The human circulatory system consists of a complex branching pipe network ranging in diameter from
the aorta (largest) to the capillaries (smallest). The average diameter and the number of these vessels are
shown in the table. Does the average blood velocity increase, decrease, or remain constant as it travels
from the Aorta to the Capillaries?

vesselaverage diameter [mm]number
Aorta25.01
Arteries4.0159
Arterioles0.061.410^7
Capillaries0.0122.910^9
This is an excercise that requires application of conservation of mass.

all the "m" are mass flow rate, p is density, n is number of vessels,A is area

m [kg/s]=m_aorta=m_arteries=npV_avgA_vessel

V(volume flow rate) [m^3/s]=m/p=V_avgA

Blood can be considered incompressible; that is, the density does not change as the blood moves from large vessesl to small vessels. Assume the density of blood is equal to 1060 kg/m^3.

from the information, let [tex]\hat{m}[/tex] be the mass flow rate, [tex]\rho[/tex] is density, n number of vessels, and A is the cross-section area for each vessel

the flow rate is constant so it is equal for all vessels,

The average velocity is related to the flow rate by:

[tex]\hat{m} = v* \rho * A * n[/tex]

we clear the side where v is in:

[tex]v = \frac{\hat{m}}{\rho A n}[/tex]

area is π*R^2 where R is the average radius of the vessel (diameter/2)

we get:

[tex]v = \frac{\hat{m}}{\rho \pi R^2 n}[/tex]

you can directly see in the last equation that if we go from the aorta to the capillaries, the number of vessels is going to increase ( n will increase and R is going to decrease ) . From the table, R is significantly smaller in magnitude orders than n, therefore, it wont impact the results as much as n. On the other hand, n will change from 1 to 2.9 giga vessels which will dramatically reduce the average blood velocity