The normal stress behaviour of suspensions with viscoelastic matrix fluids

 We investigate the variations in the shear stress and the first and second normal stress differences of suspensions formulated with viscoelastic fluids as the suspending medium. The test materials comprise two different silicone oils for the matrix fluids and glass spheres of two different mean diameters spanning a range of volume fractions between 5 and 25%. In agreement with previous investigations, the shear stress–shear rate functions of the viscoelastic suspensions were found to be of the same form as the viscometric functions of their matrix fluids, but progressively shifted along the shear rate axis to lower shear rates with increasing solid fraction. The normal stress differences in all of the suspensions examined can be conveniently represented as functions of the shear stress in the fluid. When plotted in this form, the first normal stress difference, as measured with a cone and plate rheometer, is positive in magnitude but strongly decreases with increasing solid fraction. The contributions of the first and the second normal stress differences are separated by using normal force measurements with parallel plate fixtures in conjunction with the cone-and-plate observations. In this way it is possible for the first time to quantify successfully the variations in the second normal stress difference of viscoelastic suspensions for solid fractions of up to 25 vol.%. In contrast to measurements of the first normal stress difference, the second normal stress difference is negative with a magnitude that increases with increasing solid content. The changes in the first and second normal stress differences are also strongly correlated to each other: The relative increase in the second normal stress difference is equal to the relative decrease of the first normal stress difference at the same solid fraction. The variations of the first as well as of the second normal stress difference are represented by power law functions of the shear stress with an unique power law exponent that is independent of the solid fraction. The well known edge effects that arise in cone-and-plate as well as parallel-plate rheometry and limit the accessible measuring range in highly viscoelastic materials to low shear rates could be partially suppressed by utilizing a custom- designed guard-ring arrangement. A procedure to correct the guard-ring influence on torque and normal force measurements is also presented. Received: 20 December 2000 Accepted: 7 May 2001     

Nonlinear viscoelastic properties of MR fluids under large-amplitude-oscillatory-shear

Nonlinear viscoelastic properties of the MR fluid, MRF-132LD, under large-amplitude oscillatory shear were investigated. This was accomplished by carrying out the experiments under the amplitude sweep mode and the frequency sweep mode, using a rheometer with parallel-plate geometry. Investigations under the influence of various magnetic field strength and temperatures were also conducted. MR fluids behave as nonlinear viscoelastic or viscoplastic materials when they are subjected to large-amplitude shear, where the storage modulus decreases rapidly with increasing strain amplitude. Hence, MR fluid behaviour ranges from predominantly elastic at small strain amplitudes to viscous at high strain amplitudes. Large-amplitude oscillatory shear measurements with frequency sweep mode reveal that the storage modulus is independent of oscillation frequency and approaches plateau values at low frequencies. With increasing frequency, the storage modulus shows a decreasing trend before increasing again. This trend may be explained by micro-structural variation. In addition, the storage modulus increases gradually with increasing field strength but it shows a slightly decreasing trend with temperature.     

Rhéologie des matériaux en cours de prise : Approche théorique

The rheological behavior of setting heterogeneous materials is studied from a theoretical approach by means of the homogenization technique of periodic medium. These materials considered as suspensions of gas bubbles at finite concentration in a viscoelastic matrix with low compressibility, present the macroscopic behavior of a compressible viscoelastic medium. The shear and volume macroscopic moduli are of the same order of magnitude and directly proportional to that of the fluid. The effective compressibility of the gas (out of thermal equilibrium) is added to these fluid contributions.     

Nonlinear viscoelastic response of asphalt binders: An experimental study of the relaxation of torque and normal force in torsion

Development of normal stress in the direction perpendicular to the plane of shear is an important feature of the nonlinear viscoelastic behavior of asphalt binders. Here, we study the significance of this phenomenon with the help of stress-relaxation experiments in torsion. We conducted these experiments using a dynamic shear rheometer on an unmodified binder and polymer modified binder, at different temperatures and aging conditions. The results not only illustrate the nonlinearity of the behavior but also show certain distinctive characteristics of the relaxation behavior (torque relax faster than normal force) and it is seen that new constitutive models are required to predict such behavior.     

Optimization of the vane geometry

The use of nonstandard geometries like the vane is essential to measure the rheological characteristics of complex fluids such as non-Newtonian fluids or particle dispersions. For this geometry which is of Couette type, there is no analytical simple model defining the relation between the shear stress and the torque or relating the angular velocity to the shear rate. This study consists on calibrating a nonstandard vane geometry using a finite volume method with the Ansys Fluent software. The influence of geometrical parameters and rheological characteristics of the complex fluids are considered. First, the Newtonian fluid flow in a rotative vane geometry was simulated and a parametric model is derived therefrom. The results show an excellent agreement between the calculated torque and the measured one. They provide the possibility to define equivalent dimensions by reference to a standard geometry with concentric cylinders where the relationships between shear stress (resp. shear rate) and the torque (resp. the angular rotation) are classical. Non-Newtonian fluid flows obeying a power law rheology with different indices were then simulated. The results of these numerical simulations are in very good agreement with the preceding Newtonian-based model in some ranges of indices. The absolute difference still under 5 % provided the index is below 0.45. Finally, this study provides a calibration protocol in order to use nonstandard vane geometries with various heights, gaps, and distance to the cup bottom for measuring the rheology of complex fluids like shear thinning fluids and concentrated suspensions.     

Spherical Couette flow of a viscoelastic fluid Part I: Experimental study of the inner sphere rotation

Flow of viscoelastic fluid contained between a stationary outer sphere and a rotating inner sphere is studied experimentally. In the present investigation, relatively low-concentration polyacrylamide-water solutions are used as viscoelastic fluid, and for the sake of comparison glycerin-water solutions are used as the Newtonian fluid. In experiments, measurements of the rotational torque and the velocity profile in the meridional plane of the spherical gap are made. Various transition phenomena of flow modes that are unique to viscoelastic fluids are investigated by flow visualization for a wide range of rotational Reynolds numbers. Experimental results revealed that a roll-cell-like structure (banded radial structure) caused by elastic instability is generated in the polar region, and that it propagates toward the equatorial region when rotation of the inner sphere is increased, resulting in the formation of two distinct regions: the elastic dominant region and the inertial dominant region. Flow modes are classified and the critical Reynolds numbers are obtained for different gap widths. A correlation is obtained for the torque data in the regime before the onset of instability.     

Effect of viscoelasticity on the rotation of a sphere in shear flow

When particles are dispersed in viscoelastic rather than Newtonian media, the hydrodynamics will be changed entailing differences in suspension rheology. The disturbance velocity profiles and stress distributions around the particle will depend on the viscoelastic material functions. Even in inertialess flows, changes in particle rotation and migration will occur. The problem of the rotation of a single spherical particle in simple shear flow in viscoelastic fluids was recently studied to understand the effects of changes in the rheological properties with both numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346] and experiments [Snijkers et al., J. Rheol. 53 (2009) 459–480]. In the simulations, different constitutive models were used to demonstrate the effects of different rheological behavior. In the experiments, fluids with different constitutive properties were chosen. In both studies a slowing down of the rotation speed of the particles was found, when compared to the Newtonian case, as elasticity increases. Surprisingly, the extent of the slowing down of the rotation rate did not depend strongly on the details of the fluid rheology, but primarily on the Weissenberg number defined as the ratio between the first normal stress difference and the shear stress.In the present work, a quantitative comparison between the experimental measurements and novel simulation results is made by considering more realistic constitutive equations as compared to the model fluids used in previous numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346]. A multimode Giesekus model with Newtonian solvent as constitutive equation is fitted to the experimentally obtained linear and nonlinear fluid properties and used to simulate the rotation of a torque-free sphere in a range of Weissenberg numbers similar to those in the experiments. A good agreement between the experimental and numerical results is obtained. The local torque and pressure distributions on the particle surface calculated by simulations are shown.     

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

We study an unsteady nonlinear fluid–structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action–reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain, in particular that contact between the viscoelastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, and of the existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.     

A new approach for calculating the true stress response from large amplitude oscillatory shear (LAOS) measurements using parallel plates

The parallel plates geometry is often deemed unsuitable for nonlinear viscoelasticity measurements because the strain field, and thus the nonlinear response, varies across the sample. Although cone–plate and Couette geometries are designed to circumvent this problem by ensuring a uniform strain field, it is not always easy to shape the material to the complex shapes that is required for these geometries. This has motivated the development of techniques to accurately determine the nonlinear stress response using the more convenient plate–plate geometry. Here, we introduce a new approach to obtain this true material response in large amplitude oscillatory shear (LAOS) experiments using the plate–plate geometry. By tracing the Fourier components of the torque response and their derivatives with respect to the maximum applied deformation, we accurately obtain the material’s true stress–strain response from parallel plate measurements. The approach does not require any assumptions about the material’s viscoelastic behavior. We test our approach experimentally on fibrin biopolymer gels, as well as numerically on a Giesekus model. We confirm in both cases that our approach captures the detailed shape of the true stress response in LAOS measurements. Moreover, we also show that our method is less sensitive to experimental noise present in the data than the previous standard method. Our approach for obtaining the true stress response from parallel plate measurements is directly applicable to measurements on a wide range of solid-like nonlinear materials, including biological networks, tissues, or hydrogels.     

Rheology of shear thickening suspensions and the effects of wall slip in torsional flow

The rheological characterisation of concentrated shear thickening materials suspensions is challenging, as complicated and occasionally discontinuous rheograms are produced. Wall slip is often apparent and when combined with a shear thickening fluid the usual means of calculating rim shear stress in torsional flow is inaccurate due to a more complex flow field. As the flow is no longer “controlled”, a rheological model must be assumed and the wall boundary conditions are redefined to allow for slip. A technique is described where, by examining the angular velocity response in very low torque experiments, it is possible to indirectly measure the wall slip velocity. The suspension is then tested at higher applied torques and different rheometer gaps. The results are integrated numerically to produce shear stress and shear rate values. This enables the measurement of true suspension bulk flow properties and wall slip velocity, with simple rheological models describing the observed complex rheograms.     

Compliance effects on the torsional flow of a viscoelastic fluid

The effects of transducer compliance on transient stress measurements in torsional flows of a viscoelastic fluid are investigated theoretically. The analysis is based on the torsional flow of an upper-convected Maxwell fluid between a rotating and ‘stationary’ disk, which is allowed to twist and displace axially as a result of the stresses exerted on the disk by the fluid. An approximate analytical solution to the governing equations is obtained using a standard perturbation method. Results of the analysis are used to examine how the fluid velocity is altered by the motion of the stationary disk and to gain insight on how transient stress measurements are affected by transducer compliance. The analysis shows that compliance effects increase with applied shear rate and that the effects of torsional and axial compliance are coupled in measurements of the shear stress and first normal stress difference.     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998     

Dynamic simulation of sedimentation of solid particles in an Oldroyd-B fluid

In this paper we present a two-dimensional numerical study of the viscoelastic effects on the sedimentation of particles in the presence of solid walls or another particle. The Navier-Stokes equations coupled with an Oldroyd-B model are solved using a finite-element method with the EVSS formalism, and the particles are moved according to their equations of motion. In a vertical channel filled with a viscoelastic fluid, a particle settling very close to one side wall experiences a repulsion from the wall; a particle farther away from the wall is attracted toward it. Thus a settling particle will approach an eccentric equilibrium position, which depends on the Reynolds and Deborah numbers. Two particles settling one on top of the other attract and form a doublet if their initial separation is not too large. Two particles settling side by side approach each other and the doublet also rotates till the line of centers is aligned with the direction of sedimentation. The particle-particle interactions are in qualitative agreement with experimental observations, while the wall repulsion has not been documented in experiments. The driving force for lateral migrations is shown to correlate with the pressure distribution on the particle's surface. As a rule, viscoelasticity affects the motion of particles by modifying the pressure distribution on their surface. The direct contribution of viscoelastic normal stresses to the force and torque is not important.     

Determination of thermal conductivity of liquids and polymers from batch mixer data

The paper proposes a new technique allowing the determination of thermal conductivity of liquids and polymer melts with unknown rheological behavior from batch mixer data; the mixing elements can have simple or complex geometries. The simple mixer is represented by an equivalent simple Couette with a cup and a bob having an effective hydrodynamic (or thermal) radius, and the true batch mixer is represented by two adjacent concentric cylinder viscometer. Using the universal internal effective radius, the thermal conductivity for four polymers was estimated from the batch mixer temperature?Crotor speed and torque data. A good agreement was found with the values measured by transient line source method and those of the literature. Melt activation energy was calculated from torque?Ctemperature batch mixer data, and the obtained values were found to be consistent with the values extracted from low-amplitude oscillatory shear experiments as well as with the available published data.     

A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: Application to thermally responsive gels

An elastomeric gel is a cross-linked polymer network swollen with a solvent, and certain gels can undergo large reversible volume changes as they are cycled about a critical temperature. We have developed a continuum-level theory to describe the coupled mechanical deformation, fluid permeation, and heat transfer of such thermally responsive gels. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model based on a Flory–Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical–mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. We have numerically implemented our theory in a finite element program. We show that our theory is capable of simulating swelling, squeezing of fluid by applied mechanical forces, and thermally responsive swelling/de-swelling of such materials.     

Stress communication and filtering of viscoelastic layers in oscillatory shear

We revisit a classical topic: response functions of viscoelastic layers in large amplitude oscillatory shear. Motivated by questions concerning protective biological layers, we focus on boundary stresses in a parallel plate geometry with imposed oscillatory strain or stress. These features are gleaned from resolution and analysis of coupled standing waves of deformation and stress. We identify a robust non-monotone variation in boundary stress signals with respect to all experimental controls: viscoelastic moduli of the layer, layer thickness, and driving frequency. This structure of peaks and valleys in boundary values of shear and normal stress indicates redundant mechanisms for stress communication (by tuning to the peaks) and stress filtering (by tuning to the valleys). In this paper, we first restrict to a single-mode non-linear Maxwell model for the viscoelastic layer, where analysis renders a transparent explanation of the phenomena. We then consider a Giesekus constitutive model of the layer, where analysis is supplanted by numerical simulations of coupled non-linear partial differential equations. Parametric studies of wall stress values from standing waves confirm persistence of the Maxwell model phenomena. The analysis and simulations rely on and extend our recent studies of shear waves in a micro parallel plate rheometer [S.M. Mitran, M.G. Forest, L. Yao, B. Lindley, D. Hill, Extenstions of the Ferry shear wave model for active linear and nonlinear microrheology, J. Non-Newtonian Fluid Mech. 154 (2008) 120–135; D.B. Hill, B. Lindley, M.G. Forest, S.M. Mitran, R. Superfine, Experimental and modeling protocols from a micro-parallel plate rheometer, UNC Preprint, 2008].     

Cross-stream forces and velocities of fixed and freely suspended particles in viscoelastic Poiseuille flow: Perturbation and numerical analyses

The cross-stream migration of a circular particles (or infinitely long cylinder) in two dimensional, inertia-less viscoelastic pressure-driven flows is examined through complementary finite element simulations and second-order fluid perturbation analyses for small Deborah number (De), where De is defined as the fluid relaxation time divided by the characteristic flow time. A neutrally buoyant, freely suspended particle migrates toward the center of the channel for all particle sizes and cross-stream positions due to the coupled effects of the linear and quadratic variations of the imposed velocity. A particle that is held at a fixed position, in contrast, experiences a cross-stream force directed toward the wall as a result of the coupled effects of the local shear flow and the flow relative to the particle.     

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

We study a nonlinear, unsteady, moving boundary, fluid–structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by two-dimensional incompressible Navier–Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the one-dimensional cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (two-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and the balance of contact forces at the fluid–structure interface. We prove the existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in Guidoboni et al. (J Comput Phys 228(18):6916–6937, 2009) to numerically solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk–Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.     

Migration of a sphere suspended in viscoelastic liquids in Couette flow: experiments and simulations

The intrinsically coupled effects of the curvature of the flow-field and of the viscoelastic nature of suspending medium on the cross-stream lateral migration of a single non-Brownian sphere in wide-gap Couette flow are studied. Quantitative videomicroscopy experiments using a counterrotating device are compared to the results of 3D finite element simulations. To evaluate the effects of differences in rheological properties of the suspending media, fluids have been selected which highlight specific constitutive features, including a reference Newtonian fluid, a single relaxation time wormlike micellar surfactant solution, a broad spectrum shear-thinning elastic polymer solution and a constant viscosity, highly elastic Boger fluid. As expected for conditions corresponding to Stokes flow, migration is absent in the Newtonian fluid. In the wormlike micellar solution and the shear-thinning polymer solution, spheres are observed to migrate in the direction of decreasing shear rate gradient, i.e. the outer cylinder, except when the sphere is initially released close to the inner cylinder, in which case the migration is towards it. The migration is enhanced by faster relative angular velocities of the cylinders. Shear-thinning reduces the migration velocity, showing an opposite behavior as compared to previous results in planar shear flow. In the Boger fluid, within experimental error no migration could be observed, likely due to the large solvent contribution to the overall viscosity. For small Deborah numbers the migration results are well described by an heuristic argument based on a local stress balance.     

The squeeze-film flow of a viscoelastic fluid

We report on the use of a numerical method to solve the (inertia-less) squeeze-film flow problem for a viscoelastic fluid. The method is based on a Boundary Element formulation and relies on a time-marching scheme. The viscoelastic fluid is modelled by a constitutive law that allows for a shear stress overshoot mechanism in a suddenly started shear flow. The results show convincingly that the load enhancement sometimes observed experimentally is due to stress overshoot. A simple explanation for the enhancement is suggested; the stress overshoot appears quickly and takes a long time to die away, so that steady-state viscous behaviour is not very relevant.