## Gauss Theorem

**Introduction: **A positive divergence of the vector field means

There is a net flow of material out of an infinitesimal volume. Substituting into the continuity equation gives:

But now both integrals are over the same volume. Since the equation is true for any volume CV, it follows that the equation is true for the integrands alone:

This is the differential form of the continuity equation. An incompressible material cannot change density.

Hence,

From continuity

**Example: Flow into tank through orifice with A=0.0025m2.Flow out at rate of 0.003m3/s. What is the rate of change of mass in the tank?**

We want to know an average over the control volume, so any of the integral forms of the continuity equation will do.

This is the simplest form of the equation which can be used here. Notice: