The application of this question is towards clogged arteries and blood flow. I am wondering why people with clogged arteries have high blood pressure if a smaller artery (assuming it is smooth and cylindrical) means less pressure exerted by the blood according to Bernoulli's Equation. Is this because the heart has to pump the blood harder to be able to travel through a smaller arterial volume?
3 Answers
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I am not sure if the same laws apply to the heart as that of mechanical pump, but for a given flow rate, say X gallons per minute, the mechanical pump must develop a pressure P to overcome pipe friction and any other force trying to retard flow. If the pipe in a system is reduced in size, to pump the same flow rate a higher pressure will be required. The higher pressure will also require an increase in horsepower.
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As mentioned by @Chester, Bernoulli isn't a good approximation for viscous flows which blood flow is. Instead you should use the Hagen-Poiseuille law which relates the average volumetric flowrate and the pressure gradient in the pipe. From it we find that the flowrate $Q$ is proportional to:
$$Q \propto R^4 \Gamma$$
where $R$ is the radius of the pipe and $\Gamma$ is the pressure drop. Now the function of your heart is to provide enough freshly oxigenated blood throughout your body and it does this by striving to keep the flowrate constant. From our relation it is easy to see for a constant $Q$ and a reduced $R$ (as is the case for clogged arteries), the pressure gradient $\Gamma$ has to increase to compensate.
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The Bernoulli equation is a good approximation only if viscous flow resistance is not important. In blood flow through arteries, veins and (particularly) capillaries, viscous flow resistance is very important.
answered Apr 24, 2016 at 12:56