2.1.1 Mass Conservation

The equation of mass conservation is well known and does not need a long explanation to be derived. Both Eulerian and Lagrangian forms are subsequently given. The latter is less common in fluid dynamics but it deserves some attention, as it brings in some fundamental Lagrangian concepts and relationships.

Let us begin by addressing the Eulerian description. To this end, we consider a fixed arbitrary control volume , delineated by a surface S. The total mass of the fluid is governed by the following integral balance equation:

in which ?, u, and m are the density, the velocity, and the rate of mass production, respectively. All these fields are assumed to be continuous fields in terms of time t and Eulerian and Cartesian coordinates x. In this equation, d ³ x is the elementary volume of a fluid particle, d ? is the elementary surface without ward normal, and n is the unit vector. The classical Ostrogradsky formula yields , so that the previous equation is rewritten as

For the sake of clarity, the divergence of a vector V is denoted as ( V) or, alternatively, in the following. The classical local and instantaneous counterpart of the preceding equation is the continuity equation,

In the Lagrangian description, fluid particles follow trajectories, which are given by the relationship

which links the position of the fluid particle at time t