Bernoulli's principle in stagnant situations

[intuitional]

please note in the process proving bernoulli we use the CONTINUITY equation.Consider a situation wherein a pitot tube is used to measure pressure/velocity in a steady flow(picture attached for reference).Here we use bernoulli for pressure measurement.we apply the bernoulli theorem for a streamline which has some finite velocity at one end while zero(stagnation point) at the other end.This violates continuity and hence we canot apply bernoullis theorem in the first place itself!can ayone explain the situation
PS-in almost all fluid mechanics books they just apply bernoullis directly as written,without any explanation of the above question.a name of a book explaining the situation properly would be very helpful too.

 

[rcgldr]

These might help, wiki articles:

http://en.wikipedia.org/wiki/Stagnation_pressure

http://en.wikipedia.org/wiki/Impact_pressure

 

[intuitional]

have already read them.the point here is the that continuity is essential for bernoulli(atleast from the proofs that i have seen till now).there is no point applying bernoulli if there is no continuity

 

[intuitional]

have already read them.the point here is that while proving bernoulli you use continuity in the proof(atleast the proofs which i know) and continuity is not valid here at that specific point(A) for any streamline passing through it.so you cannot use bernoulli in the first place

 

[intuitional]

have already read them.when you prove the bernoullis theorem you require continuity in some step while proving.but there is no coninuity here for the point A.so solving the problem using bernoulli is wrong which is done widely in every book and elsewhere.so how do you do it then?

 

[imiuru]

in the process proving bernoulli we use the CONTINUITY equation.

Where do you get this? I did not remember using continuity equation while deriving bernoulli's equation.

 

[imiuru]

Your post is interesting. I don't have answer for now, guess I have to investigate a bit.

Well, I learned bernoulli's equation from one of the "every book" you mentioned and I never questioned the use of bernoulli's equation when there is no continuity.

 

[boneh3ad]

You don't need the continuity equation to derive Bernoulli's equation. It can be done directly from Euler's equation assuming the flow is inviscid, incompressible and steady. In fact, just the other day I made a post where I did just that (see this thread). So in other words, whether or not continuity is satisfied, Bernoulli's equation still makes mathematical sense.