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I've been trying to reconcile the resistance component of Poisseuille's law with a mental analogy of a garden hose; specifically, I had assumed that the effects of reducing the radius of a blood vessel (of an arteriole in vasoconstriction for instance) would be the same as when you put your thumb over the opening of a hose, causing water to leave at an increased velocity. However, the law instead predicts that a decrease in radius like this would increase resistance, actually decreasing flow by a proportional amount equal to r^4. Why doesn't arteriolar flow actually end up increasing because of the pressure from the arteries, much like its "hose equivalent"? Does the hose analogy even work in this scenario? What am I missing that separates the two scenarios?

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It comes down to the distinction between velocity and flow rate.

While you are right that the water would leave a an increased velocity if you put your thumb over the end of the hose, this is deceptive, because the water is exiting the closed system, so it can go absolutely anywhere. So the fact that the velocity is increased could give one the impression that the flow rate is increased, but in fact it is the inverse.

Imagine that instead of putting your thumb on the end of the hose, you pinched the hose in the middle instead. This would result in the water flowing out of the end of the hose at a reduced flow rate (lower volume of water coming out per minute) and velocity (lower speed of the individual water particles). Why?

It might help to take a snapshot of the place where you are pinching the hose. the pressure behind the pinch zone goes up, causing the water to pass through the pinch zone at a faster velocity. However, because the hole is smaller, less water is able to pass through. When it reaches the other side of the pinch zone, it slows down again because it has a very low pressure, and the smaller volume of water must fill the entire tube. However, because the flow rate was lowered because of the pinch zone, the amount of water at the end of the pinch zone is lower. Consequently, the water flows slower and the flow rate is reduced simultaneously.

Another analogy that might help would be the comparison between a garden hose and a fire hose. Let's say they have the same flow rate (same volume coming out of the hose per unit time). You can immediately realise that for the garden hose to put out the same amount of water that the fire hose is putting out in a minute, the water would have to be flowing many times faster in terms of velocity.


Furthur reading

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At a constant volumetric flow-rate, the product of velocity and vesicle cross-section area is constant. In a real-life scenario, the volumetric flow-rate decreases when you hold your finger over the opening. So while water exits 'faster', at higher velocity, you would fill less bottles of water per minute.

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