Why does the formula for blood velocity hold true?

v = Q/A where

v = velocity (cm/s) Q = blood flow (ml/s) A = cross sectional area (cm2)

So the greater the cros sectional area, the lower the velocity, assuming that the cardiac output is a constant for a individual. But i dont understand the formula content wise, why do you get the velocity by dividing Q by A? I do understand that the units are correct, and you get cm/s as final unit. Can anyone explain this, perhaps graphically? In the capillaries the A must be extremely low, yet why is the velocity so small? My book says that you have to add all capillaries up, but you cant do that, since you would get the velocity of an extremely big pipe, which isnt that of a capillary?

The trick here is that 1 mL = 1 cm3 (or cc). So if you divide cm3/s by cm2 the result will be in cm/s. What that means is that the blood volume has three dimensions - if you picture the vessel as a simple "box", then it has widths in two dimensions, which define a cross-sectional area, and then the third dimension is the distance that the blood has moved past, through the vessel, in order to define the three-dimensional "box" of blood that has moved past as the blood moved that distance.

The principle should be familiar from the phrase "still waters run deep". A stream that flows rapidly in a shallow cascade over rocks can abruptly become quite slow at a deep hole, as a great depth and breadth of water is sharing the work of moving forward.

In the capillaries the A must be extremely low

For a single capillary yes, the area is small.

However, these statements are not referring to a single capillary! They are referring to the entire capillary bed when it says "the capillaries". There are many many many capillaries and their total cross-sectional area is large compared to other vessels. Blood doesn't flow through each capillary in series, it's flowing through them all in parallel.

• So if it flows parallel then the Q gets divided into many capillaries, making the nominator in the formula extremely low, and the area in the denominator as well since it is a capillary. So the velocity decreases, since Q decreases more than A decreases for each capillary. Jan 7 '21 at 16:47
• Is this true? c Jan 7 '21 at 16:47
• @excellence Yes, that's true, but I think the best way to think about it is to just think of the "A" as the area for all the capillaries (so, A = N * a, where N is the number of capillaries and a is the area of one capillary). The total A is larger for the capillary bed than it is for, say, the arterioles (this is what your figure shows in the blue line). It doesn't matter what N or a is to determine velocity, just N*a. Jan 7 '21 at 17:05
• @excellence No I think they're explaining it just fine, you just need to notice that the Y-axis on the right is total cross-sectional area, not for a single capillary. If you missed that point I can see how it would be confusing, though. Jan 7 '21 at 17:33
• @excellence No, you are not right. "A" here is the total cross sectional area. That's what's plotted on the Y axis and is labeled as such: "Total cross-sectional area". This total area is large in the capillaries, which is why velocity is slow there. For the purpose of calculating the velocity as presented in this chart, yes you can consider it as one big blood vessel, and that's what they are doing. They are considering the entire capillary bed together, just like they are considering all the arterioles together, all the venules, all the larger veins, etc. Jan 7 '21 at 20:02