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Beyond the Classical Stefan Problem Beyond the classical Stefan problem Francesc Font Martinez PhD thesis Supervised by: Prof. Dr. Tim Myers Submitted in full fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics in the Facultat de Matem`atiques i Estad´ıstica at the Universitat Polit`ecnica de Catalunya June 2014, Barcelona, Spain ii Acknowledgments First and foremost, I wish to thank my supervisor and friend Tim Myers. This thesis would not have been possible without his inspirational guidance, wisdom and expertise. I appreciate greatly all his time, ideas and funding invested into making my PhD experience fruitful and stimulating. Tim, you have always guided and provided me with excellent support throughout this long journey. I can honestly say that my time as your PhD student has been one of the most enjoyable periods of my life. I also wish to express my gratitude to Sarah Mitchell. Her invaluable contributions and enthusiasm have enriched my research considerably. Her remarkable work ethos and dedication have had a profound effect on my research mentality. She was an excellent host during my PhD research stay in the University of Limerick and I benefited significantly from that experience. Thank you Sarah. My thanks also go to Vinnie, who, in addition to being an excellent football mate and friend, has provided me with insightful comments which have helped in the writing of my thesis. I also thank Brian Wetton for our useful discussions during his stay in the CRM, and for his acceptance to be one of my external referees. Thank you guys. Agraeixo de tot cor el suport i l’amor incondicional dels meus pares, la meva germana i la meva tieta. Ells han sigut des de sempre la meva columna vertebral i, sense cap dubte, mai no hauria arribat on s´oc si no hagu´es estat per ells. Tamb´e vull dedicar un record especial al meu avi que, tot i ja no ser entre nosaltres, sempre estava pendent de mi i dels meus progressos, i s’il lusionava amb cada petit repte que anava aconseguint. Sempre el tindre · present. Tamb´evull dedicar unes paraules especials a la Gloria que, a part dajudar-me en la iii iv ACKNOWLEDGMENTS vessant art´ıstica de la tesi, ha esdevingut durant aquestultim ´ any la meva font permanent de felicitat, calidesa i amor. Infinites gr`acies a tots vosaltres. Tamb´etinc paraules especials per (enumero per ordre alfab`etic, perqu`ening´use m’enfadi) en David, en Guillem, en Jordi, en Josep, en Marc, en Pau, en Roger i en Sergi, els meus eterns proveidors de somriures. Potser sense adonar-vos-en, per`otamb´eheu format part de l’energia necessaria per dur a terme un projecte com aquest. Sou els millors. M’agradaria tamb´edonar les gr`acies als meus companys del CRM, l’Esther, l’Anita, el Dani, el Francesc i tota la tropa que han vingut despr´es, pels bons moments i bon ambient viscuts al despatx. Ovbiously, a special shout out to Michelle for being the best possible PhD comrade and a good friend. Tamb´evull agrair-li a l’Albert el bon humor, les estones de fer petar la xerrada a la uni, els caf`es i els dinars que hem fet. Moltes gr`acies a tots. Finalment, vull agrair al Centre de Recerca Matem`atica el finan¸cament rebut per a la re- alitzaci´od’aquesta tesi doctoral. Dono les gr`acies tamb´ea tots els membres de l’administraci´o i la direcci´odel centre per tots aquests anys de bons moments i feina ben feta. A tots vosaltres, gr`acies de tot cor. Outline The main body of this thesis is based on the research papers published since I started my PhD in September 2010. The papers listed below, from 1 to 5, correspond to the chapters 2, 3, 4, 5, 6, respectively. Paper 6 is a review of this whole body of work, which will most likely be submitted to SIAM Review. Chapter 1 is an introduction to the topic. Chapter 7 contains the conclusions. Chapters 8 and 9 are the Appendix and Bibliography, respectively. The abstract and conclusions are written in both English and Catalan. 1. F. Font, T.G. Myers. Spherically symmetric nanoparticle melting with a variable phase change temperature, Journal of Nanoparticle Research, 15, 2086 (2013). Impact factor: 2.175, [29]. 2. F. Font, T.G. Myers, S.L. Mitchell. Mathematical model for nanoparticle melting with density change, Microfluidics and Nanofluidics, DOI:10.1007/s10404-014-1423-x (May 2014). Impact factor: 3.218, [30]. 3. F. Font, S.L. Mitchell, T.G. Myers. One-dimensional solidification of supercooled melts, International Journal of Heat and Mass Transfer, 62, 411-421 (2013). Impact factor: 2.315, [28]. 4. T.G. Myers, S.L. Mitchell, F. Font. Energy conservation in the one-phase supercooled Stefan problem, International Communications in Heat and Mass Transfer, 39, 1522- 1225 (2012). Impact factor: 2.208, [74]. 5. T.G. Myers, F. Font. On the one-phase reduction of the Stefan problem, Submitted to International Communications in Heat and Mass Transfer (May 2014). Impact factor: v vi OUTLINE 2.208, [75]. 6. F. Font, T.G. Myers, S.L. Mitchell. Beyond the classical Stefan problems, To be sub- mitted to SIAM Review (July 2014). Impact factor: 8.414. Abstract In this thesis we develop and analyse mathematical models describing phase change phenom- ena linked with novel technological applications. The models are based on modifications to standard phase change theory. The mathematical tools used to analyse such models include asymptotic analysis, similarity solutions, the Heat Balance Integral Method and standard numerical techniques such as finite differences. In chapters 2 and 3 we study the melting of nanoparticles. The Gibbs-Thomson relation, accounting for melting point depression, is coupled to the heat equations for the solid and liquid and the associated Stefan condition. A perturbation approach, valid for large Stefan numbers, is used to reduce the governing system of partial differential equations to a less complex one involving two ordinary differential equations. Comparison between the reduced system and the numerical solution shows good agreement. Our results reproduce interesting behaviour observed experimentally such as the abrupt melting of nanoparticles. Standard analyses of the Stefan problem impose constant physical properties, such as density or specific heat. We formulate the Stefan problem to allow for variation at the phase change and show that this can lead to significantly different melting times when compared to the standard formulation. In chapter 4 we study a mathematical model describing the solidification of supercooled liquids. For Stefan numbers, β, larger than unity the classical Neumann solution provides an analytical expression to describe the solidification. For β 1, the Neumann solution is ≤ no longer valid. Instead, a linear relationship between the phase change temperature and the front velocity is often used. This allows solutions for all values of β. However, the vii viii ABSTRACT linear relation is only valid for small amounts of supercooling and is an approximation to a more complex, nonlinear relationship. We look for solutions using the nonlinear relation and demonstrate the inaccuracy of the linear relation for large supercooling. Further, we show how the classical Neumann solution significantly over-predicts the solidification rate for values of the Stefan number near unity. The Stefan problem is often reduced to a ’one-phase’ problem (where one of the phases is neglected) in order to simplify the analysis. When the phase change temperature is variable it has been claimed that the standard reduction loses energy. In chapters 5 and 6, we examine the one-phase reduction of the Stefan problem when the phase change temperature is time- dependent. In chapter 5 we derive a one-phase reduction of the supercooled Stefan problem, and test its performance against the solution of the two-phase model. Our model conserves energy and is based on consistent physical assumptions, unlike one-phase reductions from previous studies. In chapter 6 we study the problem from a general perspective, and identify the main erroneous assumptions of previous studies leading to one-phase reductions that do not conserve energy or, alternatively, are based on non-physical assumptions. We also provide a general one-phase model of the Stefan problem with a generic variable phase change temperature, valid for spherical, cylindrical and planar geometries. ix Resum En aquesta tesi construirem i analitzarem models matem`atics que descriuen processos de transici´ode fase vinculats a noves tecnologies. Els models es basen en modificacions de la teoria est`andard de canvis de fase. Les t`ecniques matem`atiques per resoldre els models es basen en l’an`alisi asimpt`otic, solucions autosimilars, el m`etode de la integral del balan¸cde la calor i m`etodes num`erics est`andards com ara el m`etode de les difer`encies finites. En els cap´ıtols 2 i 3 estudiarem la transici´osolid-l´ıquid d’una nanopart´ıcula, acoblant la relaci´ode Gibbs-Thomson, que descriu la depressi´ode la temperatura de fusi´oen una superf´ıcie corba, amb l’equaci´ode la calor per la fase s`olida i l´ıquida, i la condici´ode Stefan. Mitjan¸cant el m`etode de pertorbacions, per a valors grans del nombre de Stefan, el problema d’equacions en derivades parcials inicial ´es redu¨ıt a un sistema m´es senzill de dues equacions diferencials ordin`aries. La soluci´odel sistema redu¨ıt concorda perfectament amb la soluci´o num`erica del sistema inicial d’equacions en derivades parcials. Els resultats confirmen la transici´oultra r`apida de s`olid a l´ıquid observada en experiments amb nanopart´ıcules.
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