Blood Pressure v.s. Cross-sectional Area

[LINK1290]

My prep book states the following:

"Although Bernoulli's equation tells us that pressure is inversely related to cross-sectional area, it is evident that this is not the case in blood vessels."

I fail to see where in Bernoulli's equation this implication is made, but if I look at the Continuity Equation Q = Av where Q is the volume flow rate, A is the cross-sectional area, and v is the velocity of a fluid, it is easy to see that cross-sectional area is inversely related to velocity. From Bernoulli's equation P + pgh + (pv^2)/2 = K, we see that pressure is inversely related to the square of velocity. Since A is inversely proportional to v and v^2 is inversely proportional to P, doesn't this mean that P is directly proportional to the square of A? If anyone can explain the statement my book makes, I'd be very grateful.

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[jp134711]

One thing the book may not tell you is that the flow rate Q, is also equal to the pressure of the vessel over the resistance of the vessel. The resistance of the vessel is inversely related to the fourth power of the vessel radius (assume cylindrical shape). Since flow rate Q is assumed constant, you can see now that P=QR=K/r^4, where K is some "effective" constant. Thus, the smaller the radius, the higher the pressure. This may seem counter-intuitive for capillaries, which have the smallest diameter (hence, radius) of all the vessels. But because there are so many capillaries in the capillary bed, you can effectively bunch together their radii to get a large cross-sectional area. Thus, pressure is low in the capillaries.

Make sure you're aware that blood is NON-Ideal and that Bernouilli's eqn only works for ideal fluids.

 

[animasian]

Hello - sorry to bring back such an old thread but are you saying that
1) For ideal fluids, as cross-sectional area decreases pressure decreases
(using Bernoulli's Equation and Continuity Equation)
2) For non-ideal fluids, as cross-sectional area decreases pressure increases
(using Av = P/R and R = constant/r^4)

If this is indeed true, could you offer some guidance as to how we can tell whether a fluid is ideal or non-ideal? For example, if asked about "a stream of fluid flowing steadily through a horizontal pipe of varying cross-sectional diameter... Neglecting viscosity" then can we safely assume that pressure is greatest where area is greatest? On the other hand, if asked about "a stream of viscous fluid... such as blood" can we assume that pressure increases when there is vasoconstriction?

Finally - I am trying to get an intuitive feel for why the conclusions are the opposite for ideal and non-ideal fluids. Can we think of it like, there is "time lag" in non-ideal fluids which are viscous and slower to respond to changes? So when cross-sectional area decreases, the non-ideal fluid doesn't speed up its flow quickly enough, thereby resulting in an increase in pressure?

Thanks in advance for clarifying this topic - it is really very confusing for me!