# Math Modeling Holds the Key to Manufacturing Breakthroughs

Posted July 2011

Many modern manufacturing processes rely on extrusion: either pushing or drawing a thick fluid containing suspended hard particles through a dye to form parts. This process is so complicated that—until now—understanding it has been more of an art form than a science.

The way that the material being extruded behaves can vary widely, depending on factors such as temperature, pressure, the fluids, the size and shape of the particles—even the manufacturing equipment itself. Understanding how all these factors impact an extrusion process is one of the keys to developing successful extrusion processes.

These factors can be measured in a laboratory—but the results reflect only one specific mixture. Wouldn't it be better to be able to predict the effective properties of any fluid mixture? Weigang Zhong developed a mathematical model that might enable researchers to do just that.

A post-doctoral fellow at the IMA from 2008 through 2010, Zhong collaborated with Corning Incorporated to develop ways to characterize and predict the behavior of fluids in extrusion processes. Corning is the world leader in developing and manufacturing specialty glass and ceramic products, and their manufacturing processes commonly involve extrusion. Working with Corning scientist Amy Rovelstad, Weigang devised a computer simulation of flows of a fluid containing hard particles of arbitrary shapes.

In Zhong's simulation, the motion of the fluid is modeled by the Navier-Stokes equations, which are commonly used by mathematicians to describe flow. Each suspended object is tracked as it interacts with other particles, the suspending fluid, and the solid boundaries (such as the die used for the extrusion). The equations were solved using a powerful computational scheme based on the Immersed Boundary Method—a numerical method for calculating the interactions of solids and fluids developed in the late 1970s by Charles Peskin of the Courant Institute for simulating blood flow in the heart.

The research is intended to help engineers explore the effects of changing the parameters of the fluid and the suspended particles—as well as the conditions that generate the flow. Such a tool, when realized, will help lead to major improvement of manufacturing processes that involve extrusion.