I Trouble with Bernoulli's principle

1. Nov 8, 2017

Erland

According to the second Wikipedia derivation for incompressible fluids

https://en.wikipedia.org/wiki/Bernoulli's_principle

the change of the kinetic energy of the system equals the net work done on the system.

Why only kinetic energy? Why can't the work done on the system also change the potential energy? (Note: I don't mean gravitational potential energy - here, we only consider the case where the elevation is the same for the whole system.)

I'm thinking (correct me if I'm wrong) that in a region with higher pressure, there is an internal potential energy which is greater than in a region with low pressure. It seems logical that if that the harder the molecules in the fluid press against each other, the more work can be done on a neighbouring part of the fluid.

If it wasn't so, how could work be done at all on the system by the pressure of the neighbouring fluid at the ends of the system? Where does the energy doing the work come from?

And if there is such an internal potential energy, why doesn't it change when work is done on the system? Why does only the kinetic energy change?

2. Nov 8, 2017

Bernoulli'sdoesn't equation has three terms in it's classical form: pressure, dynamic pressure, and hydrostatic pressure.

Dynamic pressure ($\rho V^2/2$) is the bulk kinetic energy per unit volume of the fluid in motion.

The hydrostatic pressure term ($\rho g h#) is effectively gravitational potential energy per unit volume. The pressure ($p##) is also an energy per unit volume term. You can think of it one of two ways: it is either a stored potential energy sort of like a spring, or else it is a continuum representation of the kinetic energy due to the motion of all of the individual fluid molecules. It behaves the same way either way.

All of these vary (or potentially vary) when work is done on the system, but their sum remains constant. in that sense, Bernoulli'sdoesn't equation is a conservation of energy statement.

3. Nov 8, 2017

Jano L.

Only as a special case. In general, pressure is a function of position and usually changes in downstream direction.

4. Nov 8, 2017

Jano L.

Internal energy of the fluid can change as well. This can only happen if the fluid gets compressed or if internal friction in the fluid is significant (large gradients of velocity). When that is so, the assumption of incompressibility that the Bernoulli equation derivation is based on is no longer applicable and things get more complicated. For example, flow in thin tubes such as blood flow in capillaries manifests lots of internal friction in the fluid and the Bernoulli equation does not apply. In the simplest case the friction can be taken into account - see the Hagen-Poiseuille equation.

https://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

But there are many cases where the fluid does not compress much and where the internal friction can be neglected, if only to get a rough idea of the flow. Then the internal energy of the fluid is constant, can be ignored and the Bernoulli equation is applicable.

That is true, but usually*, if the fluid does not experience extreme pressures, the change of internal energy is very minuscule. In liquids, to change internal energy appreciably, extreme pressures would be needed since the work pressure forces can do is constrained by quite a small volume changes allowed by the liquid state. In common cases of flow of water, changes in internal energy are minuscule and can be very well ignored.

* This is not true in gas flows that experience significant changes in pressure, such as air flows around fast moving bodies, or expansion of gas from pressurized tank ,or liquid flows near sharp propeller blades where bubbles form. There, the Bernoulli equation may not be even roughly correct description, if the fluid changes temperature and volume too much.

Internal energy is not needed to explain the work of pressure forces. It is not needed to explain mechanical work on incompressible solids and similarly it is not needed for mechanical work on incompressible fluids.

The work on any fluid element is done by contact pressure forces of the surrounding fluid. The energy comes from other energies in the body pushing and moving that surrounding liquid, maintaining pressure and moving the fluid so it can do work. For example, work done on a hydraulic lever is done by a motor that pushes a piston in contact with the hydraulic liquid.