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Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow
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16. A. J. Giacomin, R. B. Bird, L. M. Johnson, and A. W. Mix, ‘ Corrigenda: “Large-amplitude oscillatory shear flow from the corotational Maxwell model,” [J. Non-Newtonian Fluid Mech. 166, 1081–1099 (2011)],’
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18. R. B. Bird, O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of Polymeric Liquids, 1st ed. ( Wiley, New York, 1977), Vol. 2. Erratum: In Problem 11.C.1 d., “and ” should be “through ;” On p. N-2, under entry “Length of a rigid dumbbell connector” should be “Bead center to center length of a rigid dumbbell.”
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27. R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd ed. ( John Wiley & Sons, Inc., New York, 1987), Vol. 2. Erratum: On p. 409 of the first printing, the in the denominator should be ; In Table 16.4-1, under entry “length of rod” should be “bead center to center length of a rigid dumbbell.” In the Figure 14.1-2 caption, “Multibead rods of length ” should be “Multibead rods of length .”
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37. R. B. Bird, A. J. Giacomin, A. M. Schmalzer, and C. Aumnate, “ Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response,” J. Chem. Phys. 140, 074904 (2014); Corrigenda: In Eq. (91), should be . In caption to Fig. 3, “ ” should be “ ” and “ ” should be “ .”
38. A. M. Schmalzer, R. B. Bird, and A. J. Giacomin, “ Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions,” PRG Report No. 002, QU-CHEE-PRG-TR–2014-2, Polymers Research Group, Chemical Engineering Department, Queen's University, Kingston, CANADA, 2014.
39. A. M. Schmalzer, R. B. Bird, and A. J. Giacomin, “ Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions,” J. Non-Newtonian Fluid Mech. (2014); Errata: Above Eqs. (14) and (25), “significant figures” should be “16 significant figures.”
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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
is zero and the Weissenberg number
is above unity), (ii) nonlinear viscoelasticity (where both
exceed unity), and (iii) linear viscoelasticity (where
exceeds unity and where
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.
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