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 application of 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 for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.
Cite this article
Shawn W. Walker, Andrea Bonito, Ricardo H. Nochetto, Mixed ﬁnite element method for electrowetting on dielectric with contact line pinning. Interfaces Free Bound. 12 (2010), no. 1, pp. 85–119