An effective medium approach for the elongational viscosity of non-colloidal and non-Brownian hard-sphere suspensions

An effective-medium fluid mechanics model based on the original idea first presented by Brinkman ["A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,"Appl. Sci. Res. 1, 27-34 (1949)] for the viscous force exerted by a flowing fluid on a dense swarm of fixed spherical particles is utilized for the prediction of the elongational viscosity of a non-colloidal, non-Brownian hard-sphere suspension in an incompressible Newtonian matrix fluid. The same model was explored by Housiadas and Tanner ["A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids,"Rheol. Acta 53, 831-841 (2014)] for the derivation of an analytical formula for the bulk shear viscosity of the suspension as a function of the volume fraction of the solid phase, a formula which is in very good agreement with widely used semi-empirical relationships and available experimental data from the literature. In the present paper, it is assumed that a spherical particle is subject, in an average sense, to a far-field uniform uniaxial elongational flow and a suitable pressure gradient. Under steady, isothermal, creeping conditions, and imposing no-slip and no-penetration conditions at the surface of a particle in a stagnation point of the fluid and the far-field velocity and pressure profiles, the solution of the three-dimensional Brinkman equations is found analytically. The solution shows a faster decay of the velocity disturbances around a reference particle than the single-particle case. A volume average of the total stress tensor gives an analytical formula for the bulk elongational viscosity of the complex system as a function of the particle concentration. A significant increase of the elongation viscosity with increasing the particle concentration is predicted. The increase is larger than the corresponding increase of the shear viscosity, in qualitative accordance with the theoretical formula of Batchelor and Green ["The determination of the bulk stress in a suspension of spherical particles to order c2" J. Fluid Mech. 56(3), 401-427 (1972)]. The new formula reduces to Einstein's expression in the infinite dilution limit and agrees well with other theoretical formulas in the semi-dilute regime. Moreover, the agreement of the new formula with recently developed semi-empirical formulas over the whole concentration regime is remarkable. Finally, the model predictions perform very well, and better than other formulas, when compared with a few experimental data for extensional measurements of hard-particle suspensions from the literature. (C) 2015 AIP Publishing LLC.