The course is of postgraduate level, and its main goal is to introduce and elaborate on methods of analysis, from first principles, of fluid flow and their application in a wide range of spatial scales, from cellular to atmospheric. The basic aim of the course is to bring together the mathematical formulation and the physical understanding of the flow. The course builds on basic undergraduate courses, such as Transport Phenomena I: Fluid Mechanics, Transport Phenomena II: Heat and Mass Transport; also, on courses that cover fluid flow phenomena and processes, such as Electro-Mechanical Process Equipment, Unit Operations, Chemical Reaction Engineering. Theoretical teaching is combined with computational exercises. The course covers the following topics: Elements of vector and tensor calculus and continuum mechanics. Differential analysis of flow. Analysis of flow-transport problems analysis with the methods of separation of variables, similarity transformations, perturbation theory and order-of-magnitude theory. Boundary layer flow - differential and Integral analysis. Elements of interfacial fluid mechanics. Analysis of simultaneous transport phenomena analysis - with convection, diffusion and chemical reaction.