Question

Arteriosclerotic plaques forming on the inner walls of arteries can decrease the effective cross-sectional area of an artery. Even small changes in the effective area of an artery can lead to very large changes in the blood pressure in the artery and possibly to the collapse of the blood vessel. Imagine a healthy artery, with blood flow velocity of v 0 =0.14m/s and mass per unit volume of ρ=1050kg/ m 3 . The kinetic energy per unit volume of blood is given by K 0 = 1 2 ρ v 2 0 . Relative to its initial, healthy state, by what factor does the velocity of blood increase as the blood passes through this blockage? Express your answer numerically.

Answer 1

Let the initial velocity of the blood be v₀ = 0.14 m/s

Let the mass per unit volume be р = 1050 kg / m³

Now we can solve the problem simply by using continuity equation.

                        A1 * V1 = A2 * V2

       where A1 and A2 are area of cross section of artery and V1 and V2 are speed of the blood before and after plaque

Actually, plaque decreases the area of the artery. But according to continuity equation for the pressure to be maintained, speed of the blood flow must increase.

Let the increased speed be V

Here area of blockage is not mentioned.

So I am taking two cases to help how to calculate the speed of the blood after blockage.

( I )Let the area of blockage be 80 %

Then the area available for the blood to flow is only 20 % of the initial area A1

So A2 = 20 % of A1 = A1 / 5

So now according to continuity equation

        A1 * v₀ = A2 * V

      A1 * v₀ = ( A1 / 5 ) * V

So             V = 5 v₀

                 V = 5 * 0.14 = 0.70 m/s

so the blood flow increases by a factor 5

( ii ) let the area of the blockage be 90 %

Then A2 = 10 % of A1 = A1 / 10

So from continuity equation V = ( A1 / A2 ) * v₀ = 10 v₀ = 1.40 m/s

So the blood flow increases by a factor 10