Blood flows through the arteries and veins of the body to carry nutrients and oxygen to tissues and to carry away waste products and carbon dioxide (along with numerous other functions). Fluids stick to surfaces (no slip boundary condition), which results in no fluid flow near any surface. For a small artery with a radius of R = 0.06 cm, blood satisfies Poiseuille's law, which states that the velocity of blood r cm from the central axis of the artery is given by
v(r) = k(R2 - r2),
where k = 4670 /(cm-sec) for this specific artery.
a. Graph the velocity of blood as a function of the radius r cm (0 < r < R) from the central axis and determine the maximum velocity of blood in this small artery.
b. The rate of flow of blood through this artery can be found by integrating the velocity over the cross sectional area of the blood vessel. In particular, if one divides the artery into small concentric rings with radii r and r + Dr, then the area of the concentric ring is
p((r + Dr)2 - r2) = 2prDr + pDr2,
which for small Dr can be approximated by 2prDr. The flow rate is approximated by taking the cross sectional area of each concentric ring times the velocity of the blood flowing through that concentric ring, then add together all the small concentric rings. In the limit of very small Dr, we obtain the following integral (where the Dr is replaced by dr):
With the information above, find the flow rate in cm3/sec for this small artery.
c. The average velocity for the blood in the artery is found by taking the flow rate and dividing by the cross-sectional area. Find the average velocity of blood for this small artery.
d. Suppose that this small artery bifurcates into two smaller arteries each with a radius of 0.045 cm. Assuming that the blood flows equally into each of these smaller arteries, then by conservation of blood in these vessels, they each have one half the flow rate found in Part b. Find the value of k for these smaller arteries, then graph the velocity profile of the blood in these smaller arteries. What is the maximum velocity of the blood in these smaller arteries?