Orientation in Large-Amplitude Oscillatory Shear
Schmalzer, Andrew M.
Giacomin, A. Jeffrey
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We examine the simplest relevant molecular model for large-amplitude oscillatory shear flow of a polymeric liquid: the dilute suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the orientation distribution function, and specifically for test conditions of frequency and shear rate amplitude that generate higher harmonics in the shear stress and normal stress difference responses. Our analysis employs the general method of Bird and Armstrong (1972) for analyzing the orientation of the rigid dumbbells in suspension in any unsteady shear flow. We have solved the diffusion equation analytically to explore the orientations of the molecules induced by the oscillatory shear flow. We see that the orientation distribution function is neither even nor odd. We find zeroth, first, second, third and fourth harmonics of the orientation distribution function, and we have derived explicit analytical expressions for these. We use our analytical solution to examine the detailed shape of the orientation distribution function. We provide a clear visualization of the orientation distribution function in large-amplitude oscillatory shear flow in spherical coordinates all the way around one full alternant cycle (at ωt = 0, π4 , π2,…,2π ). Our analysis supplements our previous results for the shear stress [Bird et al., JCP, 140, 074904 (2014)] and normal stress differences [Schmalzer et al., PRG Report No. 002, Queen’s University (April 2014)] for a suspension of rigid dumbbells in large-amplitude oscillatory shear flow. This exploration includes the Newtonian, the linear viscoelastic and nonlinear viscoelastic regimes. We find the orientation distribution for Newtonian behavior to be nearly isotropic (spherical), for the linearly viscoelastic behavior, only slightly anisotropic (only a slight ellipsoidal departure from spherical), and for the nonlinear viscoelastic regime, we find the orientation distribution function for the dumbbells to be highly anisotropic (and even peanut or dumbbell shaped, which we call lemniscoidal).