Non-local effects and size-dependent properties in Stefan problems with Newton cooling |
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| Institution: | 1. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, Barcelona 08193, Spain;2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, 08028, Spain;1. EPI DISCO Inria-Saclay, Laboratoire des Signaux et Systèmes (UMR CNRS 8506), CNRS, CentraleSupélec, Université Paris-Sud, 3 rue Joliot Curie, Gif-sur-Yvette 91192, France;2. Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile;1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam;2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam;3. School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China |
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| Abstract: | We model the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment. In the limit of a large Biot number, this condition becomes the commonly used fixed-temperature condition. It is shown that in practice the size of this non-dimensional number is very small. We study the effect of a small Biot number on the solidification process with numerical and asymptotic solution methods. The study indicates that non-local effects become less important as the Biot number decreases. |
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