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Problem 29 Hard Difficulty

Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $ 3000 dynes/cm^2, $ and viscosity $ \eta = 0.027. $
(a) Find the velocity of the blood along the centerline $ r = 0, $ at radius $ r = 0.005 cm, $ and at the wall $ r = R = 0.01 cm. $
(b) Find the velocity gradient at $ r = 0, r = 0.005, $ and $ r = 0.01. $
(c) Where is the velocity the greatest? Where is the velocity changing most?

Answer

a) $0$
b) $-185.185(\mathrm{cm} / \mathrm{s}) / \mathrm{cm}$
c) highest velocity at the center, changing most at the edge

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Video Transcript

here we have the law of laminar flow, and we're going to use it to look at the flow of blood through a blood vessel and for this problem, were given some specific numbers to plug in for the constants. So we're going to plug in 3000 for the pressure and the value of ADA, which is that one that looks kind of like an end. It's a Greek letter. Or maybe it's pronounced pita. The value of Vita is given as 0.27 The length is given us three. The radius of the blood vessel is given as 0.1 Okay, so now that we have that, we could go ahead and take a step to simplify it. The fraction in the front we could simplify, simplify a little bit to 2 50 over 0.27 And if we want to, we can simplify the number inside the parentheses as well. Not gonna make a whole lot of difference. Okay, so for party, what we're going to do is find the velocity of the blood at various points in the blood vessel at R equals zero, which would be smack dab in the middle, not close to the walls at all, but right through the middle. And that's going to be you're gonna plug that in your calculator and that's going to be approximately 0.9 to 5. And then we're going to find the velocity of the blood when it is about 0.5 centimeters from the middle and that would be approximately 0.694 Then we're gonna find the velocity when the blood is right there at the edge of the blood vessel at 0.1 units from the center. And that would be zero, So that doesn't sound good. That means the blood is not flowing at that part, and we can see that the velocity of the blood is decreasing as we go closer to the edge of the blood vessel. Okay, for part B, what we're going to do is find the derivative, and that is known as the velocity Grady int. And then we'll evaluate the derivative for various those same three values of our and that will tell us how fast the blood is flowing or actually the rate of change of the speed of the blood. Because we already have the speed, the rate of change of the speed. Okay, So the derivative of this, we can leave the constant there and when multiplied by the derivative of ah, the inside function. And that's gonna be negative to our And we could simplify this if we want to. This is negative. 500 are over 0.27 So then we're going to go ahead and substitute the various values of are in there. Starting with R equals zero, and we end up with zero, which means that the rate of change of the velocity is zero. There's no change in how fast the blood is flowing at that point. And then the prime of 0.5 is negative. 92.59 approximately and be prime of this is the prime of our, by the way, not t, there's no time involved in this will change that to and are the prime of 0.1 right there at the edge of the blood vessel is about negative 1 85.185 So what that's telling us is that the most changes happening at the edge because That's the greatest velocity, greatest rate of change of velocity. We also saw and discussed earlier from part A that the blood is flowing at the greatest speed in the center.

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