Mass Conservation

Let $\rho({\bf r},t)$ and ${\bf v}({\bf r},t)$ be the mass density and velocity of a given fluid at point ${\bf r}$ and time $t$ . Consider a fixed volume $V$ , surrounded by a surface $S$ . The net mass contained within $V$ is

$$M = \int_V \rho\,dV,$$ (1.32)

where $d V$ is an element of $V$ . Furthermore, the mass flux across $S$ , and out of $V$ , is [see Equation (1.29)]

$${\mit\Phi}_M = \oint_S \rho\,{\bf v}\cdot d{\bf S},$$ (1.33)

where $d{\bf S}$ is an outward directed element of $S$ . Mass conservation requires that the rate of increase of the mass contained within $V$ , plus the net mass flux out of $V$ , should equal zero: that is,

$$\frac{dM}{dt} +{\mit\Phi}_M=0$$ (1.34)

[cf., Equation (1.31)]. Here, we are assuming that there is no mass generation (or destruction) within $V$ (because individual molecules are effectively indestructible). It follows that

$$\int_V \frac{\partial \rho}{\partial t}\,dV + \oint_S\rho\,{\bf v}\cdot d{\bf S} = 0,$$ (1.35)

because $V$ is non-time-varying. Making use of the divergence theorem (see Section A.20), the previous equation becomes

$$\int_V\left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\,{\bf v})\right]dV = 0.$$ (1.36)

However, this result is true irrespective of the size, shape, or location of volume $V$ , which is only possible if

$$\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\,{\bf v}) = 0$$ (1.37)

throughout the fluid. The previous expression is known as the equation of fluid continuity, and is a direct consequence of mass conservation.