Math Modeling Seminar: Modeling of Thin Film Flows

Math Modeling Seminar
Modeling of Thin Film Flows

Dr. Steven J. Weinstein
Professor and Department Head
Department of Chemical Engineering, RIT

You may attend this lecture in person at 2305 Gosnell Hall or virtually via Zoom.
If you’d like to attend virtually, you may register here for Zoom link.

Abstract:
Laminar flow of thin liquid films is fundamental to coating processes. Even with modern high-powered computers, numerical solutions of the full set of governing equations are often time-intensive; this is due to the nonlinearities inherent in free surface flows, as well as the small aspect ratios typically involved. It is often the case, however, that the film thickness varies gradually in the flow direction in various regions of a coating process. Consequently, the classical boundary-layer approximation to the Navier-Stokes equation is justified. A standard approach to the boundary layer equations, attractive because of its ease of use, is to integrate the boundary layer equations across the film and introduce an assumed form of the velocity field. This velocity field is typically parabolic and self-similar, in that it does not change its basic shape in the direction of the flow. The result is a simplified equation, henceforth referred to as the film equation, involving only the unknown film thickness. In many cases, the film curvature is small and surface tension terms in the film equation are negligible; the resulting first-order nonlinear differential equation is typically straightforward to solve, oftentimes in analytical form.
Under conditions of negligible surface tension, the film equation can exhibit a singularity called a critical point, which is related to a change in the direction of the underlying wave propagation. In many problems, the singularity may be eliminated by a particular choice of parameters, and a physically correct model of the flow may be obtained. Singularity elimination can occur in configurations where the fluid accelerates in the primary flow direction (such as in weir flows or rapid dip coating) and the wave propagation thus undergoes a subcritical-to-supercritical transition. On the other hand, a non-removable singularity arises in configurations where the fluid is decelerating in the primary direction of flow, and the wave propagation thus transitions from supercritical to subcritical.  In this talk, we specifically examine decelerating flow on an inclined plane, which is perhaps the simplest configuration that can exhibit a non-removable singularity. Finite element predictions and numerical solutions of the full boundary layer equations indicate that interface solutions having small slopes should exist under conditions where the film equation fails. Using both physical and mathematical arguments, we demonstrate that the film equation must be modified for a velocity profile of changing shape. The resulting predictions are found to compare favorably with finite element solutions, the full boundary layer equations, and experiments.

Speaker Bio:
Dr. Steve Weinstein received his B.S. in Chemical Engineering from the University of Rochester and his MS/PhD in Chemical Engineering from the University of Pennsylvania. He worked for Eastman Kodak Company for eighteen years after receiving his PhD, and joined RIT in January of 2007. Dr. Weinstein is well published in the field of coating and interfacial fluid mechanics, and has focused on the dynamics and hydrodynamic stability of thin liquid films, curtain flows (flows in thin sheets of liquid), die manifold design via asymptotic methods, and web dynamics; he also has seven patents in these areas. He founded the Department of Chemical Engineering at RIT in the fall of 2008 and has continued as its department head. Dr. Weinstein has worked with colleague Nate Barlow at RIT to develop and demonstrate a novel approximant method to sum divergent series solutions that naturally arise in the analysis of various problems in mathematical physics. Dr Weinstein routinely provides mathematical analysis to support collaborators beyond his core research areas, and thus has a publication record that spans disparate fields. Read more here.

Intended Audience
Undergraduates, graduates, and experts. Those with interest in the topic.

The Math Modeling Seminar will recur each week throughout the semester on the same day and time. Find out more about upcoming speakers on the Mathematical Modeling Seminar Series webpage.
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