An aortic aneurysm is a cardiovascular abnormality where the primary artery supp
ID: 1614214 • Letter: A
Question
An aortic aneurysm is a cardiovascular abnormality where the primary artery supplying blood from the heart irregularly increases in size, usually in the abdomen region. Because 0f the aneurysm, the radius of the aorta of a patient has increased by a factor of 1.4 as compared to a healthy aorta. The speed of blood through the healthy portion of the aorta is 0.43 m/s when the patient is lying flat. The blood has a density of 1060 kg/m^3. Consider the case where blood is treated as an ideal fluid in steady flow.
(a) Consider the artery that starts at healthy size, increases size in aneurysm, then decreases back to healthy size. What is the blood velocity in the aneurysm region? Ignore any dissipative effects from viscosity.
(b) What is the magnitude of the pressure difference between the aneurysm as compared to the healthy aortic artery?
Explanation / Answer
(a)
We use the equation of continuity which says.
A1v1 = A2v2
where A1 is area of portion where fluid speed is v1 and A2 is area of portion where fluid speed is v2.
Now, if the radius of healthy portion of aorta is r, then the radius of aorta with aneurysm will be 1.4r.
If A1 is the area of cross section of healthy portion of aorta and A2 is the radius of aorta in a region with aneurysm, then we have,
A1 = r2, and A2 = (1.4r)2
Using the equation of coninuity we have
r2v1 = (1.4r)2v2
or v1 = 1.96v2
or v2 = v1/1.96
or v2 = (0.43 m/s)/1.96 (as the speed of plood through the healthy portion of aorta is 0.43 m/s)
or v2 = 0.22 m/s
So the speed of blood in the region with aneurysm is 0.22 m/s.
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(b)
For healthy region of aorta, let the pressure be P1 and for region with aneurysm let the pressure be P2. The speed of blood at the position where pressure is P1 is v1 and the speed of blood at the position where pressure is P2 is v2. Then according to Bernoulli's equation we have;
P1 + (1/2)v12 = P2 + (1/2)v22 (if we assume the person to be lying flat, the height difference between two positions would be zero.)
or, P2 - P1 = (1/2)(v12 - v22) = (0.5)(1060 kg/m3)[(0.43 m/s)2-(0.22 m/s)2]
or, P2 - P1 = 72.34 N/m2
So the pressure difference is 72.34 N/m2.
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(c) As we saw above pressure in aneurysm is higher than pressure in healthy aorta by 72.34 N/m2, so the pressure in aneurysm is higher as compared to the pressure in healthy aorta.
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This concludes the answers. Check the answer and let me know if it's correct. If you need any more clarification or correction, feel free to ask.....