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Exploration 15.1: Blood Flow and the Continuity Equation

constriction = mm

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Blood flows from left to right in an artery with a partial blockage. A blood platelet is shown moving through the artery. How does the size of the constriction (variable from 1 mm to 8 mm from each wall) affect the speed of the blood flow? Restart Assume an ideal fluid (position is given in millimeters and pressure is given in torr = mm of Hg). We can use the continuity equation and Bernoulli's equation to understand the motion:

Continuity: Av = constant        Bernoulli: P + (1/2) ρv2 + ρgy = constant.

With a 2.0-mm constriction,

  1. What is the platelet's speed before and after it passes through the constriction?
  2. What is the platelet's speed while it passes through the constriction?

Set the constriction to 8.0 mm.

  1. Does the speed of the platelet before it reaches the constriction increase, decrease, or not change?
  2. With the 8-mm constriction, is the speed of the platelet in the constriction faster, slower, or the same as with the 2-mm constriction?
  3. Assume the blood vessel and the blockage are cylindrical (circular cross-sectional area for both). Measure the radius of the artery and the radius of the flow area where the blockage is. Verify the equation of continuity to compare the 2-mm and 8-mm cases.

Now compare the 2-mm and 8-mm cases.

  1. What is the pressure inside of and outside the constriction (use the white box to measure pressure)?
  2. Does the pressure decrease or increase in the region where the blockage is?
  3. This result, (g), is surprising to many students, so let's figure out why: At the instant the platelet travels from the wide region to the narrower constricted region, what is the direction of acceleration?
  4. What, then, is the direction of the force that the platelet feels?
  5. What region should have a larger pressure?
  6. Do the same analysis for the platelet as it leaves the constricted region and goes back to the unblocked artery (sketch a diagram to show the direction of acceleration and force).
  7. Verify that Bernoulli's equation holds inside and outside the constricted region for the 2-mm and 8-mm cases (760 Torr = 760 mm of Hg = 1.01 x 105 Pa). The density of blood is 1050 kg/m3.

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Exploration authored by Anne J. Cox and Chuck Niederriter. Script authored by Chuck Niederriter and Anne J. Cox.

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