# Large-amplitude oscillatory shear: comparing parallel-disk with cone-plate flow

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## Abstract

We compare the ratio of the amplitudes of the third to the first harmonic *of the torque*, Open image in new window, measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics *of the shear stress*, |*τ*_{3}|/|*τ*_{1}|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a *correction factor* with which Open image in new window must be multiplied, to get the quantity |*τ*_{3}|/|*τ*_{1}|, where |*τ*_{3}|/|*τ*_{1}| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between Open image in new window and *τ*_{3}/*τ*_{1} using the simplest continuum model relevant to large-amplitude oscillatory shear flow: the single relaxation time co-rotational Maxwell model. We focus on the region where the harmonic amplitudes and thus, their ratios, can be fully described with power laws. This gives the expression for Open image in new window, by integrating the explicit analytical solution for the shear stress. In the power law region, we find that, for low Weissenberg numbers, for the third harmonics Open image in new window, and for the fifth harmonics, Open image in new window. We verify these results experimentally. In other words, the heterogeneous flow field of the parallel-disk geometry significantly attenuates the higher harmonics, when compared with the homogeneous, sliding-plate flow. This is because only the outermost part of the sample is subject to the high shear rate amplitude. Furthermore, our expression for the torque in large-amplitude oscillatory parallel-disk flow is also useful for the simplest design of viscous torsional dampers, that is, those incorporating a viscoelastic liquid between two disks.

### Keywords

Rheology Oscillatory shear Large-amplitude oscillatory shear## Notes

### Acknowledgments

AJG acknowledges the Ontario/Baden-Württemberg Exchange Program for the Faculty Research Exchange 2014–2015 award. AJG is also indebted to Professor Manfred Wilhelm of the Institut fuer Technische und Polymerchemie of the Karlsruhe Institute of Technology in Germany for hosting his Visiting Professorship during the summer of 2014. AJG is indebted to the Faculty of Applied Science and Engineering of Queen’s University at Kingston, for their support through a Research Initiation Grant (RIG). This research was undertaken, in part, thanks to funding from the Canada Research Chairs program of the Government of Canada of the Tier 1 Canada Research Chair in Rheology.

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