Internal viscosity dumbbell model with a Gaussian approximation

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Abstract
A noninertial Hookean dumbbell with internal viscosity (or internal friction) is studied in transient and steady shear flows by use of a Gaussian closure on the second moment equation of the configuration of the dumbbell. The model predicts shear thinning for the viscosity and first normal stress coefficients for all values of the relative internal viscosity parameter ε. A second Newtonian region is observed for the viscosity. Qualitative, but not quantitative, agreement is found with optically determined orientation angles of polymer coils in steady shear flows for dilute polymer solutions. The model greatly overestimates the amount of relative stretching of the polymer coil in steady shear flow. In startup flows, large, but finite, values of ε show shear stress overshoot at high shear rates, and oscillatory behavior at the highest shear rates studied. Transient negative values of the first normal stress difference are also predicted. The maximum in stress is attained at much lower values of strain than for the predictions at small ε. The oscillations are shown to be caused primarily by oscillations in the orientation of the polymer coil, rather than by oscillations in the size of the polymer coil. Instantaneous jumps in the shear stress at t=0 are observed in agreement with Manke and Williams. Cessation of shear flows shows a jump in stress in agreement with data on xanthan gum. The decay upon cessation is nonexponential, but does follow the Lodge–Meissner relation. The width of the polymer coil is predicted to go through a maximum during this decay. Also, the addition of internal viscosity to the dumbbell satisfactorally gives a positive asymptotic value for η’−η s (in‐phase complex viscosity minus solvent contribution) in small amplitude oscillatory shear flow.

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