Common to liquids, solids and substances in between the former two is that if a stress is applied to them, they will strain. Stress may be visualized by placing a small amount of fluid between two parallel plates. When one plate slides over the other, forces act on the fluid dependent upon the rate of the plate movement. This causes a shear stress on the liquid. Recall laminar flow of fluids through a tubular vessel. Strain is the response to the stress. If solids are elastic, they deform and return to their original shape. Since fluids are not elastic and, hence, viscous, their deformation is irreversible.

Introduction to rheology

Systems made by particles in viscoelastic liquids are ubiquitous in a variety of industrial and biological applications. Much work has been done in the last half-century in understanding the effect of non-Newtonian properties on the dynamics of the suspended particles. Theoretical predictions, experimental observations and numerical simulations highlighted peculiar phenomena induced by fluid elasticity that dramatically affect the particle motion and patterning. In this review, the existing literature on the dynamics of non-Brownian particles in viscoelastic fluids is discussed. The main part is focused on the dynamics of rigid particles passively transported in flowing viscoelastic liquids. The available results are classified by increasing level of complexity in terms of hydrodynamic interactions (single-particle problems, binary interactions, multi-body systems) and according to the flow field. Recent results on soft and active viscoelastic suspensions are also discussed.

Particle dynamics in viscoelastic liquids

Particles suspended in viscoelastic media are known to develop string-like structures when being sheared at sufficiently high shear rates. The occurrence of these strings has been attributed to normal stress effects, and it has been suggested that a critical Weissenberg number of 10 governs their appearance. This assumption is tested here, using suspensions of monodisperse polystyrene spheres. The level of viscosity and elasticity, as well as the degree of shear thinning of the suspending fluid, have been varied over a wide range of values. The critical Weissenberg number for the onset of string formation, as detected by microscopy, was not constant but ranged from 0.5 to 16.5 in contrast with literature results. Suspensions in highly elastic Boger fluids did not display any alignment at all, even at Weissenberg numbers as high as 260. Small angle light scattering (SALS) was used to calculate an alignment factor. It provides an objective and quantitative measure for the degree of alignment of the particles. It also makes it possible, in some cases, to determine precisely the interparticle spacing in the strings. The kinetics of string formation can be studied as well, they seem to be strain-controlled. SALS experiments with different gap spacings have been used to demonstrate that the string formation occurs in the bulk of the liquid rather than at the walls of the cell.

Effect of the viscoelasticity of the suspending fluid on structure formation in suspensions

We investigate the link between particle interactions and induced flow patterns around two identical spheres sedimenting along their centerline in a polymeric fluid. The fluid is strongly shear thinning and, in agreement with previous results, the spheres are observed to chain even at large initial separation distances. The wake of a single particle displays an upward motion of fluid, i.e., a “negative wake” that is commonly observed in fluids with low extensional viscosities. We show that the features of this negative wake vary only weakly with the Deborah number. In the two-sphere case, the pattern of the induced flow depends on the sphere separation distance. The change in the flow pattern does not, however, induce any significant qualitative change in the sphere interactions. Upstream of the leading sphere and downstream of the trailing one along the sedimentation axis, the variations of the fluid velocity are well described by a single master curve for different values of the sphere separation distance. The existence of such a curve indicates that non-Newtonian effects near each particle are dominated by local conditions near the sphere surfaces, and are only weakly influenced by the presence of a second sphere.

Axisymmetric two-sphere sedimentation in a shear thinning viscoelastic fluid : Particle interactions and induced fluid velocity fields

The mechanical behavior of non-Newtonian fluids can be modeled by several constitutive differential equations. The Oldroyd model is viewed as one of the successful models for describing the response of a subclass of polymeric liquids, in particular the non-Newtonian behavior exhibited by these fluids. In this paper, we are concerned with the study of the unsteady flows of an incompressible viscoelastic fluid of an Oldroyd-B type in a blood vessel acting on a Brownian force. First we derive the orientation stress tensor considering Hookean dumbbells on Brownian configuration fields. Then we reformulate the three-dimensional Oldroyd-B model with the total stress tensor which consists of the isotropic pressure stress tensor, the shear stress tensor, and the orientation stress tensor. Finally we present the numerical simulations of the model and analyze the effect of the orientation stress tensor in the vessel.

Blood flow of an Oldroyd-B fluid in a blood vessel incorporating a Brownian stress

In this research, experimental studies have been performed on the hydrodynamic interaction between two spheres by using particle image velocimetry and measuring the force between the spheres. To approach the system as a resistance problem, a servo-driving system was set-up by assembling a microstepping motor, a ball screw and a linear motion guide for the particle motion. Glycerin and a dilute solution of polyacrylamide in glycerin were used as Newtonian and non-Newtonian fluids, respectively. The polymer solution behaves like a Boger fluid when the concentration is 1000 ppm or less. The experimental results were compared with the asymptotic solution of Stokes equation. The result shows that fluid inertia and unsteadiness play important roles in the particle–particle interaction in the Newtonian fluid. This implies that the motion of two particles in suspension is not reversible even in the Newtonian fluid. In the non-Newtonian fluid, in addition to inertial effect, normal stress differences and viscoelasticity play important roles as expected. In dilute solutions weak shear thinning and the migration of polymer molecules in the inhomogeneous flow field also appear to affect the physics of the problem.

Studies on the axisymmetric sphere–sphere interaction problem in Newtonian and non-Newtonian fluids

In this research, experimental studies have been performed on the hydrodynamic interaction between a spherical particle and a plane wall by measuring the force between the particle and wall. To approach the system as a resistance problem, a servo-driving system was set-up by assembling a microstepping motor, a ball screw and a linear motion guide for the particle motion. Glycerin and dilute solution of polyacrylamide in glycerin were used as Newtonian and non-Newtonian fluids, respectively. The polymer solution behaves like a Boger fluid when the concentration is 1,000 ppm or less. The experimental results were compared with the asymptotic solution of Stokes equation. The result shows that fluid inertia plays an important role in the particle-wall interaction in Newtonian fluid. This implies that the motion of two particles in suspension is not reversible even in Newtonian fluid. In non-Newtonian fluid, normal stress difference and viscoelasticity play important roles as expected. In the dilute solution weak shear thinning and the migration of polymer molecules in the inhomogeneous flow field also affect the physics of the problem.

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Sang Wang Lee

Papers

Studies on the axisymmetric sphere–sphere interaction problem in Newtonian and non-Newtonian fluids

Experimental studies on the axisymmetric sphere-wall interaction in Newtonian and non-Newtonian fluids