bernoulli's equation vs continuity equation

[KDVMSPH]

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Im so confused?! How do you know which one to use?

I thought that:

constriction of blood vessel = increase in blood pressure; decrease in blood flow
dilation of blood vessel = decrease in blood pressure; increase in blood flow.

I recently came across a question asking what the pressure would be in a TUBE . I got the question wrong, because I used the above logic.

The question that I missed asked this: *please see attachment *

-------> The following represents an incompressible fluid in laminar flow through pipes. Where is the pressure highest?



The pressure was actually highest where the tube was the widest; and the flow was greatest where the tube was the most constricted. I do not understand. Any clarifications please?

 

[FeinMS]

A1v1=A2v2 From this equation, we can tell that as Area of the vessel increase, velocity must decrease given that volume flow rate (Av) is constant. From the diagram, they're all at the same height, so ignore rho*g*h term in Bernoulli. P+0.5*rho*v^2=constant. To maximize P, minimize v.

This kind of question comes up literally every week. So for more explanation, use search function?

 

[KDVMSPH]

Ok thanks!!

 

[The_Sunny_Doc]

Ok thanks!!

It might help you to think of fluid pressure as pressing outward on the pipe in all directions . Imagine thousands of arrows pointed from the fluid, perpendicular to the pipe's interior surface. It takes energy for the fluid molecules to bounce around pressing on the sides of the pipe, keeping it full of water. If you poke a tiny hole into a balloon and deflate it slowly, what happens? The balloons deflates because all those gas molecules that were bouncing around inside, keeping it inflated, are rushing out through the hole. There's a massive pressure drop as the air molecules increase in velocity, flowing out. The molecules have turned some of that random bouncing energy they used to keep the balloon inflated by pressing out on its sides into translational energy, to flow out of the balloon direction. The molecules can't spend their energy doing both tasks at the same time, or that would violate conservation of energy. 🙂 If you caught all the air that flowed out of the first balloon in a second balloon, it should be just as full because energy was conserved in the fluid flow. That's not possible in real life due to friction with the insides of the balloon and heat loss, but it's a good example to illustrate the relationship between pressure and velocity.

So, back in our pipe, we have fluid flowing. It's a bit different from the balloon scenario because it's more controlled. If I'm using a hose to fill a bucket of water at a rate of 10 gallons per minute, the fluid flow through the continuous circuit of the hose has to be the same in every section of the hose ("constant flow rate") or else the fluid would get all backed up and the water wouldn't be able to fill up the hose. It's like a line of conga dancers. If one person stops dead in their tracks, everyone has to stop and they're no longer making their way across the dance floor.

Now that we've established the contuinity of fluid flow in pipes, we can see why a pipe section with bigger area would have fluid flowing more slowly compared to a narrow section of pipes in the circuit. When the fluid flows from the fatter section into the narrower pipe, the pressure pushing outward on the pipe must drop in order to speed up more of the fluid molecules, moving the fluid it through the smaller section at the same rate as it flowed through the fatter section. You can also think of it by imagining those narrow pipes exploding if the fluid pressure pushing outward on them stayed the same as in the fatter pipe section that has way more interior surface area to accomodate the pressure. Finally, the fluid pressure has to be higher in the fatter pipe or else the water level would drop and the pipe would look half full. That's definitely not a continuous fluid flow. Does that help you to picture what's going on?