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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

$\displaystyle M = \int_V \rho\,dV,$ (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)]

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

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

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

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

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

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

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

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


next up previous
Next: Convective Time Derivative Up: Mathematical Models of Fluid Previous: Conservation Laws
Richard Fitzpatrick 2016-01-22