1 - Lagrangian kinematics

Summary

Conservation of particle identity

The essence of Lagrangian fluid dynamics is fluid particle identity acting as an independent variable. The identifier or label may be the particle position at some time, but could for example be a triple of the thermodyamic properties of the particle at some time. Time after labeling is the other independent variable. The fluid particle may not actually have been released into the flow at the time of labeling, but merely labeled with position or with some other properties at that time. Nevertheless, “time of release” will be used interchangeably with “labeling time.” The subsequent position of the particle is a dependent variable, even though it may coincide with the independently chosen position of an Eulerian observer at the subsequent time. The Eulerian observer also employs time, after some convenient initial instant, as the other independent variable. Of course, a particle path can be calculated in the Eulerian framework by integrating velocity on the path, with respect to time. Indeed, the suppression or implicitness of this detailed path information is the basis of the relative simplicity of the Eulerian formulation. On the other hand, fluid velocity is readily calculated from the particle position in the Lagrangian framework by the local operation of particle differentiation with respect to time after labeling.

Conservation of particle identity is not an immediately compelling consideration in the Eulerian framework, but is fundamental in the Lagrangian.