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Cal. Tech. Drivel: A Badobadop Exposť
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I was surfing the internet this week looking for something interesting to post as a 'Did You Know?'  as I haven't done one for quite some time. My search led me to a basic problem in fluid dynamics which is known as  'The Feynman Sprinkler Paradox'. It is sometimes referred to by other names such as 'The Feynman Reverse Sprinkler', or just 'The Reverse Sprinkler'.




27 October 2014
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Drivel: Nonsense.

Synonyms: twaddle, claptrap, balderdash, gibberish, rubbish, mumbo jumbo, rot, tommyrot, poppycock, phooey, hot air, eyewash, piffle, garbage, tripe, waffle, bosh, bull, bunk, blah, hogwash, baloney.
You would think that scientific papers emanating from illustrious institutions like Cal. Tech. pictured here, would be free from basic, schoolboy errors.... Not so!

The essence of the problem is this:


We know that a sprinkler turns when fluid is ejected under pressure, but what happens if fluid is sucked into the sprinkler instead?


The problem was not originally conceived by Richard Feynman, and its true origins are not known. The first reference in scientific literature  was probably made by Ernst Mach, who  published this picture of a sprinkler in his  book 'The Science of Mechanics'. He spoke of  "Reaction wheels" and stated that if air or gas is emitted from the wheel, the whole wheel will be set motion. Mach also wrote: " It might be supposed that sucking on the reaction wheels would produce the opposite motion to that of blowing. Yet this does not usually take place."


Mach's worded explanation of the behaviour of the reverse sprinkler is insightful and based on sound reasoning. 




Although the reverse sprinkler problem was discussed by Feynman, he never published a solution. I suspect that he viewed the problem as being trivial. He must have derived amusement from the confusion it created within the academic community. Trivial as it is, the problem has received much attention in recent years and many papers have been published expressing conflicting opinions.
I found a number of online explanations of the problem. Most of them arrive at the conclusion that a reverse sprinkler will not rotate. They usually cite the reason as "conservation of angular momentum" without providing any real justification or analysis. Then I came across a paper produced at Cal. Tech. by Alejandro Jenkins (click here) which provides a detailed analysis of the problem (click here). The paper was published in 2004, in the American Journal of Physics (Am. J. Phys. 72 (2004) 1276-1282), so I presume it was refereed and subject to rigorous peer review.
On the face of it, the analysis is convincing and arrives at the conclusion that the reverse sprinkler will not rotate unless some small second order effects are considered. Broadly speaking, I concur with the paper's conclusions, but I fell off my seat when I read the analysis in detail.
In addition to a flawed analysis of the forces acting at a bend in a pipe, Jenkins asserts that in the steady state, fluid within the sprinkler pipe gains momentum as it flows. This implies that there is an increase in mass flow rate within the pipe. I quote:
"Therefore, over the entire length of the pipe, the water picks up momentum at a rate A(P1 - P2), where P1 and P2 are the values of the pressure at the endpoints of the pipe."
"As water flows down a tube with a pressure gradient it picks up momentum."
what a load of old cobblers!
Every schoolboy knows that water is incompressible and that the mass flow rate of water in a pipe has to be constant throughout its length (click here). If water enters the pipe at 1 kg per second, then 1 kg per second must flow through any chosen point along the pipe's length. It follows that provided the mass flow rate is constant (steady flow), the momentum of water flowing in a straight length of pipe must remain constant. If the direction of flow is changed at a bend, then the directional property of the momentum is changed, but its magnitude is not.
Cal. Tech. is an illustrious academic institution. The American Journal of physics is the premier international physics publication. How on earth did these errors slip through the rigorous process of peer review and editorship? Everyone associated with this debacle should hang their heads in shame and eat dirt! It just goes to show that experts are often wrong. Expert opinion should always be taken with a pinch of salt and treated with a degree of skepticism. Badobadop's articles are no exception!
For those interested in the problem, here is a simple explanation of what happens:
In a reverse sprinkler, momentum gain takes place before the water enters the pipe. and can be resolved to a constant force acting on the water as it enters the pipe. The size of this force is equal to:
                                                                  A(P1 - P2)
where P1 is the ambient static pressure outside the pipe, P2 is the static pressure just inside the pipe entrance and A is the cross sectional area of the pipe entrance.
Forces act at the bend in the pipe to change the direction of flow and cancel the force acting at the entrance of the pipe. Force diagrams for the sprinkler and reverse sprinkler are shown below: 
As you can see, a torque is generated in the case of the sprinkler. For the reverse sprinkler, the sum of all forces is zero and there is no torque in either direction, so the sprinkler does not turn.
A layman's explanation for this situation is that water, which is initially motionless, is drawn into the pipe from all directions, resulting in a net zero force on the pipe end. This is distinct from a normal sprinkler where water is ejected into the surrounding volume in a particular direction. This results in a reaction force, much as a rocket is propelled through space. It is the difference between blowing out a candle and attempting to extinguish a candle by sucking instead. Historically, this explanation has been castigated by the scientific community, but Badobadop argues that this is a valid explanation, and all rigorous scientific analysis is consistent with this rather simple, qualitative approach.
As a post script, Alejandro Jenkins has made a second attempt at analysing the reverse sprinkler, 'Sprinkler Head Revisited: Momentum, Forces, and Flows in Machian Propulsion' which was published in 2011, in the European Journal of Physics (click here). I haven't had time to analyse the work in detail, but first impressions indicate that the explanation is free from the schoolboy errors made in the original paper, but perhaps a little more complicated than is necessary to illustrate the point.
As a point of interest, YouTube has video footage of just about anything, the reverse sprinkler included. Look at this one:
The results seem confusing at first, and do not support the mathematical analysis. The design of the apparatus is inspired, in that it implements the reverse sprinkler with virtually zero friction. However, this is also the reason for the strange results. It is impossible to fabricate the apparatus so perfectly that there is no bias (pipes misaligned or bent very slightly). Under zero friction conditions, the smallest bias either way will result in rotation one way or the other. See my comments on the YouTube page itself. 
Image courtesy Canon vs Nikon.
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