Hydraulic dynamics question We know the length of a pipe, say L, and the starting point pressure p

othereyeshmt4l 2022-05-18 Answered
Hydraulic dynamics question
We know the length of a pipe, say L, and the starting point pressure p1, ending point p2, cross-sectional area A. What else do we need to compute the mean velocity of flow in this pipe?
My full question actually runs as follows. Given the mean cardiac output(volume flux) be 5.5L per min, the radius of the aorta is about 1.1cm. In the systemic circulatory system, the mean radius of a capillary is about 3µm. The mean pressure at the arterial end of the capillary bed (beginning of the capillary system) is estimated to be about 30mmHg, and about 15mmHg at the venous end (end of the capillary system). The length of capillary is 0.75mm.
Calculate the mean velocity in the aorta and in a capillary.
My idea is to use the volume flux 5.5L/min and the radius of aorta, divide the flux by the crossectional area to find the mean velocity in the aorta. However, I have no idea how to deal with the velocity in a capillary. I tried to use Poiseulle's law, but given that we only know the difference of pressure and length of the capillary, it is still unsolvable. Could anyone tell me how to deal with it?
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Answers (1)

Gerardo Barry
Answered 2022-05-19 Author has 17 answers
Poiseulle's Law should work! Let's look at each quantity involved in Poiseuille's Law:
Δ p = 8 π μ L Q A 2
As you've rightfully pointed out, we know pressure difference Δ p = p 1 p 2 = 15  mm Hg and length L = 0.75  mm as specified by the question. Additionally, the cross-sectional area A can be computed in terms of radius r as A = π r 2 (assuming an approximately circular cross-section).
For dynamic viscosity μ, it is a constant that depends purely on the property of the fluid. So you could just take a literature value for this! According to Wikipedia, this ranges somewhere between μ ( 3  to  4 ) × 10 3  Pa s
Substituting these values into the equation, we can then compute the unknown volume flux Q through the capillary. Note that this value should turn out to be smaller than the volume flux through the aorta (i.e. mean cardiac output), since not all of the blood through the aorta would go through one specific capillary.
From the volume flux, you can then work out the mean velocity of blood in a capillary to be v = Q A
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