26 th Annual Workshop on Mathematical Problems in Industry Rensselear Polytechnic Institute, June 14–18, 2010 Two problems on the flow of viscous sheets of molten glass “Flows related to drawing of viscous sheets” problem presented by Dr. John Abbott Corning Incorporated Advanced Modeling and Analysis group Corning New York Participants: Sean Bohun Chris Breward Ivan Christov Linda Cummings Don Drew Ian Griffiths Russ Howes Dave Martin Javed Siddique David Sondak Alexander Volfson Tom Witelski Haley Yaple Summary Presentation given by Ian Griffiths (6/18/10) Summary Report compiled by Sean Bohun, Chris Breward, Linda Cummings and Tom Witelski Abstract The problem brought to the 2010 Mathematical Problems in Industry Workshop by Corning Incorporated involved a manufacturing process for the production of high-quality thin glass sheets for many commercial applications (large screen video displays for example). The process begins with hot molten glass flowing out of an apparatus (essentially an overflowing triangular trough) that helps distribute the material into uniform sheets of desired width and controlled thickness. The design of the apparatus is fundamentally based on exploiting fluid dynamics to control the process. The MPI Workshop studied the fluid dynamics of two distinct stages of the process during the MPI workshop. Formulation of sound mathematical models of the glass flow were the basis for numerical simulations and stability analysis that can provide important insight on improving the reliability of the sheet production process. 1 Introduction The basic sheet glass production process considered by the workshop is illustrated schematically in Figs.1 through 4. A trough is filled with molten glass via an inlet, as shown in Fig.1 (the inlet is on the left in this figure). The exterior of the trough is a symmetric triangular wedge, while its interior is a sloped, more shallow, wedge (indicated by a dashed line in Fig.1). As the trough’s interior reservoir is filled, the molten glass overflows, and cascades down the two exterior walls of the trough. The two cascading sheets meet at the trough’s underside, coalesce, and form a single viscous sheet. This hanging sheet continues to fall under gravity, and also under external drawing forces that may be applied. As the sheet falls in this “extensional flow” regime it thins, and eventually is cooled and solidified downstream, at an appropriate final thickness. Clearly the whole production process relies on the system design generating the correct fluid dynamics. The desired outcome is a sheet of desired thickness and uniform cross-section, free of impurities. Any nonuniformities (such as ripples) in the final manufactured sheet may be linked back to the fluid dynamics in the molten state. It should therefore prove very useful to study these fluid dynamics in order to see how best to control and optimize the manufacturing process. 1