The main topic of this book is an analysis and numerical simulation of a mathematical model consisting of magnetohydrodynamic equations describing the motion of two immiscible incompressible Newtonian fluids. The prototypical industrial application of this model is a simulation of reduction cells for the production of aluminium. However, the authors do not restrict themselves only to this specific application and cover a wide area of related topics. The book starts with a brief explanation of principles leading to the governing equations of magnetohydrodynamics. The resulting model consists of the incompressible Navier-Stokes equations with the Lorentz force on the right-hand side, a parabolic equation for the magnetic induction depending on the fluid velocity and appropriate boundary conditions. Chapters 2 and 3 are devoted to the analysis of the one-fluid problem and to its numerical solution by the finite element method. The emphasis is on the coupling between hydrodynamic and electromagnetic phenomena. The material is rather standard but very useful for readers not familiar with the respective techniques. Moreover, it allows the authors to concentrate only on key issues, with a special emphasis on nonlinear phenomena.

The next two chapters are the central chapters of the book and are mainly based on original research by the authors. These chapters deal with the mathematical analysis and numerical solution of multifluid magnetohydrodynamic problems, where the basic additional difficulty is a geometric nonlinearity due to the presence of free interfaces separating the immiscible fluids. The discretization is based on the popular arbitrary Lagrangian Eulerian method but other approaches are also briefly mentioned. The final chapter is devoted to the industrial application that motivated the preceding five chapters and it serves as an illustration of the efficiency of the approach presented in the book. The comprehensive text is well written and it contains many examples of numerical simulations and many references to relevant literature. It is intended for mathematicians, engineers and physicists and it will also be valuable for experts in mathematical and numerical analysis of magnetohydrodynamics as well as for those who want to learn the basic issues in this fascinating area.