On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(ϕDe) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where ϕ is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.