Long-range electrostatic fields deform the surface profile of a conductive liquid in the vicinity of the contact line. We have investigated the equilibrium profiles by balancing electrostatic and capillary forces locally at the liquid vapor interface. Numerical results show that the contact angle at the contact line approaches Young's angle. Simultaneously, the local curvature displays a weak algebraic divergence. Furthermore, we present an asymptotic analytical model, which confirms these results and elucidates the scaling behavior of the profile close to the contact line.
Electrowetting is becoming a more and more frequently used tool to manipulate liquids in various microfluidic applications. On the scale of the entire drop, the effect of electrowetting is to reduce the apparent contact angle of partially wetting conductive liquids upon application of an external voltage. Microscopically, however, strong electric fields in the vicinity of the three phase contact line give rise to local deformations of the drop surface. We determined the equilibrium surface profile using a combined numerical, analytical, and experimental approach. We find that the local contact angle in electrowetting is equal to Young’s angle independent of the applied voltage. Only on the scale of the thickness of the insulator and beyond does the surface slope assume a value consistent with the voltage-dependent apparent contact angle. This behaviour is verified experimentally by determining equilibrium surface profiles for insulators of various thicknesses between 10 and 250 µm. Numerically and analytically, we find that the local surface curvature diverges algebraically upon approaching the contact line with an exponent −1<μ<0. We discuss the relevance of the local surface properties for dynamic aspects of the contact line motion.
Electrowetting is a versatile tool to reduce the apparent contact angle of partially wetting conductive liquids by several tens of degrees via an externally applied voltage. We studied various fundamental and applied aspects of equilibrium liquid surface morphologies both theoretically and experimentally. Our theoretical analysis showed that surface profiles on homogeneous surfaces display a diverging curvature in the vicinity of the three phase contact line. The asymptotic contact angle at the contact line is equal to Young’s angle, independent of the applied voltage. With respect to the morphology of the liquid surface, contact angle variations achieved by electrowetting are equivalent to those achieved by varying the chemical nature of the substrates, except for electric field-induced distortions in a region close to the contact line. Experimentally, we studied the (global) morphology of liquid microstructure substrates with stripe-shaped electrodes. As the local contact angle is reduced by increasing the applied voltage, liquid droplets elongate along the stripe axis as expected. For droplets on a single surface with a stripe electrode, there is a discontinuous morphological transition where elongated droplets transform into translationally invariant cylinder segments with the contact line pinned along the stripe edge and vice versa. If the liquid is confined between two parallel surfaces with parallel stripe electrodes, the elongation of the droplet and its transformation into a translationally invariant morphology with pinned contact lines is continuous. Experimental results are compared to analytical and numerical models.
We analyzed the morphology of droplets of conductive liquids placed between two parallel plate electrodes as a function of the two control parameters electrode separation and applied voltage. Both electrodes were covered by thin insulating layers, as in conventional electrowetting experiments. Depending on the values of the control parameters, three different states of the system were found: stationary capillary bridges, stationary separated droplets, and periodic self-excited oscillations between both morphologies, which appear only above a certain threshold voltage. In the two stationary states, the morphology of the liquid is modified by the electric fields due to electrowetting and due to mutual electrostatic attraction, respectively. We determined a complete phase diagram within the two-dimensional phase space given by the control parameters. We discuss a model based on the interfacial and electrostatic contributions to the free energy. Numerical solutions of the model are in quantitative agreement with the phase boundaries found in the experiments. The dynamics in the oscillatory state are governed by electric charge relaxation and by contact angle hysteresis.
Plane Couette flow, the flow between two parallel plates moving in opposite directions, belongs to the group of shear flows where turbulence occurs while the laminar profile is stable. Experimental and numerical studies show that at intermediate Reynolds numbers turbulence is transient and that the lifetimes are distributed exponentially. However, these studies have remained inconclusive about a divergence in lifetimes above a critical Reynolds number. The extensive numerical results for flow in a box of width 2 and length 8 presented here cover observation times up to 12 000 units and show that while the lifetimes increase rapidly with Reynolds number, they do not indicate a divergence and therefore no transition to persistent turbulence. DOI: 10.1103/PhysRevE.81.015301 PACS number͑s͒: 47.27.Cn, 05.45.Ϫa, 47.27.ed, 47.52.ϩj Plane Couette flow ͑pCf͒, the flow driven by two parallel plates moving in opposite direction, belongs to a class of shear flows where transition to turbulence may be observed at flow speeds where the laminar profile is linearly stable against infinitesimal perturbations ͓1͔. In these systems, which also include pressure driven plane Poiseuille and pipe flow, laminar and turbulent dynamics coexist for the same flow speed ͓2͔. Linear stability of the laminar flow implies that a finite amplitude perturbation is needed to drive the system out of the laminar fixed point's basin of attraction and to trigger turbulence.In the transition region the reverse process, whereby the turbulent flow relaminarizes without external stimulus or any noticeable precursor, has been seen both experimentally and numerically ͓3,4͔. Therefore, the turbulent state must be associated not with a turbulent attractor but rather with a turbulent saddle. An important question is whether there is a critical Reynolds number above which the lifetimes diverge and the system undergoes a transition to an attractor ͓5͔. Previous evidence in pCf indicated a divergence ͓3,6͔, but the reanalysis of data in ͓7͔ failed to reproduce the divergence. Pipe flow is another flow where the situation is undecided ͓7-12͔. Beyond the immediate interest of the transition to turbulence in linearly stable shear flows, these studies may also contribute to our understanding of transiently chaotic systems in general ͓13͔.The critical Reynolds number in pCf has long been studied experimentally. Ever present fluctuations and perturbations in the flow can trigger a "natural" transition without explicit external stimulus. Experiments such as ͓14͔ indicate that this happens near Re= 370, with the usual definition of the Reynolds number as Re= U 0 h / with U 0 is the half the velocity difference between the plates, h half the gap width and the kinematic viscosity of the fluid. However, since the "natural" transition depends on uncontrolled background fluctuations, experiments with reproducible finite amplitude perturbations were developed. Early experiments ͓15͔ using a jet injection perturbation give Re c = 370Ϯ 10 above which turbulent spots are repo...
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