Speaker: Vaughan R. Voller, Civil, Environmental, and Geo- Engineering, University of Minnesota

Time : 15:00 - 16:00 CET (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

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Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Fractional Time Stefan Problem, Conservative Formulation, Convergent Numerical Solution

Abstract:  The Stefan problem [1] is the classical model for the heat conduction controlled melting of a solid. In a one-dimensional setting, where an initially solid material is melted by the application of a fixed temperature at x = 0, the solution of the Stefan problem exhibits a moving melt front s(t) that advances as a square root in time (s ∼ t^1/2 ).  Here, motivated by a physical example of melting a material containing a fractal distribution of insulating elements, we consider a Stefan problem where the time derivative in the governing Fourier heat conduction equation is replaced by a fractional derivative of order 0 < β < 1. In a one dimension domain this results in a melt front that advances as s ∼ t ^β/2 – sub-diffusion. In this seminar two core features of this problem are emphasized. In the first place, by considering an arbitrarily fixed control volume and defining a Memory Enthalpy function, the fractional time Stefan problem (the Memory Stefan Problem) is formulated to explicitly satisfy a conservation of heat through time [2]. Secondly, a time explicit finite difference scheme for the solution of a one-dimensional Memory Stefan Problem is presented [3]. This scheme is verified by demonstrating convergence to a classic memory free Stefan problem, and to a recently presented closed analytical solution [4] for a one-phase Memory Stefan Problem–a problem where the initial solid is an the melt temperature. Additional numerical solutions also indicate convergence of the proposed numerical solution for cases where the initial solid is below the phase change temperature –two phase Memory Stefan Problems.

Contributions to this work from Sabrina Roscani, Lucas Venturato, Universidad Austral, Argentina and Nahuel Caruso, Universidad Nacional de Rosario, Argentina are acknowledged.

Biography:  Vaughan Voller received his PhD in applied mathematics from the University of Sunderland, UK, in 1980. Since 1985 he has been a professor in the Department of Civil, Environmental and Geo- Engineering at the University of Minnesota, USA. His work centers on the study and development  of analytical and numerical methods for free and moving boundary problems, with an emphasis on phase change problems. In this endeavor, he has considered a wide swath of applied problems, including alloy solidification processes, sequestration of CO2, and the formation of sedimentary ocean deltas. In terms of theoretical contributions, Prof. Voller had a leading role in the development of  so called ``Enthalpy-Porosity Methods", a common method of choice for  phase change algorithms in computational fluid dynamics codes. Recently, through applications of fractional calculus, Prof. Voller has directed attention towards phase change problems that exhibit memory and non-local effects. Here, his main focus is directed at developing conserved physically realizable models and robust numerical solutions.

Bibliography

[1] J. Crank. Free and Moving Boundary Problems. Clarendon Press, Oxford, 1984.
[2] V. Voller and S. Roscani. “A general non-Fourier Stefan problem formulation that accounts for memory effects”. In: Int. J. Heat and Mass Transfer 209 (2023), 124094.
[3] V. R. Voller. “Fractional Stefan problems”. In: Int. J. Heat and Mass Transfer 74 (2014), pp.269-277.
[4] N. Caruso, S. Roscani, L. Venturato, and V. Voller. “On computation of prefactor of free boundary in one dimensional one-phase fractional Stefan problems”. In: Fractal and Fractional 9.7 (2025), 397.