Hydrodynamic Interaction for Rigid Dumbbell Suspensions in Steady Shear Flow
Piette, Jourdain H.
Giacomin, A. Jeffrey
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From kinetic molecular theory, we can attribute the rheological behaviors of polymeric liquids to macromolecular orientation. The simplest model to capture the orientation of macromolecules is the rigid dumbbell. For a suspension of rigid dumbbells, subject to any shear flow, for instance, we must first solve the diffusion equation for the orientation distribution function. From this distribution, we then calculate the first and second normal stress differences. To get reasonable results for the normal stress differences in steady shear flow, one must account for hydrodynamic interaction between the dumbbell beads. However, for the power series expansions for these normal stress differences, three series arise. The coefficients for two of these series, (ck,dk), are not known, not even approximately, beyond the second power of the shear rate. Analytical work on many viscoelastic material functions in shear flow must be checked for consistency, in their steady shear flow limits, against these normal stress differences power series expansions. For instance, for large-amplitude oscillatory shear flow (LAOS), we must recover the power series expansions in the limits of low frequency. In this work, for (ck,dk), we arrive at exact expressions for the first 18 coefficients.