Steady-shear rheological properties for suspensions of axisymmetric particles in second-order fluids

- Slender rod dynamics in a second-order fluid.
- Complete rheological constitutive equation for dilute and semidilute slender rod suspensions.
- Numerical solutions of the associated Fokker-Planck equation.
- Adaptation of the original Pipkin diagram to the case of rod suspensions in viscoelastic media.

Following Leal who gave the motion of a slender axisymmetric rod in a second-order fluid, we derived a complete rheological constitutive equation for dilute and semidilute slender rod suspensions in a viscoelastic solvent based on a cell model. Numerical solutions for the Fokker-Planck equation are obtained for simple shear flows at low and large Peclet numbers using a finite volume method, hence avoiding the need for closure approximations. The second normal stress difference coefficient of the solvent plays a non-negligible role in the particle contribution to the stress as well as on the rod orientation dynamics: a spread of the particle orientation in the flow-vorticity plane and an enhancement of the alignment along the vorticity direction are predicted when increasing the second normal stress difference coefficient. Brunn extended the Leal analysis to dumbbells and tri-dumbbells, for which both normal stress difference coefficients have to be considered. The original Pipkin diagram is finally modified to help guide the choice of relevant constitutive equations for particles in viscoelastic fluids.