# Do you need to increase pressure to pump a fluid from a large pipe through a small pipe?

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?

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.