Shawn W. Walker

Modeling the fluid dynamics of electrowetting on dielectric (EWOD)

This paper discusses the modeling and simulation of a parallel-plate Electrowetting On Dielectric (EWOD) device that moves fluid droplets through surface tension effects. We model the fluid dynamics by using Hele-Shaw type equations with a focus on including the relevant boundary phenomena. Specifically, we show that contact angle saturation and hysteresis are needed to predict the correct shape and time scale of droplet motion. We demonstrate this by comparing our simulation to experimental data for a splitting droplet. Without these boundary effects, the simulation shows the droplet splitting into three pieces instead of two and the motion is over 15 times faster than the experiment. We then show how including the saturation characteristics of the device, and a simple model of contact angle hysteresis, allows the simulation to better predict the splitting experiment. The match is not perfect and suffers mainly because contact line pinning is not included. This is followed by a comparison between our simulation, whose parameters are now frozen, and a new experiment involving bulk droplet motion. Our numerical implementation uses the level set method, is fast, and is being used to design algorithms for the precise control of microdroplet motion, mixing, and splitting

Electrowetting with contact line pinning: Computational modeling and comparisons with experiments

This work describes the modeling and simulation of planar electrowetting on dielectric devices that move fluid droplets by modulating surface tension effects. The fluid dynamics are modeled by Hele-Shaw type equations with a focus on including the relevant boundary phenomena. Specifically, we include contact angle saturation and a contact line force threshold model that can account for hysteresis and pinning effects. These extra boundary effects are needed to make reasonable predictions of the correct shape and time scale of liquid motion. Without them the simulations can predict droplet motion that is much faster than in experiments (up to 10–20 times faster). We present a variational method for our model, and a corresponding finite element discretization, which is able to handle surface tension, conservation of mass, and the nonlinear contact line pinning in a straightforward and numerically robust way. In particular, the contact line pinning is captured by a variational inequality. We note that all the...

Optimization of chiral structures for microscale propulsion.

Recent advances in micro- and nanoscale fabrication techniques allow for the construction of rigid, helically shaped microswimmers that can be actuated using applied magnetic fields. These swimmers represent the first steps toward the development of microrobots for targeted drug delivery and minimally invasive surgical procedures. To assess the performance of these devices and improve on their design, we perform shape optimization computations to determine swimmer geometries that maximize speed in the direction of a given applied magnetic torque. We directly assess aspects of swimmer shapes that have been developed in previous experimental studies, including helical propellers with elongated cross sections and attached payloads. From these optimizations, we identify key improvements to existing designs that result in swimming speeds that are 70-470% of their original values.

A control method for steering individual particles inside liquid droplets actuated by electrowetting

An algorithm is developed that allows steering of individual particles inside electrowetting systems by control of actuators already present in these systems. Particles are steered by creating time varying flow fields that carry the particles along their desired trajectories. Results are demonstrated using an experimentally validated model developed in ref. . We show that the current UCLA electro-wetting-on-dielectric (EWOD) system contains enough control authority to steer a single particle along arbitrary trajectories and to steer two particles, at once, along simple paths. Particle steering is limited by contact angle saturation and by the small number of actuators that are available to actuate the flow in practical electrowetting systems.

Shape optimization of peristaltic pumping

Transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this; it is also important to many industrial processes. We present a variational method for optimizing the wave shape of a peristaltic pump. Specifically, we optimize the wave profile of a two dimensional channel containing a Navier-Stokes fluid with no assumption on the wave profile other than it is a traveling wave (e.g. we do not assume it is the graph of a function). Hence, this is an infinite-dimensional optimization problem. The optimization criteria consists of minimizing the input fluid power (due to the peristaltic wave) subject to constraints on the average flux of fluid and area of the channel. Sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming (SQP) method to find a solution of the first order KKT conditions. We also use a merit-function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape. Our numerical implementation uses a finite element method for computing a solution of the Navier-Stokes equations, adjoint equations, as well as for the SQP method when computing perturbations of the channel shape. The walls of the channel are deformed by an explicit front-tracking approach. In computing functional sensitivities with respect to shape, we use L^2-type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature; we show error estimates for the boundary stress and tangent field approximations. As a result, we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature. Specifically, we see highly asymmetric wave shapes that are far from being sine waves. Many examples are shown for a range of fluxes and Reynolds numbers up to Re=500 which illustrate the capabilities of our method.

Mixed finite element method for electrowetting on dielectric with contact line pinning

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.

The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative

Many things around us have properties that depend on their shape-for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a shape variable. This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts. Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included. Audience: This book is a convenient reference for various shape derivative formulas and should be of value to anyone interested in surface geometry and shape optimization. Graduate students can use it to quickly get up to speed on the machinery of shape differential calculus. Scientists studying continuum mechanics, fluid mechanics, numerical analysis, and PDEs will find the book helpful for problems in which surface geometry is critical and/or geometry evolves in time. Those who want to learn the basics of shape differentiation will also find it useful.

A hybrid variational front tracking-level set mesh generator for problems exhibiting large deformations and topological changes

We present a method for generating 2-D unstructured triangular meshes that undergo large deformations and topological changes in an automatic way. We employ a method for detecting when topological changes are imminent via distance functions and shape skeletons. When a change occurs, we use a level set method to guide the change of topology of the domain mesh. This is followed by an optimization procedure, using a variational formulation of active contours, that seeks to improve boundary mesh conformity to the zero level contour of the level set function. Our method is advantageous for Arbitrary-Lagrangian-Eulerian (ALE) type methods and directly allows for using a variational formulation of the physics being modeled and simulated, including the ability to account for important geometric information in the model (such as for surface tension driven flow). Furthermore, the meshing procedure is not required at every time-step and the level set update is only needed during a topological change. Hence, our method does not significantly affect computational cost.

A Finite Element Method for Nematic Liquid Crystals with Variable Degree of Orientation

We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field

A Mixed Finite Element Method for EWOD That Directly Computes the Position of the Moving Interface

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