Mass conservation -- Momentums conservation -- Derivatives for the equations of state -- On the variety of notations of the energy conservation for single-phase flow -- First and second laws of the thermodynamics -- Some simple applications of the mass and energy conservation -- Exergy of multi-phase multi-component systems -- One-dimensional three-fluid flows -- Detonation waves caused by chemical reactions or by melt-coolant interactions -- Conservation equations in general curvilinear coordinate systems -- Type of the system of PDEs -- Numerical solution methods for multiphase flow problems -- Numerical methods for multi-phase flow in curvilinear coordinate systems.Multi-phase flows are part of our natural environment such as tornadoes, typhoons, air and water pollution or volcano activities as well as of industrial technology such as power plants, combustion engines, propulsion systems, or chemical and biological industry. The industrial use of multi-phase systems requires analytical and numerical strategies for predicting their behavior. In its second extended edition this monograph contains theory, methods and practical experience for describing complex transient multi-phase processes in arbitrary geometrical configurations, providing a systematic presentation of the theory and practice of numerical multi-phase fluid dynamics. In the present first volume the fundamentals of multiphase dynamics are provided, as well as various interactive multimedia demonstrations on an accompanying CD-ROM. This second edition includes various new material such as a new chapter devoted to the mathematical tools and types of systems of partial differential equations, a completely rewritten chapter presenting the derivatives for the equations of state, as well as various extensions and improvements in all the other book chapters, including additional experiments and movies in the accompanying CD.