Bulk rheology of dilute suspensions in viscoelastic liquids

A theoretical relation is derived for the bulk stress in dilute suspensions of neutrally buoyant, uniform size, spherical drops in a viscoelastic liquid medium. This is achieved by the classic volume-averaging procedure of Landau and Lifschitz which excludes Brownian motion. The disturbance velocity and pressure fields interior and exterior to a second-order fluid drop suspended in a simple shear flow of another second-order fluid were derived by Peery [9] for small Weissenberg number (We), omitting inertia. The results of the averaging procedure include terms up to orderWe ². The shear viscosity of a suspension of Newtonian droplets in a viscoelastic liquid is derived as $$eta _{susp} = eta _0 left[ {1 + frac{{5k + 2}}{{2(k + 1)}}varphi - frac{{psi _{10}^2 dot gamma ^2 }}{{eta _0^2 }}varphi f_1 (k, varepsilon _0 )} right],$$ whereη ₀, andω ₁₀ are the viscosity and primary normal stress coefficient of the medium,ε ₀ is a ratio typically between ?0.5 and ?0.86,k is the ratio of viscosities of disperse and continuous phases, and (dot gamma ) is the bulk rate of shear strain. This relation includes, in addition to the Taylor result, a shear-thinning factor (f ₁ > 0) which is associated with the elasticity of the medium. This explains observed trends in relative shear viscosity of suspensions with rigid particles reported by Highgate and Whorlow [6] and with drops reported by Han and King [8]. The expressions (atO (We ²)) for normal-stress coefficients do not include any strain rate dependence; the calculated values of primary normal-stress difference match values observed at very low strain rates.