ABSTRACT
To understand macroscopic phenomena involving motion of fluids (both liquids and gases), we work with mathematical models which characterize them by their kinematic and dynamic properties such as velocity, mass density, and pressure. In mathematical models for solid mechanics, the description of these properties (field variables) are for each material point of the body and their evolution over time. This type of description is known as the Lagrangian representation. In contrast, mathematical theories for fluid mechanics are more conventionally formulated in an Eulerian description instead, although Lagrangian representation is also used sometimes. In the Eulerian representation, the field variables are given as functions of position in space x and time t. In particular, v(x,t) is the velocity vector of a fluid particle occupying the point x in space at time t; ρ{x,t) and ρ(x, t) are the corresponding mass density of a fluid element which happens to occupy x at time t and the pressure acting on the element.
