What is pressure accourding to Bernoulli's theorem?

[Chestermiller]

For the record, Munson defines an ideal fluid as inviscid and incompressible. So does Nakayama. Durst only requires the fluid to be inviscid.

I don't know the it is right to say that the Euler equation requires the fluid to be adiabatic, as it is simply a momentum balance for a flowing inviscid fluid. Why would you consider heat transfer when doing a momentum balance? Of course a complete description of the flow will necessarily account for any heat transfer, so that the flow parameters will be coupled through continuity, momentum and energy equations (and an entropy equatoin of course).

If the fluid is inviscid, the coupling of the momentum balance (and Bernoulli equation - the mechanical energy equation )with the heat balance is only through the effect on density. Of course, for an incompressible fluid, there is no coupling at all.

In undergraduate mechanical engineering fluids courses, we usually use some form of the three in the image below (from White) as the "complete description" of flows.

View attachment 229986

With an inviscid flow the last term of the momentum balance (and also the energy balance) is zero, yielding Euler's equation. (The differential of the velocity is the total.)

I totally agree with what you are saying here; it is basically what I have been saying all along, although you have presented it much more elegantly.

 

[vanhees71]

For the record, Munson defines an ideal fluid as inviscid and incompressible. So does Nakayama. Durst only requires an ideal fluid to be inviscid.

I don't know that it is right to say the Euler equation requires the fluid to be adiabatic, as it is simply a momentum balance for a flowing inviscid fluid. Why would you consider heat transfer when doing a momentum balance? Of course a complete description of a general flow will necessarily account for any heat transfer, so that the flow parameters will be coupled through continuity, momentum and energy equations (and an entropy equation of course). In undergraduate mechanical engineering fluids courses, we usually use some form of the three in the image below (from White) as the "complete description" of flows, with further simplifications starting from these.

View attachment 229986

With an inviscid flow the last term of the momentum balance (and also the energy balance) is zero, yielding Euler's equation. (The differential of the velocity is the total derivative.)

Well, I've seen this discrepancy too in the German textbook literature. Some define an ideal fluid to be inviscit and incompressible. Of course, if you assume incompressibility you don't need an equation of state.

When I learned hydrodynamics the first time, I attended a lecture given by an engineering professor (Kolumban Hutter at TU Darmstadt). He defined an ideal fluid to be inviscid but not necessarily incompressible. Then, of course, you need an equation of state, and he also defined an ideal fluid as one where not only friction but also heat conductance can be neglected. Then the flow is adiabatic or equivalently isentropic.

In my community (relativistic heavy-ion collisions) we use a lot of relativistic hydrodynamics to describe the evolution of the fireball created in the collisions. In this community an ideal fluid by definition has no friction nor heat conductivity, i.e., the flow is adiabatic. For a better description of course we need viscous hydro with shear (and sometimes also bulk) viscosity. Of course in relativistic viscous hydro you have to work at at least 2nd order in the moment expansion, because first order (Navier Stokes) leads to acausalities, as is well known for decades.

Usually hydrodynamics is derived as a limit of the Boltzmann equation, where the fluid is close to local thermal equilibrium and relaxation times (or equilibration times) are small compared to the typical macroscopic time scales of the fluid. The ideal-fluid limit is the limit, where you neglect all relaxation times completely, i.e., the fluid is considered to be instantly in local thermal equilibrium all the time, and then it's clear from the very foundations that there is neither friction nor heat conduction and thus the flow is autormatically adiabatic/isentropic.

I think, this debate is just about semantics. So we have to be more careful in defining what we mean with our definitions (in this case what we precisely understand under an "ideal fluid"). The only way is of course to give a complete set of equations stating clearly all assumptions.

 

[Zeusss Dude]

Its velocity increases because its pressure was suddenly released. This accounts for both principles.

When water is forced into a smaller tube (a nozzle), the pressure increases and the velocity decreases due to the particles colliding more and more. It like two people trying to run through a door way. They don’t both fit even if you have a lot of pressure. The same happens here. When they reach the end of the nozzle, the pressure is gone because they are released to do whatever. They speed up.