Shear and extensional rheology of particle-laden viscoelastic suspensions
Keywords: Chemical Engineering.

Recent discoveries have allowed a mathematical and computational foundation for understanding the rheology of particles suspended in viscoelastic fluids. We employ these new tools to understand the shear and the extensional rheology of such suspensions. In the first project, we study the time dependent evolution and steady values of the bulk shear stress in non-Brownian rigid particle suspensions during start-up of shear flow via experiments and nu- merical simulations. We compute the per-particle "extra" stress contribution to suspensions due to the interaction of the particles with the elastic fluid as well as the particle-particle hydrodynamic interactions. Previously, non-colloidal suspensions in viscoelastic fluids have been studied in steady shear flows mostly through experiments for relatively concentrated suspensions but there are very few computational simulation studies that may shed light into these experiments. We are not aware of any theoretical or 3D direct numerical studies in the literature that compute the viscometric functions of viscoelastic suspensions in the start-up of shear flow. We are also not aware of any ex- perimental data regarding rigid particle suspensions in polymeric fluid undergoing start-up of shear flow appearing in any previous work. Thus, there is a great opportunity to use high performance computing to study the evolution of stress in viscoelastic suspensions starting from rest and compare with transient shear experiments. First, we compute the viscometric functions (viscosity and first normal stress difference coefficient) of dilute suspensions as a function of shear strain for a wide range of Weissenberg numbers. We show that the "extra" per-particle stress contribution can be decomposed into two components: 1) the direct contribution from the rigid particles as they resist deformation in the flow, known as the stresslet; and 2) the contribution from the polymeric fluid as it deforms around the particles, leading to extra stresses in the fluid phase, known as the particle-induced fluid stress (PIFS). We validate our "single particle" numerical simulations in the weak elastic regime by comparing with small Weissenberg theory that we develop for the time-dependent evolution of average stress in a dilute particle suspension. We perform numerical simulations using both the Oldroyd-B equation and the Giesekus equation. We find that the per-particle viscosity and the primary normal stress coefficient evolve monotonically to steady state with strain at all Weissenberg number studied. The steady viscosity values show shear-thickening with Weissenberg number but the steady primary normal coefficient values are non-monotonic with Weissenberg number. We also study non-dilute suspensions in Boger fluids via numerical simulations to elucidate the effect of particle-particle hydrodynamic interactions on the stress contributions. We use an imple- mentation based on the class of Immersed Boundary (IB) methods to simulate multiple moving particles in computational domain. These simulations include fully resolved particle-scale hydrody- namics and fluid stresses. The IB method has a main disadvantage, the loss of resolution near the particle boundaries due to interpolation of information between the Lagrangian and Eulerian meshes - consequently the solid-fluid interface is not sharp. Therefore we develop a new method to compute the stresslet and PIFS that doesn't require finding the particle surface. We find the time-dependent evolution for different particle volume fraction suspensions φ in IB simulations is qualitatively similar to the evolution for "single particle" dilute suspensions. However, the per-particle viscosity values increase with particle volume fraction and the primary normal stress coefficient is independent of particle volume fraction in suspension. We also perform experiments of different particle volume fraction suspensions in a Boger fluid that include small amplitude oscillatory shear (SAOS), stress relaxation, steady shear and transient shear to characterize Boger fluid and suspensions. We compare single particle and multiple particle IB simulations with experiments and obtain excellent quantitative agreement. The second project is a study of the time dependent uniaxial extensional rheology of viscoelastic suspensions. We first calculate the renormalized particle contribution to the extensional viscosity in such a suspension in the dilute particle limit over a wide range of extensional Weissenberg number and Hencky strain. The models we use for the suspending fluids are the simplest dumbbell models — the Oldroyd-B, FENE-P, and Giesekus models — such that our results are general for polymer solutions which exhibit strong strain hardening at values of the Weissenberg number above those which engender the coil-stretch transition, Wi> = 0.5. We demonstrate that the effect of particles on the "extra elongational viscosity" relative to the fluid is nonmonotonic in strain (increasing for small strain and then decreasing for large strain). Thus at a fixed strain, the particle "extra viscosity" relative to the fluid may increase or decrease with Wi. We demonstrate that this interesting behavior is due to the interplay between the two contributions of the particle-induced fluid stress and the stresslet to the extra viscosity. The contribution of the particle-induced fluid stress to the suspension viscosity increases at small strain but plateaus and then decreases at higher values of the strain. On the other hand, the stresslet contribution to the viscosity relative to the fluid decreases monotonically with strain. We find that there are regions of large polymer stretch relative to the fluid, along the principal axis of extension near the particle surface. However, at larger strains we also observe the presence of regions around the particle where the polymer is significantly less stretched compared to the polymer stretch in the fluid alone. We believe the non-monotonicity in ratio of particle viscosity to fluid viscosity is universal for spherical particles in dilute polymer solutions subject to (uniaxial) extensional flow. We also perform experiments to measure extensional rheology of Boger fluid and different particle volume fraction suspensions in Boger fluid using commercially available filament stretching rheometer "Versatile Accurate Deformation Extensional Rheometer" (VADER 1000)