Integrating 3 equations together

[ch5112]

There are 3 equations

1/ Continuity equation

Av=constant

2/ Bernoulli's equations

P+1/2*p*V^2+pgy=constant

3/ Poiseiulle's equation


Question1:

If a single blood vessle is narrowed by the build-up of plaque so that its inner radius is reduced. How can I use above three equations to apply in this situation to provide insight into the varation of speed and pressure as the blood vessel become narrower.

Question 2:
A single small artery branches into about six smaller blood vessels, which in turn branch a fewmore times into a total of about 1000 arterioles. The radius of the initial, single small artery is 0.7 mm, and the average radius of each of the 1000 arterioles is 70 μm. Explain how the three equations above could be applied in this situation to provide insight into the variation of speed and pressure as blood travels from the single artery to 1000 arterioles. In particular, why does the Continuity equation together with only Bernoulli’s equation not provide an adequate explanation? What would be wrong with such a prediction, and what is it about this system that means Poiseiulle’s equation is crucial to a fuller explanation.

 

[Harrisonized]

Let's call the radius of the normal blood vessel r₁ and the reduced radius r₂. Assume that the blood is of constant viscosity. The continuity equation is (as you stated):

A₁v₁ = A₂v₂
A is the area of the cross-section
v is the velocity across that cross-section

The area of a circle is πr². Substitute that in. The equation becomes:

πr₁²v₁ = πr₂²v₂

v₂ = (r₁²/r₂²) v₁

v₂² = (r₁²/r₂²)² v₁²

Now we need to replace v₁ with something more descriptive. Use Bernoulli's equation:

P₁+ρv₁²/2+ρgh₁=P₂+ρv₂²/2+ρgh₂

ρv₁²/2=P₂-P₁+ρgh₂-ρgh₁+ρv₂²/2

v₁²=2(P₂-P₁)/ρ+2(gh₂-gh₁)+v₂²

Subbing into the above:

v₂² = (r₁²/r₂²)² * [2(P₂-P₁)/ρ+2(gh₂-gh₁)+v₂²]

Isolate v₂.

By now you probably get the point. You can use Poiseiulle's equation for even better (more descriptive) substitutions. Substitute it in for the quantity P₂-P₁.