JDP Student Ramakrishnan Thirumalaisamy and His Advisor Dr. Amneet Bhalla Solved a 130+ Year Old Unsolved Fluid Mechanics and Heat Transfer Problem Known as the Stefan Problem

Stefan's problem describes the evolution of the boundary between two phases of a material undergoing a phase change, such as ice melting into water. Josef Stefan proposed the Stefan problem in a set of four papers in 1890. A similar problem was first encountered in 1831 by Lame and Clapeyron while studying the solidification of the earth's crust.  A mathematical solution to Lame and Clapeyron's problem was provided by Neumann in 1860. Stefan consolidated phase change problems of his time into precise mathematical statements involving free-moving boundaries, which now bear his name. 

In the literature, Stefan's problems have been analyzed analytically (and numerically as well) by ignoring density variations in the solid and liquid phases. The reason for this is that whenever density changes during phase change, fluid flow is induced. Thus, the heat transfer problem is coupled with fluid mechanics, making it difficult to solve analytically. Without density changes, the Stefan problem is a standalone heat transfer problem. In a recent International Journal of Multiphase Flow paper, authored by Thirumalaisamy and Bhalla, the researchers solved the Stefan problem analytically by considering density variation between the two phases. As such, Thirumalaisamy and Bhalla solved the problem of coupled heat transfer and fluid mechanics. In addition, they developed a novel low Mach equation for the computational fluid dynamics (CFD) algorithm to capture the volume change of the material during phase transitions. The CFD algorithm is more general and includes a gas phase as well. The theoretical and computational framework is being developed to model metal additive manufacturing problems as part of Prof. Bhalla's NSF CAREER award.