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Mass Conservation
Let
and
be the mass density and velocity of a given fluid at point
and time
. Consider a
fixed volume
, surrounded by a surface
. The net mass contained within
is
![$\displaystyle M = \int_V \rho\,dV,$](img252.png) |
(1.32) |
where
is an element of
.
Furthermore, the mass flux across
, and out of
, is [see Equation (1.29)]
![$\displaystyle {\mit\Phi}_M = \oint_S \rho\,{\bf v}\cdot d{\bf S},$](img253.png) |
(1.33) |
where
is an outward directed element of
. Mass conservation requires
that the rate of increase of the mass contained within
,
plus the net mass flux out of
, should equal zero: that is,
![$\displaystyle \frac{dM}{dt} +{\mit\Phi}_M=0$](img254.png) |
(1.34) |
[cf., Equation (1.31)].
Here, we are assuming that there is no mass generation (or destruction) within
(because individual molecules
are effectively indestructible).
It follows that
![$\displaystyle \int_V \frac{\partial \rho}{\partial t}\,dV + \oint_S\rho\,{\bf v}\cdot d{\bf S} = 0,$](img255.png) |
(1.35) |
because
is non-time-varying. Making use of the divergence theorem (see Section A.20),
the previous equation becomes
![$\displaystyle \int_V\left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\,{\bf v})\right]dV = 0.$](img256.png) |
(1.36) |
However, this result is true irrespective of the size, shape, or location of volume
, which is only possible if
![$\displaystyle \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\,{\bf v}) = 0$](img257.png) |
(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: Convective Time Derivative
Up: Mathematical Models of Fluid
Previous: Conservation Laws
Richard Fitzpatrick
2016-03-31