normal vector field n. The small arrows represent the spatial velocity."/>
Figure 2.5 A time‐independent region
having oriented boundary
and unit outward normal vector field
. The small arrows represent the spatial velocity.
2.2.1 Mass Balance
Consider first the mass balance. We associate with each body a nonnegative, integrable function , called the mass density. This function gives the mass contained in any region of three‐dimensional Euclidean space as the volume integral
(2.2)
having physical dimension . Here, denotes the element of volume integration. Since is nonnegative, so is the mass. The expression (2.2) requires that .
The mass balance arises from a simple observation: The rate of change in the mass inside any region of three‐dimensional space exactly balances the rate of movement of mass across the region's boundary. In symbols,
(2.3)
This equation is the integral mass balance. Here, denotes the boundary of ; denotes the unit‐length vector field orthogonal to and pointing outward, as Figure 2.5 depicts; and denotes the element of surface integration. We call the function in the integral on the right side of Eq. (2.3) the mass flux per unit area; the integrand is the component of mass flux per unit area in the direction of the unit vector , that is, outward from . The surface integral itself, together with the negative sign, is the net flux of mass inward across .
Often of greater utility than the integral equation (2.3) is a pointwise form of the mass balance, valid when the density and velocity are sufficiently smooth. To derive this form, consider a region that does not change in time. In this case,
(2.4)
Also, by the divergence theorem,
(2.5)
where denotes the divergence operator. With respect to an orthonormal basis ,
Applying the identities (2.4) and (2.5) to the integral mass balance (2.3) yields the equivalent equation
(2.6)
valid for any time‐independent region .
If the integrand in Eq. (2.6) is continuous, then the integrand must vanish:
(2.7)
