An elementary treatment of the reverse sprinkler

Abstract

We discuss the reverse sprinkler problem: How does a sprinkler turn when submerged and made to suck in water? We propose a solution that requires only a knowledge of mechanics and fluid dynamics at the introductory university level. We argue that as the flow of water starts, the sprinkler briefly experiences a torque that would make it turn toward the incoming water, while as the flow of water ceases it briefly experiences a torque in the opposite direction. No torque is expected when water is flowing steadily into it unless dissipative effects, such as viscosity, are considered. Dissipative effects result in a small torque that would cause the sprinkler arm to accelerate toward the steadily incoming water. Our conclusions are discussed in light of an analysis of forces, conservation of angular momentum, and the experimental results reported by others. We review the conflicting published treatments of this problem, some of which have been incorrect and many of which have introduced complications that obscure the basic physics involved.

The ‘acoustic scallop’: a bubble-powered actuator

“…a rotary garden sprinkler in which fluid is sucked in rather than pushed out. A recent review of the copious literature on the subject and a nice explanation of the issues involved is provided by Jenkins (2004). The principle behind the regular sprinkler was already discovered by Hero of Alexandria in the second century BCE.…”

“…2 A short 'pulse' of fluid will generate a vortex ring upon exiting the tube which, just as a jet, also carries momentum. 3 There is a connection between the different behavior of fluid entering or exiting a tube and the so-called Feynman sprinkler (Feynman 1985, Jenkins 2004, i.e. a rotary garden sprinkler in which fluid is sucked in rather than pushed out.…”

Mini and Micro Propulsion for Medical Swimmers

“…The propulsion force was calculated, based on the observation that the fluid patterns for the bubble shrinking and expansion processes were not symmetric, as shown in Figure 3a,b. A similar explanation was made by Jenkins [46,47], called "Machian Propulsion". If the Reynolds number UL/v is defined using the speed and size of the body, Re ~5, one may be understood that the inertia propulsion is not effective.…”

Sprinkler head revisited: momentum, forces, and flows in Machian propulsion

“…As explained in Sec. II of [14] and then cleverly demonstrated experimentally in [25], for the reverse sprinkler the relevant torques are produced by a "pressure difference effect," which imparts to the sprinkler an angular momentum opposite to that of the incoming fluid, and a "momentum transfer effect," by which the aspirated fluid transfers its angular momentum to the sprinkler when it impinges on the tube's inner wall. In the absence of viscosity, these torques cancel each other out exactly in the steady state, as required by the conservation argument given in Sec.…”

“…The conservation argument described in Sec. II suffices to establish that the magnitude of the "momentum transfer" effect that is derived in [14] is correct, but the simplified treatment of the forces that was presented there does not account for the fact that the shape of the flow can, in practice, be quite complicated. 3 This seems an important issue to clarify, because many discussions of Machian propulsion have offered the difference in the forms of the in-and outflows as the fundamental explanation of the phenomenon.…”