28.3: Mass Continuity Equation
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A set of streamlines for an ideal fluid undergoing steady flow in which there are no sources or sinks for the fluid is shown in Figure 28.3.


We also show a set of closely separated streamlines that form a flow tube in Figure 28.4 We add to the flow tube two open surface (end-caps 1 and 2) that are perpendicular to velocity of the fluid, of areas A1 and A2, respectively. Because all fluid particles that enter end-cap 1 must follow their respective streamlines, they must all leave end-cap 2. If the streamlines that form the tube are sufficiently close together, we can assume that the velocity of the fluid in the vicinity of each end-cap surfaces is uniform.

Let v1 denote the speed of the fluid near end-cap 1 and v2 denote the speed of the fluid near end-cap 2. Let ρ1 denote the density of the fluid near end-cap 1 and ρ2 denote the density of the fluid near end-cap 2. The amount of mass that enters and leaves the tube in a time interval dt can be calculated as follows (Figure 28.5): suppose we consider a small volume of space of cross-sectional area A2 and length dl1=v1dt near end-cap 1. The mass that enters the tube in time interval dt is
dm1=ρ1dV1=ρ1A1dl1=ρ1A1v1dt
In a similar fashion, consider a small volume of space of cross-sectional area A2 and length dl2=v2dt near end-cap 2. The mass that leaves the tube in the time interval dt is then
dm2=ρ2dV2=ρ2A2dl2=ρ2A2v2dt
An equal amount of mass that enters end-cap 1 in the time interval dt must leave end-cap 2 in the same time interval, thus dm1=dm2. Therefore using Equations (28.3.1) and (28.3.2), we have that ρ1A1v1dt=ρ2A2v2dt. Dividing through by dt implies that
ρ1A1v1=ρ2A2v2 (steady flow )
Equation (28.3.3) generalizes to any cross sectional area A of the thin tube, where the density is ρ, and the speed is v,
ρAv=constant (steady flow )
Equation (28.3.3) is referred to as the mass continuity equation for steady flow. If we assume the fluid is incompressible, then Equation (28.3.3) becomes
A1v1=A2v2 (incompressable fluid, steady flow )
Consider the steady flow of an incompressible with streamlines and closed surface formed by a streamline tube shown in Figure 28.5. According to Equation (28.3.5), when the spacing of the streamlines increases, the speed of the fluid must decrease. Therefore the speed of the fluid is greater entering end-cap 1 then when it is leaving end-cap 2. When we represent fluid flow by streamlines, regions in which the streamlines are widely spaced have lower speeds than regions in which the streamlines are closely spaced.