Bio: Lyudmyla Barannyk is an Associate Professor in the Department of Mathematics at the University of Idaho. Barannyk received a masters in Applied Mathematics from the New Jersey Institute of Technology and a PhD in Mathematics Sciences from the New Jersey Institute of Technology and Rutgers the State University of New Jersery. She is currently a visiting Associate Professor of Mathematics at the University of Michigan.
We study a simple model for the evolution of the solid-liquid interface during melting and solidification (Stefan problem) of a material with constant internal heat generation and prescribed heat flux at the boundary in the cylindrical geometry. The problem is motivated by the need to control the behavior of nuclear fuel rods in a potential meltdown scenario. The equations are solved by splitting them into transient and steady-state components and then using separation of variables. This results in an ordinary differential equation for the interface that involves infinite series. The initial value problem is solved numerically, and solutions are compared to the previously published quasi-static solutions. We show that when the internal heat generation and boundary heat flux are close in value, the motion of the phase change front takes longer to reach steady-state than when the values are farther apart. As the difference between the internal heat generation and boundary heat flux increases, the transient solutions become more dominant and the phase change front does not reach steady-state before the outer boundary or centerline is reached. Hence the difference between the internal heat generation and boundary heat flux can be used to control the motion and speed of the solid-liquid interface. Limitations of the present model and possible future extensions will be discussed.
This is joint work with Sidney Williams (Georgia Tech), Irene Ogidan (University of Idaho), John Crepeau (University of Idaho), and Alexey Sakhnov (Kutateladze Institute of Thermophysics, Novosibirsk, Russia).