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I have read that in an artery, the blood pressure is greater in an opene artery than in a blocked/constricted one.

Why is this the case?

Looking at the Bernoulli equation,

Pressure + 1/2*density * velocity^2 + density * gravity * height is constant. In a constriction, area goes down, and as flow-rate = velocity * area = constant, as area decreases, velocity increases. As density and gravity and height are assumed to be constant, Pressure must decrease in this constriction.

However, I can't physically udnerstand why this is the case. If I blocked off a part of a pipe, wouldn't pressure increase at that area, as fluid couldn't pass through as easily and e.g. would be putting a force on the constriction?

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It is the gradient in pressure that accelerates the fluid. Mass conservation dictates that the fluid must speed up in the constriction. In order to speed up, a little cube of fluid must feel higher pressure behind it than in front of it. Thus where speed increases, pressure decreases.

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