Suppose an occlusion in an artery reduces
its diameter by 22%, but the volume flow rate of blood in the artery remains the
same. By what factor has the pressure drop
across the length of this artery increased?
Suppose an occlusion in an artery reduces its diameter by 15%, but the volume flow
rate of blood in the artery remains the same. By what factor has the pressure drop across the length of this artery increased?
(1/0.78)^4=2.7
To find the factor by which the pressure drop across the length of the artery has increased, we need to consider the relationship between flow rate, diameter, and pressure in a pipe.
According to the Bernoulli's principle, the pressure drop across a pipe is directly proportional to the square of the velocity of fluid flow. In this case, the volume flow rate stays the same, which means the velocity of fluid flow remains constant.
However, the diameter of the artery has reduced by 22%. The relationship between diameter and velocity is inversely proportional, meaning that as the diameter decreases, the velocity increases. Since the velocity is constant, the decrease in diameter has resulted in an increase in velocity.
Now, the relationship between diameter and pressure is also inversely proportional according to the Bernoulli's principle. As the diameter decreases, the pressure increases to maintain continuity of flow rate.
So, the pressure drop across the length of the artery has increased by the same factor as the reduction in diameter. In this case, the artery diameter reduced by 22%, so the pressure drop across the length of the artery has increased by 22%.
Therefore, the factor by which the pressure drop has increased is 22%.
To determine by what factor the pressure drop across the length of the artery has increased, we need to understand the relationship between pressure drop, diameter, and volume flow rate in a pipe or tube.
According to the Hagen-Poiseuille equation, the pressure drop across a pipe or tube is directly proportional to the length of the pipe and the viscosity of the fluid, while it is inversely proportional to the fourth power of the diameter of the pipe and the volume flow rate.
Let's represent the initial diameter of the artery as D and the final reduced diameter as D'. Using the given information, we know that D' = 0.78D (22% reduction).
Since the volume flow rate of blood remains the same, we can assume that the viscosity of blood remains constant as well. Therefore, we can algebraically deduce the relationship between the pressure drop across the artery before (P) and after (P') the occlusion as follows:
P' = P × (D/D')^4
Substituting the values we know, the equation becomes:
P' = P × (D/(0.78D))^4
Simplifying further:
P' = P × (1/0.78)^4
P' = P × 1.872
Therefore, the pressure drop across the length of the artery has increased by a factor of 1.872.