Gerbeau J.-F., Bris Le C., Lelievre T. Mathematical methods for the Magnetohydrodynamics of Liquid Metals

Oxford: Oxford University Press. – 2006. – 320 p. (Numerical mathematics and scientific computation) This book focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamics phenomena. The emphasis is laid on the magnetohydrodynamics of liquid metals, and on a prototypical industrial application. Starting from a good understanding of the physics at play, the approach is mathematical in nature, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the algorithmic strategies used in the simulations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature. The guideline followed throughout the book is the study of a particular setting of the MHD equations that models a real industrial case. This specific industrial case is the main topic addressed in the final chapter of the book. In a nutshell, one could probably say that this book starts from the balance equations of continuum mechanics, passes through the notion of weak convergence in Hilbert spaces, and goes all the way to the simulation of a process of metal industry. Chapter 1 presents how some standard MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modeling the electromagnetic phenomena. The matter of Chapter 2 is the modeling of one-fluid problems. At the numerical level, the difficulties due to nonlinearities translate into several numerical challenges. Dedicated finite elements methods, with ad hoc mixed formulations or stabilization techniques must be designed. This is the purpose of Chapter
3. Chapters 4 and 5 are the central chapters of the book. They deal with multifluid MHD problems, on the theoretical side and on the numerical side, respectively. The closing chapter of the book, Chapter 6, is entirely devoted to one industrial application, the simulation of the industrial production of aluminum in electrolytic cells.The magnetohydrodynamics equations
Mathematical analysis of one-fluid problems
Numerical approximation of one-fluid problems
Mathematical analysis of two-fluid problems
Numerical simulation of two-fluid problems
MHD models for one industrial application