**Project leader:** Professors Thomas Witelski and Jeffrey Wong**Project manager:** Jingzhen Hu**Team members:** Daniel Hwang and Elliott Kauffman

**Project summary:**

We investigated the problem of a thin viscous fluid film flowing down an inclined plane and how this fluid is affected by the introduction of a rigid plate, or a “slider”, floating on the surface. The fluid’s motion is governed by the Navier-Stokes equations, reduced using lubrication theory. The problem involved combining two common fluid dynamics problems: the slider bearing problem, defined by the fluid-slider interface, and the free-flow problem, defined by the fluid-air interface. The slider has three potential degrees of freedom: horizontal movement, vertical movement, and tilting. For convenience, we decided to focus on the former two. The solution to the overall problem involved coupling parameters between the different regions to ensure consistency. We used finite difference schemes and Euler’s method to simulate this problem in MATLAB. The height of the fluid in the free-flow regions was calculated using the convective porous medium equation. Since the boundary is moving, we applied a change of variables to map onto a fixed computational domain. A force balance equation was derived to describe the horizontal motion of the slider. The vertical motion of the slider was determined by ensuring compatibility with the pressure boundary conditions. We found that as the slider density or size decreases, the maximum speed of the slider approaches the surface speed of the undisturbed flow. Test points following the velocity field were observed within the fluid to demonstrate the effect of the slider on the fluid’s motion.