Rheology of semi-dilute suspensions with a viscoelastic matrix

This work explores the rheology of suspensions of spheres in a viscoelastic matrix. The volume fraction varies from zero to 0.2. Steady shearing, small strain sinusoidal shearing and uniaxial elongation are considered. We conclude that the matrix response in shear and elongation is fairly well described by a single-mode Oldroyd-B model, but the small-strain storage modulus (G’) response is less well represented by such a model. The use of a two-mode Oldroyd-B model gives significant improvement. For the suspensions, the viscometric functions η, N₁ and N₂ are given for volume fractions of 5, 10 and 20%, plus the oscillatory responses G’, G” at 1% strain amplitude. The uniaxial elongation data show a very large increase in flow resistance relative to the matrix; for the same applied force, the rate of elongation decreases from about 17.5 s⁻¹ for the matrix to about 3 s⁻¹ for the suspensions. It appears that this large increase in resistance is due to areas of intense extension attached to two adjacent spheres, as has been demonstrated numerically. It is shown that a single-mode Oldroyd-B model cannot describe suspension behaviour. A two-mode Oldroyd-B model can capture the macroscopic behaviour of the suspensions but only if an initial Hencky strain of order 4 is present in the extending suspension filaments. A two-mode model of the matrix fluid also allows one to understand the suspension response from the microscopic view.