Having introduced our continuum, or hydrodynamic, variables, we now
need to construct equations of motion for these variables. We do this
by appealing to conservation laws, such as conservation of mass,
momentum, and energy. In this section we will focus on the conservation
of mass and derive the *equation of continuity* for the mass
density. Consider an arbitrary, fixed volume *V*, inside the fluid
(see Fig. 2.3).

**Figure 2.3:** Fluid volume used for the derivation
of the continuity equation.

where the minus sign has been added since the mass is decreasing, and the
last equality follows from the divergence theorem. Now the total mass in the
volume *V* is

and the rate at which this mass is changing with time is

Since mass is conserved in an arbitrarily small volume *V*, the integrands
of Eqs. (2.1) and (2.3) must be equal, and we have

This is the equation of continuity, which is a mathematical statement of conservation of mass.

Mon Aug 11 22:46:35 EDT 1997