Mixed finite element method for electrowetting on dielectric with contact line pinning
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We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The applicationof interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization.We also prove the convergence of an Uzawa algorithm forsolving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence. medium equation. We prove that starting with an initial datum with some kind of suboptimal - concavity property, the convexity of the positivity set can be lost in a short time. European Mathematical Society 2010.
