I give you some things to think on.
First Bernoulli’s Eqn is applicable when you assume the fluid is invisid (or the viscosity equals zero).
I assume the continuty equation you are using is for an imcompressible fluid. I know air is compressible but at small velocities it can act fairly imcompressible. We mostly use the imcompressible version of the contuity equation. Basically it says if you push a fluid into a cube that fluid in that cube must react to offset it. For compressible fluid this is not entirely true. Stick to imcompressible though, I think VA=VA pretty much assumes it.
For my uses I’ve been using the Bernoulli equation in the form
Using this equation twice and subtracting will get rid of the constant.
This will give you the pressure difference, delta§, difference when the velocities, v1 and v2, over the tube are different (imagine blowing over a cup with a ping pong ball in it)
This is where I think maybe the equation of continuity you use, VA=VA, comes into play.
Think about having a small tube that expands into a large tube. When the air goes from the small area to the large area the velocity goes down.
A typical problem we do for the bernouillis is a prarie dog (I am from Kansas) tunnel where one hole is lower and the other is at the top of the hill. As the air flows from the lower hole to higher hole it speeds up. This creates a pressure difference at the two holes and let air flow through the tunnel. Basically keeping the prarie dogs from suffocating.
Anyways, using continuity you can find the two velocities. Then you can use Bernouillis equation to determine pressure difference, delta§.
Now the fun part is finding the velocity through the tube.
Here is the equation I use for a pressure driven flow in a tube
delta§=change in pressure (from Bernouillis)
radius=radius of the tube
u=viscosity of the fluid (at 20C it is .01813 milliPascals*seconds)
L=length of tube
That is how I’ve been doing it lately. If this was fun for you look into to Chemical engineering or go out and buy Transport Phenomena by Robert Bird. It really is a fun class.
I hope this helps I wasn’t sure of the exact question so I tried to answer as many as possible.