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A recently published experimental study by Xia and Steen examines the connection between the contact line velocity and contact angle of liquid drops on an oscillating substrate that display contact angle hysteresis. Of particular interest in this experimental study is the analysis of the dependence of contact angle deviation on contact line velocity. Indeed it is found that for small angular displacements, there is a linear relationship between the two. Moreover, the oscillating drop exhibits contact angle hysteresis that is much greater than that measured from quasi-static experiments. Here we use a phase-field model of dynamic wetting which directly includes the contact angle hysteresis to simulate the results of the aforementioned authors. A thorough derivation of the governing equations is presented, starting from the pioneering work of Cahn and Hilliard. Our model is unique due to the explicit inclusion of contact angle hysteresis, a phenomenon that has proven quite difficult to model. We demonstrate that by choosing appropriate parameters, our model can achieve very good agreement with experimental data. Further, we compare to our results to those computed using another model, further validating our method. We then investigate the effects of contact line friction and the hysteresis window, which are otherwise very difficult to explore experimentally. / Master of Science / When a drop of liquid comes into contact with a solid surface, it forms a semi-spherical cap having a fixed contact angle defined as that between the wall and the line tangent to the liquid drop, at the point where the liquid meets the solid. In recent decades, researchers in fields as varied as mathematics, physics, chemistry and materials science have studied the spreading and contraction of liquid drops on solid surfaces of various chemical compositions. Of particular interest to this paper is a phenomenon called `contact angle hysteresis' (CAH), in which a liquid drop can remain stationary at multiple distinct contact angles. The presence of hysteresis in physical systems significantly complicates the analysis of the physical problem and has been a motivating factor for the development of mathematical models that can contend with CAH. Here we present a model for describing the motion of liquid drops which explicitly takes CAH into account.

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analyzed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The center-to-center distance between adjacent cylinders is a constant. The Biot-Savart law of induced velocities is used to determine the velocity field due to the free vorticity in the surrounding fluid and the bound vorticity distributed on the surface of each cylinder. The bound vorticity is needed to enforce the no-penetration condition and to account for the production of free vorticity in the solid surfaces. It is governed by a Fredholm integral equation of the second kind. This equation is solved by numerical techniques. The transport of free vorticity in the flow field is governed by the vorticity transport equation. This equation is discretized for a control volume and is solved numerically. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder, but necessitates use of a non-orthogonal grid. In order to test the numerical techniques, the simpler case of unsteady flow over a single circular cylinder at various Reynolds numbers if first considered. Results compare favorably with previous experimental and numerical data. Three cases for Reynolds numbers of 10², 10³, and 10⁴ are presented for the array of cylinders. The center-to-center distance is fixed at three diameters. The time development of constant vorticity contours as well as drag, lift, and moment coefficients are shown for each Reynolds number. The motion of stagnation and separation points with time is also given. It is found that the drag for a cylinder in the array may be as low as five percent of that for flow over a single cylinder at the same Reynolds number.

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