Explore the theoretical principles of fluid dynamics in multiphase flows and their applications in various chemical engineering processes. Discuss the fundamental equations governing multiphase flow behavior, including the Eulerian$-$Eulerian and Eulerian$-$Lagrangian approaches, and their significance in modeling phenomena such as bubble columns, slurry reactors, and pneumatic conveying systems. How do factors such as phase distribution, interfacial forces, and flow regimes influence the overall performance and efficiency of multiphase flow systems?