## Exploration 15.1: Blood Flow and the Continuity Equation

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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) ρv^{2} + ρgy = constant.

With a 2.0-mm constriction,

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

Set the constriction to 8.0 mm.

- Does the speed of the platelet before it reaches the constriction increase, decrease, or not change?
- 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?
- 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.

- What is the pressure inside of and outside the constriction (use the white box to measure pressure)?
- Does the pressure decrease or increase in the region where the blockage is?
- 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?
- What, then, is the direction of the force that the platelet feels?
- What region should have a larger pressure?
- 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).
- 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 10
^{5}Pa). The density of blood is 1050 kg/m^{3}.

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