# Pressure in piezometers with ideal fluid

1. Dec 11, 2016

### fog37

Hello,
I am trying to understand how to apply Bernoulli's equation (which is valid along a single streamline and in the case of zero viscosity) to the situation below.

The ideal fluid moves at constant speed along the horizontal pipe. Because the fluid is ideal (zero viscosity), the ideal fluid will reach zero height inside any of the vertical piezometers. Why? I know there is no pressure drop along the horizontal pipe since the viscosity is zero.

If the viscosity was not zero, the fluid would actually reach a nonzero height along the first piezometer and a lower, nonzero height in the second piezometer (to indicate the pressure drop). How come the viscous fluid would manage to go up the piezometric tubes while the ideal fluid does not?

Thanks.

2. Dec 12, 2016

### Staff: Mentor

For an inviscid fluid, the pressure at the exit of the tank is equal to atmospheric pressure, so there is no pressure variation along the horizontal pipe. The pressure at the base of each piezometer is atmospheric, so the fluid cannot rise.

If the fluid is viscous, then there is a pressure drop from the inlet end to the exit end of the horizontal pipe, and the pressure varies linearly from the inlet end to the outlet end. This means that the pressure at the exit of the tank is higher than atmospheric pressure. The pressure has to be higher than atmospheric at the tank exit to overcome the viscous drag in the horizontal pipe.

3. Dec 13, 2016

### fog37

Thanks Chestermiller,

So, in the case of an ideal inviscid fluid, we have the following pressures:

The fluid must accelerate between point 1 (pressure higher than atmospheric pressure) and point 2 (atmospheric pressure). After that, the fluid proceeds at constant speed (continuity equation). There must therefore be a pressure gradient from point 1 to point 2. Does the pressure become atmospheric even before point 2 along the pipe?

Why would the pressure at points 2,3,4 be atmospheric and not equal $(P_{atm}+\rho g h_{0})$, i.e. the same pressure value as in point 1? I understand that on the right side of the exit orifice of the horizontal pipe the water stream is free and is the open atmosphere so the pressure on it is atmospheric. But to the right of the orifice, the pressure should not be atmospheric pressure. Why is it? If we cut the horizontal pipe (i.e. perform the Torricelli experiment), the pressure at the drain is always taken as atmospheric if the drain hole is very small and we are considering a point just outside the hole, in the open atmosphere.

In some textbooks we see an ideal non viscous fluid flowing in a pipe and the piezometers along the pipe have fluid reaching the same height (to indicate no loss in pressure) but those heights are not zero, like in the figure below (point c and d):

What am I missing? I know that Bernoulli's equation needs to be applied to a single streamline, inside an ideal fluid and that the sum of the three heights (geodetic, piezometric and kinetic) must be constant.

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4. Dec 13, 2016

### Staff: Mentor

In the first figure, the pressure at point 1 is not given by the hydrostatic equation, even for an inviscid fluid. Within the tank, in the region approaching the exit hole, the flow is accelerating toward the hole and the pressure is decreasing. The effective region where this occurs is within a few hole diameters of the exit hole. So the pressure at the exit hole is atmospheric, not hydrostatic.

In the 2nd figure, there are decreases in cross sectional area in the horizontal tube, so, by Bernoulli's equation, the pressures at e and f are higher than atmospheric, and the pressures at b and c are higher than at e and f.

5. Dec 13, 2016

### fog37

Ok, great, thanks! I am making some progress.

So, just to make sure, still talking about the ideal inviscid fluid, if we cut the horizontal pipe to exclude the section with the constriction (where pressure is reduced) the fluid height in the two piezometric tubes at point c and d would go down to zero and would NOT look like

but like

However, if the fluid was viscous, the figure would look like

At the exit hole, the pressure is still atmospheric. The pressure $P_b > P_c> P_d >P_{atm}$. The fact that a the liquid can climb up the piezometers in the case of a viscous fluid is due to the fact that the pressure gradient exist along the entire horizontal pipe and the pressure differences are manifested as nonzero heights along the piezometers...

6. Dec 13, 2016

### Staff: Mentor

Yes. Perfect, except that the pressures at the base of the vertical tubes (not the pressure differences) are manifested as nonzero heights along the pizometers.

7. Dec 13, 2016

### fog37

Great!
I am quite satisfied with this minor achievement and relieved to know that, in the case of an inviscid fluid, the two figures are not incompatible with each other and both correct:
(no restrictions or expansions in the horizontal pipe).

What threw me off was the nonzero height reached by the inviscid fluid in the first figure and the zero height in the 2nd figure. The difference is due to the different container configurations.

If we closed the exit hole in the horizontal pipe the fluid would soon reach the same height in all the piezometric tubes in both configurations. Of course, the longer the horizontal tube the lower the height reached by the fluid will be in that static situation (communicating vessels principle).