Question

The velocity of the blood in the aorta is about 0.5 m s-1 , and the velocity of the blood in a capillary is about 0.001 m s-1 . We have only one aorta, with a diameter of 20 mm, but many capillaries in parallel, each with a diameter of 8 ?m. Estimate how many capillaries are typically open at any one time. Assume now that the diameter of the aorta has decreased by 15%. What needs to happen so that it is able to supply the same amount of blood to the capillaries? What does this mean for the pressure gradient?

(This is from the textbook University physics for Medicine and Biology by Russle Hobbie) Problem 41. Please do not ask me further for equations because they will most likely be in this book.Thanks!!

Answer 1

V_a = speed of blood in aorta = 0.5 m/s

V_c = speed of blood in capillary = 0.001 m/s

d_a = diameter of aorta = 20 mm = 0.020 m

A_a = area of cross-section of aorta = d² /4 = (3.14) (0.02)² /4 = 3.14 x 10^-4 m²

d_c = diameter of capillary = 8 x 10^-6 m

A_c = area of cross-section of capillary= d² /4 = (3.14) (8 x 10^-6)² /4 = 50.24 x 10^-12 m²

let the number of capillaries be ''n''

using equation of continuity

A_a V_a = n A_c V_c

(3.14 x 10^-4 ) (0.5) = n (50.24 x 10^-12 ) (0.001)

n = 3.125 x 10⁹

when diameter of aorta decreases

new diameter = d_new = 20 - 0.15 x 20 = 17 mm = 17 x 10^-3 m

A_new = d² /4 = (3.14) (17 x 10^-3)² /4 = 2.27 x 10^-4 m²

new speed = V_new = ?

using equation of continuity

A_a V_a = A_new V_new

(3.14 x 10^-4) (0.5) = (2.27 x 10^-4 ) V_new

V_new = 0.692 m/s

speed has to be increased and for that pressure gradient has to be increased