Mass cannot disappear or be created as fluid moves through a flow system. This gives the continuity principle.

For steady one-dimensional flow through a streamtube,

$$\dot m=\rho Av$$

is constant from one cross-section to another, so

$$\rho_1A_1v_1=\rho_2A_2v_2.$$

Incompressible flow

If density is effectively constant,

$$A_1v_1=A_2v_2.$$

A narrowing passage therefore requires a larger average velocity if the same volumetric flow rate passes through it.

Control-volume viewpoint

More generally, choose a fixed region called a control volume. The rate at which mass inside changes equals mass entering minus mass leaving.

For steady flow, the total mass flow into the control volume equals the total mass flow out.

Continuity is a conservation law. It constrains possible fluid motions independently of the forces or energy changes that produce them.