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MATHEMATICS - CASE STUDY

    Introduction


Blood flowing in a Vessel
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Velocity of the Blood is a Function of r

 

Amanda works at a biomedical engineering lab. For a particular project, a sensor needs to be installed in a blood vessel at a location where the blood velocity has the same magnitude as the average velocity of the whole vessel cross-sectional area.

What is known:

  • The vessel can be considered as a cylindrical tube with a radius of 0.08 mm and a length of 20 mm.
  • The blood in the vessel is considered as a fully developed laminar flow with a velocity function as

    v(r) = P/(4μl)(R2 - r2)

    where P is the pressure difference between the ends of the vessel, P = 400 Pa, μ is the viscosity of the blood, μ = 10x10-3 kg/(m-s), R is the radius of the vessel, and l is the length of the vessel.
     
    Questions

 

 
  • Where does the blood velocity have the same magnitude as the average velocity of the whole cross-sectional area of the vessel?
  • What is the average velocity of the whole vessel cross-sectional area?
     
    Approach

 

 

Use the Mean Value Theorem for Integrals.