1. Elements of Continuum Mechanics for Fluid Flow and General Stress–Strain System

The fundamental physical laws in fluid flows are obtained from the conservation of mass, translational momentum, and energy, with the latter being nothing but the first law of thermodynamics applied to a control volume system. While for solids and liquids, the properties vary continuously in their macroscopic states, it is not so apparent for gases, which is characterized by mobility at the molecular level. One describes the flow of gases from the average behavior of the molecular ensemble in the control volume. This approach presupposes the existence of very many large number of molecules in the control volume, so that the statistical description is feasible. This is the essence of the continuum assumption that despite the presence of discrete molecules constituting the system, one can approximate the assembly by continuous variation of macroscopic properties. Continuous variation of properties arises due to momentum and energy exchanges by incessant collisions of molecules taking place. One of the most relevant non-dimensional numbers, determining whether a gaseous flow can be considered continuum or one must use statistical approaches, is the Knudsen number.