Fluid Flow

Volumetric flow rate is just the rate at which volume goes through a given surface per unit of time. If you have a pipe like the one depicted below, the way you can measure how much water is going through the given point is by measuring the volumetric flow rate of the water. On this diagram, it's essentially how much water flows through in a given period of time, as stated before. The mathematical definition of volumetric flow rate is:

This gives another definition of volumetric flow rate: the time derivative of volume flowing through a given point. If the cross-sectional area isn't curved, then the volumetric flow rate can be expressed like this:

If we take the same picture of the pipe with fluid flowing through it above, even if the cross-sectional area of the pipe is different at different parts, the same volume of water flows through. Since the fluid flowing through is incompressible(its volume doesn't change) and the mass is conserved, the volumetric flow rate must be the same. However, the cross-sectional area the fluid flows through is decreased so the only way the area is smaller but the flow rate is conserved is if the fluid velocity is greater. This is the underlying principle of fluid flow continuity which can be mathematically stated like this:

Thus, using the pipe above, once the fluid enters the narrower pipe, it will speed up because the cross-sectional area it goes through is smaller. This is why if you partly cover water going into filling a bucket, you'll notice that, for the most part, the bucket fills up at the same rate.