Physics in Motion

Continuity


The conservation of mass implies an important relation between the speed of a fluid and the cross-sectional area it is flowing through. Consider a long tube with cross-sectional area of \( A_1 \) on one end and \( A_2 \) on the other. Using the usual relation between mass and density, the amount of mass moving into the tube through a distance \( d x_1 \) is,

$$ d m = \rho l A_1 d x_1 $$

The rate at which the mass is flowing into the tube is given by,

$$ R_1 = \frac{dm}{dt} = \rho l A_1 \frac{d x_1}{dt} = \rho l A_1 v_1 $$

Similarly, the rate at which mass is flowing out of the tube is,

$$ R_2 = \rho l A_2 v_2 $$

Since mass is conserved, the rate at which mass is flowing into the tube must be equal to the rate at which mass is flowing out of the tube. This is the statement of continuity. Therefore, we must have

$$ R_1 = R_2 $$

In the case of incompressible fluids we consider here, the density is unchanged throughout the tube and so we can drop it on both sides to arrive at the conservation of volume flow rate

$$ A_1 V_1 = A_2 v_2 $$

Thus, as the fluid moves from a more constricted tube to a large tube it slows down. Similarly, as the fluids moves from a more spacious tube to a more constricted one it speeds up.