Abstract
When liquid nitrogen is set into rotation in a warm pot, the surface, initially forming an axis symmetrical, toroidal shape, spontaneously re-structures into a series of rotating polygons with a diminishing number of corners, while slowing down. This spontaneous ordering occurs despite the violently turbulent and boiling state of the fluid. We show experimentally that these shapes are well-described as a sum of a few Fourier modes, and we present experimental results for the development of the frequencies and amplitudes of these wave-modes during the transient process. We compare our results with the theoretical results for the instabilities of a potential vortex flow and argue that the first polygon formed in the transient process should be described by this theory. The paper is dedicated to the memory of Yves Couder, colleague, friend and a life-long source of inspiration.
Highlights
Introduction by Tomas BohrYves Couder was a pioneer in the field of pattern formation that emerged through the 1980’ies from nonlinear dynamics, chaos, statistical mechanics and fluid mechanics
His background included condensed matter physics, and electronic band structures and his first work was on cyclotron resonance and Landau levels in very pure Tellurium – a single author paper in Physical Review Letters [1]
We do not believe that they are much related: the polygonal hydraulic jumps are seen at intermediary Reynolds numbers – of the order of 102 or less – and are believed to be caused by surface tension [14, 15], while “rotating polygons” seen on swirling flows have Reynolds numbers in the 105-regime, and are believed to be caused by wave resonances [16, 17], as we shall describe in the present paper
Summary
Yves Couder was a pioneer in the field of pattern formation that emerged through the 1980’ies from nonlinear dynamics, chaos, statistical mechanics and fluid mechanics. At that time he was trying to understand viscous fingering, and I remember my surprize that one could solve (or reproduce) the result that the limiting stable viscous finger has the width of half the channel in the presence of surface tension – a strongly nonlinear fluid dynamics problem – by looking at the statistical physics of random walkers [4] Both the style of Couder’s work and the style of the schools themselves made a deep impression on me and inspired me to organize summer schools in Denmark as well. We do not believe that they are much related: the polygonal hydraulic jumps are seen at intermediary Reynolds numbers – of the order of 102 or less – and are believed to be caused by surface tension [14, 15], while “rotating polygons” seen on swirling flows have Reynolds numbers in the 105-regime, and are believed to be caused by wave resonances [16, 17], as we shall describe in the present paper
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