An important class of electrokinetic, microfluidic devices aims to pump and control electrolyte working liquids that have spatial gradients in conductivity. These high-gradient flows can become unstable under the application of a sufficiently strong electric field. In many of these designs, flow channels are thin in the direction orthogonal to the main flow and the conductivity gradient. Viscous stresses due to the presence of these walls introduce a stabilizing force that plays a major role in determining the overall instability. A thin channel model for fluid flow is developed and shown to provide good agreement with a complete three-dimensional model for channel aspect ratios less than or similar to 0.1.
We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. This configuration may lead to the phenomena of ‘flooding’, i.e. the transition from a countercurrent regime to a cocurrent regime. This transition is marked by a variety of transient behaviour, such as the development of largeamplitude waves that impede the flow of one of the fluids to the reversal of the flow of the denser fluid. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system, which assumes fluids with disparate density and dynamic viscosity ratios, includes the effects of viscosity stratification, inertia, shear and capillarity. Since the experimental constraints for this effective system are unclear, we consider two ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel. The latter forcing results in a single evolution equation whose dynamics depends non-locally on the interfacial shape. From both of these driven systems, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. These criteria, through a numerical verification, do not depend significantly on the inertial effects within the more dense layer. The choice of the local/nonlocal constraints appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.
We consider the evolution of a thin viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions for rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location.