## Conservation Of Mass

In addition to Newton's laws there is a further constraint on the fluid motion: conservation of mass. Consider a fixed fluid volume as illustrated in Fig. 6.4. The volume has dimensions (Sx, Sy, Sz). The mass of the fluid occupying this volume, p Sx Sy Sz, may change with time if p does so. However, mass continuity tells us that this can only

"W (x - j Sx, y - j Sy, z - -g- Szj (x + -g- Sx, y - -g- Sy, z - g- Szj

FIGURE 6.4. The mass of fluid contained in the fixed volume, pSx Sy Sz, can be changed by fluxes of mass out of and into the volume, as marked by the arrows.

(x - j Sx, y - j Sy, z - -g- Szj (x + -g- Sx, y - -g- Sy, z - g- Szj

FIGURE 6.4. The mass of fluid contained in the fixed volume, pSx Sy Sz, can be changed by fluxes of mass out of and into the volume, as marked by the arrows.

3It might appear from Eq. 6-7c that \Dw/Dt\ << g is a sufficient condition for the neglect of the acceleration term. This indeed is almost always satisfied. However, for hydrostatic balance to hold to sufficient accuracy to be useful, the condition is actually \Dw/Dt\ <<gAp/p, where Ap is a typical density variation on a pressure surface. Even in quite extreme conditions this more restrictive condition turns out to be very well satisfied.

occur if there is a flux of mass into (or out of) the volume, meaning that dt (p Sx Sy Sz) = dp Sx Sy Sz = (net mass flux into the volume).

Now the volume flux in the x-direction per unit time into the left face in Fig. 6.4 is u (x - 1/2 Sx, y, z) Sy Sz, so the corresponding mass flux is [pu] (x - 1/2 Sx, y, z) Sy Sz, where [pu] is evaluated at the left face. The flux out through the right face is [pu] (x + 1/2 Sx, y, z) Sy Sz; therefore the net mass import in the x-direction into the volume is (again employing a Taylor expansion)

Similarly the rate of net import of mass in the y-direction is d

Therefore the net mass flux into the volume is -V ■ (pu) Sx Sy Sz. Thus our equation of continuity becomes dp

This has the general form of a physical conservation law:

d Concentration dt

in the absence of sources and sinks.

Using the total derivative D/Dt, Eq. 6-1, and noting that V ■ (pu) = pV- u + u-Vp (see the vector identities listed in Appendix A.2.2) we may therefore rewrite Eq. 6-9 in the alternative, and often very useful, form:

6.3.1. Incompressible flow

For incompressible flow (e.g., for a liquid such as water in our laboratory tank or in the ocean), the following simplified approximate form of the continuity equation almost always suffices:

dx dy dz 