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Chapter 9: Problem 101

The diameter of a certain artery has decreased by \(25 \%\) due to arteriosclerosis. (a) If the same amount of blood flows through it per unit time as when it was unobstructed, by what percentage has the blood pressure difference between its ends increased? (b) If, instead, the pressure drop across the artery stays the same, by what factor does the blood flow rate through it decrease? (In reality we are likely to see a combination of some pressure increase with some reduction in flow.)

Short Answer

Expert verified
Answer: The blood pressure difference between the ends of the artery increases by 177.78%. Question: If the pressure drop across the artery remains the same, by what factor does the blood flow rate decrease? Answer: The blood flow rate decreases by a factor of approximately 31.64%.

Step by step solution

01

Calculate the initial and final radius of the artery

Since the diameter of the artery decreased by \(25 \%\), we have the final radius \(r_f = 0.75 r_i\), with \(r_i\) being the initial radius of the artery.
02

Apply Poiseuille's Law for the initial and final states

Poiseuille's Law in the initial state is given by: \(Q_i = \frac{\pi \Delta P_i r_i^4}{8 \eta L}\). While for the final state, with radius \(r_f\), it is as follows: \(Q_f = \frac{\pi \Delta P_f (0.75 r_i)^4}{8 \eta L}\).
03

Equate blood flow rates and find the pressure difference ratio

Since the blood flow rate remains the same in both cases, we can write: \(Q_i = Q_f\). So: \(\frac{\pi \Delta P_i r_i^4}{8 \eta L} = \frac{\pi \Delta P_f (0.75 r_i)^4}{8 \eta L}\). Simplifying the equation and solving for \(\frac{\Delta P_f}{\Delta P_i}\), we find: \(\frac{\Delta P_f}{\Delta P_i} = \frac{1}{(0.75)^4} = 2.7778\).
04

Calculate the percentage increase in pressure difference

Now we can find the percentage increase in pressure difference: \(\frac{\Delta P_f - \Delta P_i}{\Delta P_i} \times 100 \% = (2.7778 - 1) \times 100 \% = 177.78\%\). The blood pressure difference between the ends of the artery has increased by \(177.78 \%\). (b) Pressure drop across the artery remains the same
05

Relate the initial and final blood flow rate with Poiseuille's Law

Here, we have to find the factor by which the blood flow rate decreases if the pressure drop remains the same. We can write the ratio of the final to the initial blood flow rate as: \(\frac{Q_f}{Q_i} = \frac{\pi \Delta P (0.75 r_i)^4}{8 \eta L} \times \frac{8 \eta L}{\pi \Delta P r_i^4}\).
06

Simplify the equation and find the flow rate ratio

Simplifying the equation, we find: \(\frac{Q_f}{Q_i} = (0.75)^4 = 0.3164\). The blood flow rate through the artery decreased by a factor of \(0.3164\) or approximately \(31.64\%\).

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