Cellular Stokes flow induced by rotation of a cylinder in a closed channel

Abstract

The evolution of the cellular structure of the two-dimensional creeping flow induced by a rotating circular cylinder set in the centre of a rectangular channel is studied numerically and experimentally when the aspect ratio A increases from 1 to 7. In the calculations, depending on the value of A, either only series in terms of polar coordinates, or both matched polar and Cartesian coordinates series are employed to represent the stream function and an efficient least-squares method, very easy to program, is selected to satisfy some of the boundary conditions. For the experiments, a special technique which visualizes intermittently the paths of solid tracers during long times of exposure permits us to observe the fluid motion in the whole domain, even in the regions where the velocities are very small. An excellent measure of agreement between the numerical and experimental results is found. Thus it is clearly shown how, in the region beyond the rotating flow directly driven by the cylinder, the two main corner cells visualized at A = 1, develop with increasing A and then coalesce, to finally merge and give rise to a single central cell. This central cell develops in its turn, tending finally to the unbounded channel reference cell, after passing through a maximum length however. Owing to the very high precision of the calculations, many details of the flow development have been clearly shown, in particular the periodicity, with increasing A, of all the different phases, progressively inducing a succession of cells. The prediction that the angle of separation of the fluid boundaries of the cells tends towards the theoretical limit of 58.61° when the aspect ratio becomes large is also confirmed.

Stokes flow in a rectangular cavity by rotlet forcing

“…By plotting the stream function for several values of the aspect ratio of the cavity, a multicellular flow structure becomes visible for narrow cavity geometries. The existence of similar structures was already reported by Hellou and Coutanceau 21 and Jana, Metcalfe, and Ottino. 22 Hellou and Coutanceau 21 describe a configuration that is similar to the rotlet configuration in that the agitation of the fluid is executed by an internal forcing mechanism.…”

“…The existence of similar structures was already reported by Hellou and Coutanceau 21 and Jana, Metcalfe, and Ottino. 22 Hellou and Coutanceau 21 describe a configuration that is similar to the rotlet configuration in that the agitation of the fluid is executed by an internal forcing mechanism. There, a rotating circular cylinder is set in the center of the cavity and is responsible for an array of single-core cells which number is determined by the aspect ratio A = b / a ജ 1, with 2a the width and 2b the length of the cavity.…”

“…In their numerical study, the cylinders were situated on a line perpendicular to the channel boundaries. A similar numerical study of Stokes flows and associated mixing properties induced by rotating cylinders in square and rectangular cavities might provide an interesting connection with the work by Hellou and Coutanceau 21 and the present study.…”

Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders

“…The series-expansion method for solving both the biharmonic and harmonic equations are studied in [14] where the velocity field for two-dimensional flow in a batch stirring device is determined following the work of Price et al [15]. Hellou and Coutanceau [16] devised a hybrid polar/Cartesian system to investigate the cellular flow induced by a rotating cylinder in a rectangular channel. In all of the last three articles the domain is bounded.…”

Flow Patterns in a Two-Roll Mill

“…It is perhaps not too surprising that considerable attention has been paid to generalisations of this basic viscometer that are also amenable to small scale experiments. For example, the classical Taylor-Couette arrangement has been extended by considering elliptic containers ( [4]), single cylinders in a rectangular or square domain ( [5], [6], [7]) and eccentric circular cylinders ( [8]). However, in all of these studies, the experimental convenience of the original Couette experiment is preserved largely at the expense of analytical simplicity.…”