Nonlinear viscoelasticity of a dilute suspension of Brownian spheroids in oscillatory shear flow

Abstract

The nonlinear viscoelasticity of a dilute suspension of Brownian spheroids subject to an oscillatory shear deformation is calculated numerically. This is achieved by determining the suspension microstructure, parameterized via the orientation distribution function. Specifically, the long-time periodic orientation distribution function is obtained via a numerical solution to the Fokker–Planck equation by combining a finite-difference approximation in space with a Fourier series in time. From an ensemble average of the particle stresslet, weighted by the orientation distribution function, the entire stress tensor and relevant birefringence parameters, namely, the average orientation angle and linear dichroism, are calculated; this is done over a range of the Weissenberg number (Wi) and the Deborah number (De), or dimensionless strain-rate amplitude and oscillation frequency, respectively. Detailed calculations are presented for prolate spheroids of aspect ratio r=20; however, our methodology is general and can be applied to spheroids of arbitrary aspect ratio. We provide results in four viscoelastic regimes: linear viscoelastic (Wi≪1), quasilinear viscoelastic (Wi>1 and Wi/De≪1), quasisteady viscoelastic (De→0), and finally, the nonlinear viscoelastic regime (Wi ≳ 1 and Wi/De ≳ 1), which is our main emphasis. In this last regime, where the nonlinear and unsteady viscoelasticity of the material is probed, multiple overshoots are observed in the shear stress and first normal stress difference. The mechanistic origin of these overshoots can be understood from the periodic orientation dynamics (i.e., Jeffery orbits) of a particle under steady shear in the absence of Brownian rotation (Wi→∞). This is achieved by simultaneously analyzing the microstructure, shear stress, first normal stress difference, and birefringence parameters specifically at Wi=20 and De=1. For these values of Wi,De, and r, the period of a Jeffery orbit is comparable to the period of an oscillation cycle, allowing sufficient time for a single Jeffery orbit (and subsequent overshoot) to occur during an oscillation cycle. We contrast this behavior to recent work by Khair [J. Fluid. Mech. 791, R5 (2016)] on nearly spherical particles, for which many more overshoots are observed during an oscillation cycle, due to the shorter period of the Jeffery orbit. We briefly provide results for oblate spheroids of aspect ratio r=0.05 and compare to the results for r=20. Finally, we comment on the relevance of the present micro-mechanical analysis to the nonlinear viscoelasticity of other complex fluid materials.