The nature of dilute viscoelastic shear flow around freely-mobile particles

A generic fundamental question of viscoelastic fluid mechanics is concerned with why viscoelastic creeping flows differ from their Newtonian counterparts.; Since viscoelastic fluids differ from Newtonian fluids by the presence of polymer molecules, it is fitting both to pursue this question from the perspective of the polymer dynamics within viscoelastic flows and to restrict one's initial attention to dilute viscoelastic fluids, which are formulated with sparing amounts of polymer dissolved in a viscous solvent such that the dynamics of a polymer molecule are determined primarily by the flow of the surrounding solvent. In the limit of infinite dilution, the polymer dynamics are determined entirely by a Newtonian flow field. Departing from this limit, one may consider how such "infinitely-dilute" polymer dynamics modify the Newtonian flow, how the modification to the Newtonian flow modifies the infinitely-dilute polymer dynamics, and how such "self-influenced" polymer dynamics further modify the Newtonian flow. In this manner, before ultimately considering the thornier nature of truly interdependent polymer dynamics and flow, one may begin to understand dilute viscoelastic flow with a logical succession of cause-and-effect mechanisms based on knowledge of a Newtonian flow field.; In this study, we pursue this generic question with this general approach in the context of dilute viscoelastic shear flow around freely-mobile particles. It has been observed that particles suspended in viscoelastic fluids under shear flow aggregate and align in the direction of flow. No such aggregation occurs in Newtonian suspensions and one is therefore provoked to ask why it occurs in viscoelastic suspensions.; In this study, we use a FENE dumbbell model to describe the dynamics of polymer molecules in dilute solutions. For the limiting case of a linear or Hookean dumbbell, which is equivalent to single-mode Rouse model and Oldroyd-B constitutive equation, one may obtain a closed form solution for the dumbbell configuration tensor and the viscoelastic stress for steady planar and axisymmetric flows. This closed-form solution greatly facilitates a thorough analysis of dilute viscoelastic shear flow around a freely-mobile circle. And such a thorough analysis serves as point of reference from which to begin an investigation of more complicated viscoelastic shear flows, such as around two-freely mobile circles or a freely-mobile sphere and subject to sophisticated dumbbell dynamics including finite-extensibility, shape effects, and inter-molecular hydrodynamic interactions.