## Relationship between Flow Rate and Radius

hml12@duke.edu
Posts: 1
Joined: Tue Jun 11, 2019 4:07 pm

### Relationship between Flow Rate and Radius

Hello,

I am a little confused about the relationship between flow rate of fluids and the radius (and thus cross sectional area) if anyone has a better grasp on the topic. According to poiseuille's law, it seems as though flow rate is directly proportional to radius so that fluids flow faster in wider cross-sectional areas; however doesn't the conservation law state that fluids in smaller cross sectional areas will have a greater velocity (i.e. capillary example)? These two laws seem contradictory and I cannot seem to figure out why.

NS_Tutor_Mathias
Posts: 257
Joined: Sat Mar 30, 2019 8:39 pm

### Re: Relationship between Flow Rate and Radius

Conservation holds in closed systems. But not all systems are closed.

According to Poiseuille's law, flow rate is exponentially proportional to how big your pipe is (radius), directly proportional to how hard you're sending stuff through (pressure), but inversely proportional to how goopy your liquid is (viscosity, greek letter Eta) and how long your pipe is (uh... length, l). This makes intuitive sense.

However, to send more stuff through, you must have more stuff - it follows that Poiseuille's law describes an open system, where more fluid is available. In a self-contained system like circulation, that would not normally be true (we can of course pump things into or out of people with devices).

In closed systems however, conservation is logically required: If I have less space to move fluid through, it must move faster - because Newtonian fluids are by definition non-compressible. Therefore the smaller a vessel becomes in this kind of scenario, the faster fluid would flow. To tie this back to physiological scenarios, keep in mind that the sum total diameter of smaller vessels is often much larger than that of the greater vessels though - so fluid is still moving more slowly through these than the great vessels.