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Blood FlowSuppose a blood vessel has radius R and length /. Let P represent the pressure difference between the ends of the vessel, and let n (eta) represent the vi...

Question

Blood FlowSuppose a blood vessel has radius R and length /. Let P represent the pressure difference between the ends of the vessel, and let n (eta) represent the viscosity of the blood: The velocity of a blood molecule that flows along this vessel depends on how close or far it is from the vessel's central axis. Blood in the center of a vessel flows much faster than blood near a vessel wall. The velocity of a blood molecule positioned at a distance r from the central axis is as follows. Thi

Blood Flow Suppose a blood vessel has radius R and length /. Let P represent the pressure difference between the ends of the vessel, and let n (eta) represent the viscosity of the blood: The velocity of a blood molecule that flows along this vessel depends on how close or far it is from the vessel's central axis. Blood in the center of a vessel flows much faster than blood near a vessel wall. The velocity of a blood molecule positioned at a distance r from the central axis is as follows. This is known as the law of laminar flow: P v(r) = (R2 _r2) 4nl The flux is the volume ofa blood that passes a cross-section per unit of time: This can be found by multiplying the area of a cross-section 2rr by the velocity V, and then by integrating all of this from 0 to R with respect to the variable r. Complete this integration to show that the flux is proportional to the fourth power of the radius of the blood vessel,



Answers

Blood flow The shape of a blood vessel (a vein or artery) can be modeled by a cylindrical tube with radius $R$ and length $L .$ The velocity $v$ of the blood is modeled by Poiseuille's law of laminar flow, which expresses $v$ as a function of five variables:
$v=f(P, \eta, L, R, r)=\frac{P}{4 \eta L}\left(R^{2}-r^{2}\right)$
where $\eta$ is the viscosity of the blood, $P$ is the pressure difference between the ends of the tube (in dynes/cm $^{2} ), r$ is the distance from the central axis of the tube, and $r, R,$ and $L$ are measured in centimeters.
(a) Evaluate $f(4000,0.027,2,0.008,0.002)$ and interpret it.
(These values are typical for some of the smaller humanarteries.)
(b) Where in the artery is the flow the greatest? Where is it least?

We have the function V equals p over four n l multiplied by capital R squared minus little r squared. And we're giving the given the data that l equals one e equals 1500 and equals 0.27 And at a particular point in time our equals 0.0.1 and the derivative of our equals minus 0.5 And we want to find d v with respect to d. T. At this given point in time, where also were given that we're in the center of the blood vessel and since little R is the distance from the center, we get that little are equal zero. So in fact, we get that Devi over DT at this particular point in time is the derivative of pee over four and oh, just times are squared. Now we can substitute in the appropriate values the derivative Well, let's take the drew first. So we have this constant on the outside and then the derivative of r squared by the chain rules. First we bring the two down by taking the director of the outer layer. Then we multiplied by the derivative of the inner layer which is just our prime. And now we can Oprea values 1500 over four times point to his dupes point 0 to 7 times one times two times 0.1 times minus 0.5 and that's our answer.

So this is a rather late the Western, so I'm just going to go ahead and you only have the relevant information for what we are being asked to do. So you can pretty much summarize that paragraph into the total flow. And a layer of our artery is defined to be a part of the lost city and a cross section area which is of our is equal to two pi r k Time's capital R squared minus R squared times Delta are where Kay is just some miracle constant. Our is the radius of the artery, and little R is the distance from a given layer to the center. And also they tell us that Delta are is equal to D R Heard differential. So what we're tasked with doing is to set up a definite interval, find the total float in the artery, and we want to evaluate, definitely doable. So what we want to do for part A. What this is saying is, we want to integrate. You are with respect to our and then we have to see what our bounds on this air going to be. So here we're told that our is the rainiest of the artery. So if I were to just call all of it if I weren't just draw all the artery here. The largest it can be is capital R, and then they have some, like layers here. So say this is one of layers and then this one here, this would be one, two or so This distance, right there would be little r so we could have impossible radius of zero, but no plane, No one's artery is literally zero, but at least from the center will be zero and then going all the way out, it can go to capital. Or so let's go ahead and read. Write this as so this here, our belts are remember is supposed to be our This part is actually equal to D. R. Here, so we just need to write the two pi rk times capital R squared minus are spared. So this would be zero to capital R oh, two pi r, okay, Capital R squared minus R squared d R. And so this here is setting up the definite novel and then, for part being, we just want to evaluate this. So to pi K is a constant with respect to the little are so we can go ahead and move that out. Fronts to pie Hey! And then I was going zero the capital are of and I'm going to distribute this are inside here. So how capital r squared times little arm minus are gr So you have to pi Hey! And now Capitol our is a constant with respect to little r since capital are just speedy total distance which will never change So integrating this here will end up with So it's our square capital R c And then the power for our here is once too. So we divide that by its new power and then is tracked all you really are to the fourth power divided by or and then we want to evaluate this from zero to capital No so we have to Hi, Kay. So plugging in our or capital are into this So we have so capital r squared will be expired and then times are square will beat are to the board delighted by two and then minus will be put r squared there are into our to the fourth which should be for over four then we need to see cracked off reporting zero but plenty of men. Zero here would be zero minus zero. And so this here I mean this up a little bit. So in two pi k, this is so are to the fourth over to minus forty fourth over four is going to be our to the fourth over for war because one half miles from forthe support and so this will be hi k r to the four over. And so this here will be the total flow in the lawyer here, so this is always going to do

In problem 70. We have an equation relating the velocity off the blood in the blood vessel with its distance from the center off the blood vessel or small, where p l envy our constant physical constants And are is the radios off the blood vessel, which is constant. All we can consider it is constant, and we want to find Q. Where. Q Is a total blood flow given by this integral we can substitute by. We are in this integral we want to integrate from zero toe are two boys multiplied boy p divided boy for l V multiply by our square minus are small squared but the boy boy are the are We can take all conscience out of the integral to boy p divided by four LV and we integrate We have the different integral from zero to our we have our squared multiply by r minus r cubed you are equals to buoy B divided by four LV The integration of all small is R squared divided by two we have r squared multiplied by r squared, small, divided by two minus integration of our cubed is our is our four divided by four and we integrate from zero to our we start by the hour bound our capital. The service you buy arse mode equals our capital toe boy be divided by four LV multiplied boy first is so stupid by are small equals our capital We have our squared multiplied by r squared, divided by two minus are is about four divided by four minus we suggested by our small equals zero which gives zero equals This bracket here sums up to be are the bar four Divide by two minus all the war four, divided by four gives arts of our four divided by four Then the final answer is to buy be divided by four LV. Applied by are two of our four divided by four. For simplification, it equals boy Pete. Art is about four divided by it l v Then the total blood flu equals boy P R. Capital to the Bar four divided by it L V, which is the final answer off our problem

Kenema, 42 were given the relation between the speed we off the blood inside the blood vessel were p l. And we are constants. It's pressure length on bond. The V is the viscosity, while R is the radius of the blood vessel and the smaller is the distance of the blood from the center of the vessel. Now, treating only are as a variable we have toe find a relation between TV over entity and are when it has already given that ls Peter Beattie. He is 500 on V is 0.0 treat Yeah, keeping these things in mind, we need to find, uh, the rate off change off velocity. So let's differentiate this. When we do that, we have all these terms will come outside. Okay, Uh, when we go inside, differentiation off, capital last, wherever we do, our d r over, DT uh, will be capital are here on the differentiation of smaller is zero because that should be treated as a constant as well. Uh, now let's substitute the values be is 500 Allah's 80 the 0.0 tree here we have to as well, and then we have our d r over DT as it is. If you simplify those, we get 1000 on top. And here we have 3 20 times 0.3 r d Roodt. So finally this comes out as you're talking about, right? So we get the answer as over here we got the final answer has gone 41.67 times are there over DT on in part. We were just given these values and we need to substitute those in part ex. So this is what we're getting after substitution. So using a calculator, the final answer, which for getters it's negative. 0.15 6 to 5. This is the rate off change off velocity.


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