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## Continuity Equation

If the mass is neither created or destroyed in the volume, then the mass inside a fixed surface, , bounding the closed volume will increase if mass (or density) flows into the volume and decrease if it flows out. This can be expressed mathematically as
• rate of increase of mass inside the volume + rate at which mass is flowing out across the surface = zero.

 (2.3)

By the divergence theorem (1.18), this becomes

This holds for an arbitrary volume and so must be true at each point in space. Thus,
 (2.4)

This is the mass continuity equation.

In 1-D problems this reduces to

 (2.5)

and we can obtain this equation another way by simply considering how a fluid behaves in a small region lying between the positions and . Consider a fluid with a spatially and temporally varying density, and velocity . The mass within this small region, as shown in Figure 2.2,

The mass in the region at time is simply

Similarly, the mass in this region at the later time is

So why has the mass changed? It changes because there is a flow across the ends of this region. Consider the mass flowing through the left hand edge in the time interval . This is given by

It is straightforward to check that the units do indeed give density times a length. At the other edge the mass flowing out of the system is

Thus the change in mass in the time is

This change is due to the mass flowing out through the right hand edge - the mass flowing into the region through the left hand edge, namely

Equating this two expressions and dividing by results in

Taking the limit of and we obtain

which gives (2.5).

For a steady state and do not change in time and so

and

Hence, the mass flowing into the region is equal to the mass flowing out of the region.

Next: Time derivatives following the Up: Fluids and Nonmagnetic Plasmas Previous: Fluids and Nonmagnetic Plasmas
Prof. Alan Hood
2000-11-06