Numerical investigation of dilute suspensions of rigid rods in power-law fluids

Highlights

• The viscosity of suspensions of rigid rods in a power-law fluid is computed.

• The rheological coefficients in the TIF equation are determined.

• Batchelors prediction for the largest coefficient underestimates the numerical result.

• Models for rods in power-law fluid strongly overpredict the orientation dependence.

• The particle velocity is the same as in a Newtonian fluid for large aspect ratios.

Abstract

Polymer melts filled with rod-like particles like glass and carbon fibers have high practical importance. Here we numerically investigate the properties of power-law fluids filled with rigid rods of different aspect ratios. For that we compute the rheological coefficients of the transversely isotropic fluid (TIF) equation and compare our results with analytical models, specifically with the model of Souloumiac and Vincent (1998). Here we found, that the additional orientation dependence predicted by the model for the nonlinear regime is too strong and overpredicts our numerical results. The rheological coefficient A in the TIF equation depends on the thinning exponent of the power-law model and decreases strongly with increasing nonlinearity. In the Newtonian case we found that Batchelor (1970, 1971) considerably underpredicts our numerical results. Comparing our current data with previous results for spheroids we found, that there is no similarity between the rheological coefficients for these particles at large aspect ratios. We further analyze the angular and translational velocities of the particles. We found that there are negligible differences in the angular velocities between the Newtonian and power-law matrix fluids, especially for large aspect ratios. The translational velocities are exactly the same for the Newtonian and power-law fluids.

Introduction

Rod-like particles are often added to polymers in form of short glass or carbon fibers to enhance the mechanical properties of the product. The addition of the fibers inevitably changes the properties of the melt to be processed. As such polymers are often processed in injection molding or extrusion, they are subjected to high shear rates and hence the polymer is in its nonlinear regime, i.e. shear thinning. The polymer-solid particle mixtures in the nonlinear regime were studied theoretically mostly for the spherical or spheroidal particle shapes. For example, the shear thinning behavior of spherical particle suspensions has been studied analytically in [4], [5], [6]. The studies [4], [5] use analytical approximations of the average fields to estimate the overall properties for highly shear thinning fluids. Datt and Elfring [6] consider suspensions of spherical particles in an asymptotically weak shear thinning Carreau fluid. The first numerical simulations of spherical particle suspensions with Carreau matrix fluids were made by us in 2015 [7], when we proposed a modification of Carreau equation for filled systems.

Spheroidal particles in non-Newtonian fluids very recently gained more attention. Most studies focus on the dynamics of spheroids. For example in works of D’Avino et al. [8] and Wang et al. [9] the rotation of spheroids in viscoelastic fluids in confined geometries is numerically investigated. Abtahi and Elfring [10] give analytical solutions of Jeffery orbits for spheroids in asymptotically weak shear thinning fluids. All those studies do not compute overall mechanical properties. In our previous paper [11] we simulated different flows of a Carreau fluid around spheroidal particles and used numerical homogenization to obtain the intrinsic viscosity of the suspension as function of applied rate of deformation, thinning exponent and aspect ratio. We were able to show that the dilute suspension of spheroidal particles can be described by the TIF equation not only in the Newtonian regime but also in strongly shear-thinning regime. This is explained by the fact, that in non-Newtonian regime the matrix fluid is isotropic and the particles have a rotational symmetry, so the suspension is transversely isotropic at any shear rate. By approaching aspect ratios of about 20, we confronted some technical problems with meshing due to acute shape at the ends of the spheroids. Also, it should be noticed that at such aspect ratios the shape of a spheroid is far away from the actual shapes of rod-like particles (glass/carbon fibers) used in polymer processing. Therefore, we realized that to describe the particle suspensions at even higher aspect ratios, real rod objects with constant cross-section should be investigated.

Interestingly, despite their practical importance the literature on rod-like particles in non-Newtonian fluids is rather scarce. One of the first publications on the topic is from 1976 by Goddard [12]. Here the author considered aligned particles in an elongational flow of a power-law fluid. Quite some time later Souloumiac and Vincent [1] in 1998 and also Gibson and Toll [13] in 1999 reconsidered the problem of rods in a power-law fluid. While the authors of the two publications used different methods, their finial results have a very similar form. Recently also clusters of rods and deformable rods have been considered [14], [15] and inertia effects on the orientation of rods have been examined by Scheuer et al. [16]. Férec et al. [17] studied also influence of shear-thinning on the orientation of rods and rods in Bingham [18] and second-order fluids [19]. Even in the case of dilute suspensions of rods in a Newtonian fluid we are not aware of numerical calculations that test Batchelor’s seminal calculations [2], [3] for the effective stress tensor. As we will show in the present study his calculation underestimates the largest coefficient of the TIF equation by up to approximately 20%, due to certain approximations.

In this work we consider rigid rods in a power-law fluid. Our aim is to use numerical simulations to extract all rheological coefficients of the TIF equation and check whether the model of Souloumiac and Vincent [1] is adequate. This model has been the basis of more recent models of rod suspensions in shear-thinning fluids [17]. In the next three sections we shortly introduce the TIF equation and the numerical homogenization procedure used. For a more elaborate explanation of the simulations we refer the reader to our previous paper [11]. After this, in Section 5 we introduce results from literature, with which we are comparing our numerical results in Section 6. In Section 7 we compare the results for rods to our previously obtained results for spheroids [11].