12) The Aorta splits up into arteries that get thinner and become arterioles that finally lead blood to capillaries. Knowing that the blood flow is, for a person at rest, of 5 L/min and that the radius decreases from 10 mm for the aorta to 0,008 mm for the capillaries, being the total area of the capillaries of 2.000 cm², determinate:

a) Number of capillaries and the total flow on each one of them
b) The velocity of the blood in the aorta and in each capillary

As in almost every Biophysics problem, you have to deal with unit conversion so as to homogenize the information and perform calculations with it. I will start by that, converting all the data to the international system (SI)… and scientific notation!

Flow in the aorta, Q_a = 5 L/min =  8,33 x 10^-5 m³/s

Radius of the aorta, r_a = 1 x 10^-2 m

Radius of 1 capillary, r_1c = 8 x 10^-6 m

Total area of 1 capillary, S_Tc =  2 x 10^-1 m²

Let’s go to the questions: the number of capillaries comes up easily by dividing the total area of the capillaries by the area of a single capillarity. That’s easy. We don’t know the area of one capillarity, but we know its radius.

S_1c = π r_c² = 3,14 (8 x 10^-6 m)²

S_1c = 2 x 10^-10 m²

Now Idivide both areas

#caps = S_Tc / S_1c

#caps = 2 x 10^-1 m² / 2 x 10^-10 m²

Finding out the flow in each capillarity is super easy: the total flow in the total area of the capillaries is the same as in the aorta, so the flow in 1 capillarity is that one divided by the number of capillaries.

Q_1c = Q_a / #caps = 8,33 x 10^-5 m³/s / 10⁹

Let’s see the velocity: bear in mind the continuity equation and it’s a piece of cake

Q = A . v

v_a = Q _a / S_a

v_a = Q_a / π r_a²

v_a = 8,33 x 10^-5 m³/s / 3,14 (1 x 10^-2 m²)

The velocity in the capillaries, the same way, and we could choose…

v_1c = Q_1c / S_1c

v_1c = Q_c / S_c

v_1c = 8,33 x 10^-14 m³/s / 2 x 10^-10 m²

600 times less than in the Aorta! It makes sense: In the arteries and veins the blood is just transported. In order to be efficient, transportation must be fast, and it is indeed. However, in the capillaries, the function is the exchange: that’s where the blood fulfils its biological function of nutrient delivery and waste collection. So as to do this job it needs time, it needs to move slowly and to stick on capillary epithelial cells as long as it takes.