating the viscous resistance to low-Mach compressible viscous gas flows through shallow non-uniform micro-fluidic configurations. Our recent study of gas flows through constricted shallow micro-channels indicates that the failure of the standard Hele-Shaw approximation to satisfy the no-slip boundary condition at the sidewalls severely restricts its applicability. To overcome this we have extended the asymptotic scheme to incorporate an inner solution in the vicinity of the sidewalls (which, in turn, allows for the characterization of the effects of channel cross-section geometry) and its matching to an outer correction. We have compared the results of the present asymptotic analysis to existing exact analytic and numerical results for straight and uniform channels and to finite-element simulations for a 90° turn and a symmetric T-junction, which demonstrate a remarkably improved accuracy relative to the standard Hele-Shaw approximation. This suggests the present scheme as a viable alternative for the rapid performance estimate of micro-fluidic devices. © 2009 Copyright Cambridge University Press.

Description: We have studied the temporal evolution of electro-kinetic flows in the vicinity of polarizable dielectric solids following the application of a 'weak' transient electric field. To obtain a macro-scale description in the limit of narrow electric double layers (EDLs), we have derived a pair of effective transient boundary conditions directly connecting the electric potentials across the EDL. Within the framework of the above assumptions, these conditions apply to a general transient electro-kinetic problem involving dielectric solids of arbitrary geometry and relative permittivity. Furthermore, the newly derived scheme is applicable to general transient and spatially non-uniform external fields. We examine the details of the physical mechanisms involved in the relaxation of the induced-charging process of the EDL adjacent to polarizable dielectric solids. It is thus established that the time scale characterizing the electrostatic relaxation increases with the dielectric constant of the solid from the Debye time (for the diffusion across the EDL) through the 'intermediate' scale (proportional to the product of the respective Debye- and geometric-length scales). Thus, the present rigorous analysis substantiates earlier results largely obtained by heuristic use of equivalent RC-circuit models. Furthermore, for typical values of ionic diffusivity and kinematic viscosity of the electrolyte solution, the latter time scale is comparable to the time scale of viscous relaxation in problems concerning microfluidic applications or micro-particle dynamics. The analysis is illustrated for spherical micro-particles. Explicit results are thus presented for the temporal evolution of electro-osmosis around a dielectric sphere immersed in unbounded electrolyte solution under the action of a suddenly applied uniform field, combining both induced charge and 'equilibrium' (fixed charge) contributions to the zeta potential. It is demonstrated that, owing to the time delay of the induced-EDL charging, the 'equilibrium' contribution to fluid motion (which is linear in the electric field) initially dominates the (quadratic) 'induced' contribution. © 2009 Cambridge University Press.