Large-Amplitude Oscillatory Shear: Comparing Parallel-Disk with Cone-Plate Flow
Giacomin, A. Jeffrey
Gilbert, Peter H.
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We compare the ratio of the amplitudes of the third to the first harmonic of the torque, |T3|/|T1|, measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics of the shear stress, |T3|/|T1|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a correction factor with which |T3|/|T1| must be multiplied, to get the quantity |τ3|/|τ1|, where |τ3|/|τ1| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between |T3|/|T1| and τ3/τ1 using the simplest continuum model relevant to large-amplitude oscillatory shear flow: the single relaxation time corotational 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 |T3|/|T1|, 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 |T3|/|T1| = 2/3|τ3|/|τ1 , and for the fifth harmonics, |T5|/T1| = 1/2|τ5|/|τ1|. 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. Further, 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.