How Does Plaque Constriction Affect Blood Velocity in Arteries?

[semitope]

Homework Statement

An artery with a 3 mm radius is partially blocked with plaque. In the constricted region the effective radius is 2 mm and the average blood velocity is 0.5 m/s. What is the average velocity in the unobstructed region? Assume no changes to η, L, and ΔP. Ans; 0.22 m/s

Homework Equations

Flow rate = ΔP(π/8)(1/η)(R^4/L)
= (PA– PB)(π/8)(1/η)(R^4/L)

The Attempt at a Solution

rate = [(ΔPπ)/8ηL] * R^4

Turned the middle section into x and solved for x. then used x to get the flow rate with a diameter of 3mm. I got something around 2.5m/s. Doesn't seem right, but at the same time the answer given by the professor doesn't seem right either. the flow rate in the larger vessel is less than that in the obstructed portion if that answer is correct. If the 0.22 is correct I'd love an explanation of how it is solved

 

[LawrenceC]

rho1*A1*V1=rho2*A2*V2 which is basic flow continuity

Therefore V2=(A1/A2)*V1

Area is proportional to square of radius.

 

[semitope]

Thanks. Would what I was doing have worked if the systems were separate with the same parameters but different radii?

 

[LawrenceC]

You have the Hagen-Poiseuille equation that relates pressure drop to discharge in a circular tube of length L for laminar flow. The problem is that whenever the velocity changes, the pressure changes so parameters do not remain the same.