2.4 Conservation of mass

The law of conservation of mass can be written

|----------------| @-|-+ r ( u) = 0| -@t--------------- \relax \special {t4ht=
where eqn is the mass density of the fluid. The equation can be derived by considering a volume eqn fixed in space (note !!), bounded by a surface eqn. If the volume is filled by a fluid with density eqn, its mass is eqn

The rate of increase of mass inside the volume must match the rate of inflow of mass across the volume’s surface. The latter is calculated by integrating the mass flux eqn over the surface, noting the negative sign due to eqn pointing out of the volume.

PICT\relax \special {t4ht=

Gauss’s Theorem relates surface and volume integrals by

Z Z (dS u) = r ( u) dV: S V \relax \special {t4ht=
Equating the rate of mass increase to rate of inflow and applying Eq. (2.9 ) gives
Z @--- V @t + r ( u) dV = 0: \relax \special {t4ht=
Since the integral is valid for any volume eqn, it follows that the integrand (in eqn) must equal 0, resulting in Eq. (2.8).


Divergence, denoted by eqn, indicates the tendency of a vector field to point outward of a closed surface. For example, When the divergence of velocity eqn is positive, the fluid is expanding; negative divergence indicates contraction. Imagine a volume of fluid with a complex distribution of eqn at its bounding surface, below left. The eqn calculation will isolate the diverging component (right) from uniform flow (centre).

PICT\relax \special {t4ht=

The divergence of velocity is calculated by integrating — i.e. summing — fluxes over the closed surface. To define divergence at a point, we consider the limiting case where the volume tends to zero. For a surface eqn that encloses a volume eqn, divergence is the flux across the surface per unit volume, as eqn, e.g.

 Z r u = lim --1- (dS u): V!0 V S \relax \special {t4ht=

The nabla operator

The nabla symbol eqn can be considered a vector operator

 @---@- -@- r @x;@y ;@z : \relax \special {t4ht=
Divergence is the inner product with eqn, e.g.
 @ux- @uy- @uz- r u = @x + @y + @z : \relax \special {t4ht=
Notes on CFD: General Principles - 2.4 Conservation of mass