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Integrating 3 equations together

  1. Sep 15, 2011 #1
    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.
     
  2. jcsd
  3. Sep 15, 2011 #2
    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.
     
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