Stress and strain amplification in a dilute suspension of spherical particles based on a Bird–Carreau model Jan Domurath a,b,⇑ , Marina Saphiannikova a , Julien Férec b , Gilles Ausias b , Gert Heinrich a,c a Leibniz-Institut für Polymerforschung Dresden e. V., Hohe Straße 6, 01069 Dresden, Germany b LIMATB, Université de Bretagne-Sud, BP 92116, 56321 Lorient, France c Technische Universität Dresden, Institut für Werkstoffwissenschaft, 01069 Dresden, Germany article info Article history: Received 13 November 2014 Received in revised form 21 February 2015 Accepted 8 April 2015 Available online 16 April 2015 Keywords: Polymer melt Rigid filler Constitutive equations Elongational flow abstract A numerical study of a dilute suspension based on a non-Newtonian matrix fluid and rigid spherical par- ticles was performed. In particular, an elongational flow of a Bird–Carreau fluid around a sphere was sim- ulated and numerical homogenization has been used to obtain the effective viscosity of the dilute suspension g hom for different applied rates of deformation and different thinning exponents. In the Newtonian regime the well-known Einstein result for the viscosity of dilute suspension is obtained: g hom ¼ð1 þ½guÞg with the intrinsic viscosity ½g¼ 2:5. Here u is the volume fraction of particles and g is the viscosity of the matrix fluid. However in the transition region from Newtonian to non- Newtonian behavior lower values of the intrinsic viscosity ½g are obtained, which depend on both the applied rate of deformation and the thinning exponent. In the power-law regime of the Bird–Carreau model, i.e. at high deformation rates, it is found that the intrinsic viscosity ½g depends only on the thin- ning exponent. Utilizing the simulation results a modification of the Bird–Carreau model for dilute sus- pensions with a non-Newtonian matrix fluid is proposed. Ó 2015 Elsevier B.V. All rights reserved. 1. Introduction In many applications polymers are filled with particles to improve usage properties of a final product. Those properties include: flame retardancy, UV resistance and mechanical proper- ties (like Young’s modulus). Yet often the filled polymer is pro- cessed in its molten state, thus the flow properties of the polymer–particle mixture are of great interest to the processing of the polymer and the final properties of the product. Filled poly- mers are very complex systems and have been a research topic for many decades. The complexity in describing those systems stems from a variety of interactions that particles can have with the sur- rounding polymer and with each other. In filled polymer systems it is, for example, observed that the particles influence the dynamics of the polymer chains close to their surface [1,2]. It is also not uncommon that filler particles build fractal agglomerates [1,3]. Since the seminal work of Einstein in 1906 [4,5] it is known that the viscosity of a liquid increases due to the presence of a small amount of spherical particles as: g hom ¼ð1 þ 2:5uÞg ð1Þ with g hom the viscosity of the suspension, g the viscosity of the matrix fluid and u the volume fraction of particles. Equation (1) holds in experiments up to approximately 3% volume fraction of particles [6,7]. Einstein’s work was later extended by Batchelor and Green [8] to volume fractions of about 10%. For a Newtonian fluid filled with rigid spherical particles in a uniaxial extensional flow their result reads: g hom ¼ð1 þ 2:5u þ 7:6u 2 Þg: ð2Þ All the aforementioned equations are derived for a Newtonian matrix. However, polymers are, in general, viscoelastic and often show non-linear (e.g. shear-thinning) behavior. Some authors have tried to apply the relations computed for a Newtonian matrix, e.g. (1) and (2), to non-Newtonian fluids. For that purpose one gener- ally defines the so called hydrodynamic amplification factor: X ¼ g hom g ; ð3Þ which is then used to amplify the resulting stress. Such a stress amplification approach has been proposed by Leonov [1]. Another approach – referred to as strain amplification approach – can be found in the work of Sarvestani and Jabbari [2], where the authors multiply the strain with the amplification factor X. In a recent paper [9] we proposed a different approach, named the stress and strain http://dx.doi.org/10.1016/j.jnnfm.2015.04.002 0377-0257/Ó 2015 Elsevier B.V. All rights reserved. ⇑ Corresponding author at: Leibniz-Institut für Polymerforschung Dresden e. V., Hohe Straße 6, 01069 Dresden, Germany. E-mail address: domurath@ipfdd.de (J. Domurath). Journal of Non-Newtonian Fluid Mechanics 221 (2015) 95–102 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm