Fourier Decomposition of Polymer Orientation in Large-Amplitude Oscillatory Shear Flow
Giacomin, A. Jeffrey
Gilbert, Peter H.
Schmalzer, Andrew M.
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In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number λω is zero and the Weissenberg number λγ0 is above unity), (ii) nonlinear viscoelasticity (where both λω and λγ0 exceed unity), and (iii) linear viscoelasticity (where λω exceeds unity and where λγ0 approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise, tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.