v(r) = P 4l R 2 r 2 this function to model the velocity of blood moving through

Question

v(r) = P 4l R 2 r 2

this function to model the velocity of blood moving through a blood vessel where r is the distance from the center of the vessel. Here, P = the pressure difference between ends of the vessel, R = the radius of the vessel, = the viscosity (resistance), and l = the length of the vessel. In practice, these numbers represent fixed numbers. The only variable is r. Notice that this function is a quadratic. (a) Determine the value of r where v(r) achieves a maximum value. Explain this result with respect to the blood vessel model. (b) Consider the specific model (omitting units) where l = 1 meter: v(r) = 1.3 × 104 4(1.5 × 103 ) (.01)2 r 2 . Determine the value v(.0025) and explain what this value represents.

Explanation / Answer

V(r)= PnlR2- r2

For maximum of velocity dV/dr= 0

dV/dr= -2r= 0

r= 0

At the centre maximum velcoity is there since this is the point where resistnace is less

V(r)= 1.3*104 *4*(1.5/1000)- r2

V(0.0025)= 1.3*104*4*(1.5/1000)*(0.01)2 -(0.0025)2 =0.007793 velocity units

Considering the blodd vessels to be a pipe, the velocity of blood at 0.0025 radius is 0.007793 velocity units

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