The rheology of non-dilute dispersions of highly deformable viscoelastic particles in Newtonian fluids

Abstract

We present a model for the rheological behaviour of non-dilute suspensions of initially spherical viscoelastic particles in viscous fluids under uniform Stokes flow conditions. The particles are assumed to be neutrally buoyant Kelvin-Voigt solids undergoing time-dependent finite deformations and exhibiting generalized neo-Hookean behaviour in their purely elastic limit. We investigate the effects of the shape dynamics and constitutive properties of the viscoelastic particles on the macroscopic rheological behaviour of the suspensions. The proposed model makes use of known homogenization estimates for composite material systems consisting of random distributions of aligned ellipsoidal particles with prescribed two-point correlation functions to generate corresponding estimates for the instantaneous (incremental) response of the suspensions, together with appropriate evolution laws for the relevant microstructural variables. To illustrate the essential features of the model, we consider two special cases: (i) extensional flow and (ii) simple shear flow. For each case, we provide the time-dependent response and, when available, the steady-state solution for the average particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspensions. The results exhibit shear thickening for extensional flows and shear thinning for simple shear flows, and it is found that the volume fraction and constitutive properties of the particles significantly influence the rheology of the suspensions under both types of flows. In particular, for extensional flows, suspensions of particles with finite extensibility constraints are always found to reach a steady state, while this is only the case at sufficiently low strain rates for suspensions of (less realistic) neo-Hookean particles, as originally reported by Roscoe (J.

Macroscopic rheological behavior of suspensions of soft solid particles in yield stress fluids

“…They considered applications of their method under shear flow [17] and extensional flow [18] conditions. Most recently, building on the work of Gao et al [16], Avazmohammadi and Ponte Castañeda [19] developed a homogenization model to estimate the time-dependent rheological behavior of non-dilute suspensions of soft viscoelastic spheres in a Newtonian fluid, again, under Stokes flow conditions. In this study, the authors formulated the homogenization problem for the suspensions in terms of the corresponding problem for a composite system whose local behavior is mathematically analogous to "thermoelastic" composites.…”

“…Based on this analogy, the authors made use of the Hashin-Shtrikman-Willis (HSW) homogenization theory [20,21] and provided homogenized estimates for the instantaneous response of the suspensions, together with appropriate evolution laws for the relevant microstructural variables. The distinct advantage in the exact solution of Gao et al [16] and, similarly, in the associated estimates of Avazmohammadi and Ponte Castañeda [19], is that they account for the simultaneous deformation and rotation of particles in the macroscopic time-dependent behavior of the suspensions of the particles, when subjected to uniform flows. The results of these works in the steady state limit were successfully compared with those provided in the earlier work of Roscoe [13].…”

“…Towards our goal, we make use of the variational approach of Ponte Castañeda [5] to replace the homogenization problem for finding the effective (instantaneous) behavior of suspensions of viscoelastic particles in yield stress fluids by a corresponding problem for a (fictitious) LCC, consisting of the same viscoelastic particles suspended in a Newtonian fluid with a suitably-chosen viscosity, denoted by η (1) . The latter problem is, in fact, the very same problem addressed in [19]. Therefore, we recall pertinent estimates for the LCC from [19], which together with optimality conditions for the unknown value of η (1) , can be used to obtain corresponding estimates for the instantaneous response of the suspensions of actual interest.…”

“…The latter problem is, in fact, the very same problem addressed in [19]. Therefore, we recall pertinent estimates for the LCC from [19], which together with optimality conditions for the unknown value of η (1) , can be used to obtain corresponding estimates for the instantaneous response of the suspensions of actual interest. Finally, these estimates, when combined with nonlinear equations for the evolution of the stress field in the particles, as well as for the shape and orientation of the particles, can be used to characterize the macroscopic, time-dependent rheological behavior of the suspensions in uniform flows.…”

“…where C = F T F is the right Cauchy-Green tensor. Avazmohammadi and Ponte Castañeda [19] presented a fairly general form of the function ψ(C), describing generalized neo-Hookean behavior, to characterize the elastic part of the isotropic KV particles. In this work, we make use of a special case of this form known as the Gent constitutive model [25].…”