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

We compare the ratio of the amplitudes of the third to the first harmonic of the torque, , measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics of the shear stress, |tau (3)|/|tau (1)|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a correction factor with which must be multiplied, to get the quantity |tau (3)|/|tau (1)|, where |tau (3)|/|tau (1)| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between and tau (3)/tau (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 , 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 , and for the fifth harmonics, . 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.