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Pattern formation of granules on the bottom of a differentially rotating tank

Published online by Cambridge University Press:  26 April 2006

P. J. Thomas
P. J. Thomas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
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Abstract

The formation of patterns of granules on the bottom of a differentially rotating tank, partially filled with a fluid, is described. The pattern formation is initiated when the tank is spun up from an initial constant rotation rate by a sufficiently large increment to another constant higher rotation rate. The granules are observed to be set into motion, slide across the bottom of the tank and organize themselves into two distinct, superposed geometrical patterns. The first pattern constitutes a system of tightly wound equi-angular spirals and is identified as being simply a visualization of the well-known stationary Class B waves of the unstable boundary layer. This pattern is not considered further here. The second pattern, which is considered here, is shown not to be a visualization of some other known stationary wave mode. It consists of a number of large-scale spiral arms extending outward towards the wall of the tank and originating at a patch of salt in the tank's centre. The presence of Class B waves is apparently a necessary condition for the spiral arm formation. The experiments show that the radius r0 of the inner salt patch is inversely proportional to the increment Δω of the rotation rate for the larger values of Δω. The angle ε(r) between the direction of the spiral arms and the tangential direction is observed to decrease with the distance r from the centre as rb where b = −0.64 ± 0.1. The relationship between the number n of spiral arms and the associated Reynolds number Re and Rossby number Ro is found from the experimental observations and dimensional considerations to be \[n^2 = Re \, Ro. \] Various aspects related to the mechanism involved in the pattern formation are discussed. However, no conclusive model for the spiral arm formation can at present be suggested.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 274 , 10 September 1994 , pp. 23 - 41
Copyright
© 1994 Cambridge University Press

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