When an incompressible fluid flows through a tube of varying cross-section, the rate at which mass flows past any point in the tube is constant. If this flow rate varied, fluid would build up at points where the flow rate is low.

The mass flow rate is the total mass flowing past a point in a given time interval, divided by that time interval.

At a point where the flow is in the x direction and the tube has a cross-sectional area A:

mass flow rate

=

Dm Dt

=

r DV Dt

=

rA Dx Dt

=

r A v

The continuity equation reflects the idea that the mass flow rate is constant:

rAv = constant         or         r₁A₁v₁ = r₂A₂v₂

In an incompressible fluid the density is constant, so the continuity equation is:

A₁v₁ = A₂v₂

Moral of the story: The fluid flows faster in narrow sections of the tube.