Pascal's law and kinetic energy in pipes

In summary, the conversation discusses the behavior of fluid in a U-shaped pipe and how it levels itself out due to Pascal's Law. The kinetic energy of the fluid comes from the potential energy in the taller column. The question then arises about what happens when the bottom of the U-shaped pipe is made longer and how energy conservation is not violated. The answer involves the differential equation governing the oscillations and how in the limit of an infinitely long pipe, the period of oscillation also goes to infinity, ensuring energy conservation.
  • #1
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If you have a U-shaped pipe like the attached image, with more fluid in the left column than the right, then the fluid will level out basically because of Pascal's Law. Force on the fluid is exerted on every part of it in equal direction etc..

So when the fluid levels itself it flows across to the right hand side to fill the empty area. The fluid inside the bottom part of the U-shape flows to the right with a certain velocity or kinetic energy. The kinetic energy of that flow comes from the potential energy of the fluid in the tall left column.

My question is what happens when you make the bottom of the U-shape very long? Now you are moving a lot more fluid with that little bit of potential energy. How is that energy being 'created from nothing'?

I understand flow is not that simple and there are eddies etc.. But it must be true that fluid is basically flowing across the whole pipe, otherwise you couldn't fill that empty space in the right column.
 

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  • #2
Interesting question. Assuming inviscid flow, one can show that, if the total length of liquid contained within the U-shaped pipe is ##L## and that the system is subject to a uniform gravitational field ##g##, then the differential equation that governs the oscillations is given by
$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{2g}{L}x = 0.$$
This means that the period of oscillation is
$$T = 2 \pi \sqrt{\frac{L}{2g}},$$
which means that the longer the fluid is, that is, the longer the bottom part of the tube is, the period of oscillation increases proportionally to ##\sqrt{L}##. You can see then that for longer tubes, the motion becomes slower and slower, and for very long tubes, the motion of the fluid will be really slow. In the limit of an infinitely long tube, the period also goes to infinity, so there's no energy conservation issues.

If you wish to see the derivation of the above differential equation of motion, have a look at this page.

It's in Spanish but I think the main ideas can be understood.

Thinking out loud now, this derivation is fine and dandy for an ideal inviscid fluid... however, at very slow speeds, I'm not sure that kind of approximation would be valid, as it would be creeping flow and the viscous terms are no longer negligible in the Navier-Stokes equation. But this first argument should make you see why energy conservation doesn't get violated.
 
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1. What is Pascal's law?

Pascal's law is a fundamental principle in fluid mechanics that states that pressure applied to a confined fluid is transmitted equally in all directions. This means that if pressure is applied to one point of a fluid in a closed system, the same pressure will be exerted on all other points of the fluid.

2. How does Pascal's law relate to kinetic energy in pipes?

Pascal's law is closely related to the concept of kinetic energy in pipes. When fluid is flowing through a pipe, the pressure exerted by the fluid at any given point is determined by its velocity and density. This is because the faster the fluid is moving, the more kinetic energy it has, and therefore the greater the pressure it exerts on the walls of the pipe.

3. How do changes in pipe diameter affect kinetic energy and pressure in a fluid?

According to Pascal's law, the pressure exerted by a fluid in a closed system is transmitted equally in all directions. This means that if the diameter of a pipe decreases, the fluid must speed up in order to maintain the same volume flow rate. As a result, the kinetic energy and pressure of the fluid will increase in the narrower sections of the pipe.

4. Can Pascal's law be applied to all types of fluids?

Yes, Pascal's law can be applied to all types of fluids, including liquids and gases. However, it is important to note that the density and compressibility of the fluid may affect the magnitude of the pressure exerted, as these factors impact the fluid's ability to transmit pressure equally in all directions.

5. How is Pascal's law used in practical applications?

Pascal's law is used in a variety of practical applications, particularly in hydraulic systems. Hydraulic systems use Pascal's law to transmit and amplify forces through the use of fluids. This allows for the efficient transfer of large amounts of energy, making hydraulic systems useful in a range of industries, from construction to manufacturing.

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