Steady shear banding in complex fluids

Shear banding (SB) is manifested by the abrupt “demixing” of the flow into regions of high and low shear rate. In this paper, we first relate analytically the rheological parameters of the fluid with the range of shear rates and stresses of SB occurrence. For this, we accept that the origin of shear banding is constitutive, and adopt a non-linear viscoelastic expression able to accommodate the double-valuedness of the stress with flow intensity, under certain conditions. We then implement the model for the case of pressure driven flow through a cylindrical pipe; we derive approximate expressions for the velocity profile in the two-banded regions (core and outer annular), the overall throughput in the presence or absence of “spurt”, and the radial location limits of the shear rate discontinuity.     

Rheology and flow-birefringence from viscoelastic polymer-clay solutions

 The shear orientation of viscoelastic clay-polymer solutions was investigated by means of rheology and flow birefringence (Δn). The polymer chains are in dynamic adsorption/desorption equilibrium with the clay particles to form a “network”. The elastic behavior of the network was characterized by constant stress, oscillatory shear, and stress relaxation experiments. Constant stress experiments indicated a yield stress upon which shear flow started and no strain recovery could be observed. Oscillatory shear experiments showed a broad elastic region followed by flow when a critical strain was reached. Stress relaxation experiments showed several relaxation times when the same critical strain was reached. Experiments under steady flow characterized the transient behavior of the network. With increasing steady shear rate a pronounced minimum in birefringence was observed at a critical shear rate. The shear rate dependent viscosity showed near power law behavior and no corresponding critical feature. While birefringence detects orientational effects on a microscopic length scale, rheology averages over macroscopic changes in the sample. The same degree of orientation could be achieved under constant shear rate or constant stress conditions. Received: 25 January 2001 Accepted: 22 May 2001     

Elastic flow-front fingering instability in flowing polymer solutions

An experimental investigation of the flow-front behavior of dilute and semi-dilute polymer solutions has been carried out to gain a better understanding of the underlying mechanisms leading to the occurrence of an unstable flow at the advancing flow-front during the filling of a rectangular Hele-Shaw cell. Our experimental results have revealed the existence of an elastic finger-like instability at the advancing flow-front that develops in semi-dilute solutions of high molecular weight polymers, with an onset time of approximately a few hundred milliseconds. Although at shear rates above critical, narrow finger patterns develop at the flow-front, their amplitude and number remain roughly constant throughout the flowing. At critical condition, no secondary flow was observed in the vicinity of the front region where the unstable flow develops. Transient response of the normal stress difference and the shear stress in the plate-and-plate geometry at shear rate above critical (for the elastic fingering instability in the Hele-Shaw cell) did not reveal any anomalous that could lead to the formation of such finger-like instabilities. These instabilities were observed for both the ideal elastic Boger fluids and shear thinning viscoelastic fluids.     

Elastic properties of polyethylene melts at high shear rates with respect to extrusion

At high shear rates a steady state of shear flow with constant shear rate, constant shear stress, and constant recoverable shear strain is observed in the short-time sandwich rheometer after some few shear units already. The melt exhibits rather high elastic shear deformations and the recovery occurs at much higher speed than it is observed in the newtonian range. The ratio of first normal stress difference and twice the shear stress, being equal to the recoverable strain in the second-order fluid limit, significantly underestimates the true elastic shear strains at high shear rates. The observed shear rate dependence of shear stress and first normal stress difference as well as of the (constrained) elastic shear strain is correctly described on the basis of a discrete relaxation time spectrum. In simple shear a stick-slip transition at the metal walls is found. Necessary for the onset of slip is a critical value of shear stress and a certain amount of elastic shear deformation or orientation of the melt.     

Elongational flow behavior of polymeric fluids according to the Leonov model

The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.     

On constitutive equations for electrorheological materials

Constitutive equations for electrorheological (ER) fluids have been based on experimental results for steady shearing flows and constant electric fields. The fluids have been modeled as being rigid until a yield stress is reached. Additional stress is then proportional to the shear rate. Recent experimental results indicate that ER materials have a regime of solid-like response when deformed from a rest state. They behave in a viscoelastic-like manner under sinusoidal shearing and exhibit time-dependent response under sudden changes in shear rate or electric field. In this work, a constitutive theory for ER materials is presented which accounts for these recent experimental observations. The stress is given by a functional of the deformation gradient history and the electric field vector. Using the methods of continuum mechanics, a general three-dimensional constitutive equation is obtained. A sample constitutive equation is introduced which is then used to determine the response of an ER material for different shear histories. The calculated shear response is shown to be qualitatively similar to that observed experimentally.     

Instabilities of micellar systems under homogeneous and non-homogeneous flow conditions

The rheological behavior of a cetylpyridinium chloride 100 mmol l^–1/sodium salicylate 60 mmol l^–1 aqueous solution was studied in this work under homogeneous (cone and plate) and non-homogeneous flow conditions (vane-bob and capillary rheometers), respectively. Instabilities consistent with non-monotonic flow curves were observed in all cases and the solution exhibited similar behavior under the different flow conditions. Hysteresis and the sigmoidal flow curve suggested as characteristic of systems that show constitutive instabilities were observed when running cycles of increasing and decreasing stress or shear rate, respectively. This information, together with a detailed determination of steady states at shear stresses close to the onset of the instabilities, allowed one to show unequivocally that "top and bottom jumping" are the mechanisms to trigger the instabilities in this micellar system. It is shown in addition that there is not a true plateau region in between the "top and bottom jumping". Finally, the flow behavior beyond the upturn seemed to be unstable and was found accompanied by an apparent violation of the no-slip boundary condition.     

Mathematical characteristics of the pom-pom model

 Recently, in order to describe the complex rheological behavior of polymer melts with long side branches like low density polyethylene, new constitutive equations called the pom-pom equations have been derived in the integral/differential form and also in the simplified differential type by McLeish and Larson on the basis of the reptation dynamics with simplified branch structure taken into account. In this study, mathematical stability analysis under short and high frequency wave disturbances has been performed for these constitutive equations. It is proved that the differential model is globally Hadamard stable, as long as the orientation tensor remains positive definite or the smooth strain history in the flow is previously given. However both versions of the model are Hadamard unstable if we neglect the arm withdrawal in the case of maximum backbone stretch. It is also dissipatively unstable, since the steady shear flow curves exhibit non-monotonic dependence on shear rate. Additionally, in the flow regime of creep shear flow where the applied constant shear stress exceeds the maximum achievable value in the steady flow curves, the constitutive equations exhibit severe instability that the solution possesses strong discontinuity at the moment of change of chain dynamics mechanisms. Received: 14 August 2001 Accepted: 18 October 2001     

Time dependent flow in equimolar micellar solutions: transient behaviour of the shear stress and first normal stress difference in shear induced structures coupled with flow instabilities

Recently we studied time dependent structural changes that are coupled with flow instabilities (Fischer 1998; Wheeler 1998; Fischer 2000). Within a stability analysis, a classification scheme for the feedback circuit of coupled shear-induced structure and flow instabilities was derived by Schmitt et al. (1995) and applied to our samples. Here, inhomogeneous flow layers of different concentration and viscosity are generated by shear-induced diffusion (spinodal demixing) and, as consequence, one no longer observes a homogeneous solution but a type of shear banding that is seen here for the first time. In this paper we present the behaviour of the first normal stress difference observed in the critical shear-rate regime where transient shear-induced structure is coupled with flow instability. Similar to the oscillations of the shear stresses (strain-controlled rheometer) one observes oscillations in the first normal stress difference. This behaviour indicates that elastic structures are built up and destroyed while the shear-induced structures occur and that the induced phase is more elastic than the initial one. Oscillations of shear stress and first normal stress difference are in phase and indicate that both phenomena are caused by the same mechanism. Received: 30 June 1999/Accepted: 14 December 1999     

Shear rheology of polymer solutions near the critical condition for elastic instability

The use of constant viscosity, highly elastic polymer solutions, so called Boger fluids, has been remarkably successful in elucidating the behavior of polymeric materials under flowing conditions. However, the behavior of these fluids is still complicated by many different physical processes occurring within a narrow window of observation time and applied shear rate. In this study, we investigate the long-time shear behavior of an ideal Boger fluid: a well characterized, athermal, dilute, binary solution of high molecular weight polystyrene in oligomeric polystyrene. Rheological measurements show that under an applied steady shear flow, this family of polymer solutions undergoes a transient decay of normal stresses on a timescale much longer than the polymer molecule's relaxation time. Rheological and flow visualization results demonstrate that the observed phenomenon is not caused by polymer degradation, phase separation, viscous heating, or secondary flows from elastic instabilities. Although the timescale is much shorter than that associated with polymer migration in the same solutions (MacDonald and Muller, 1996), the appearance of this phenomenon only at the rates where migration has been observed suggests that it may be a prerequisite for observing migration. In addition, we note that through sufficient preshearing of the sample, the normal stress decrease suppresses the elastic instability. These results show that there is considerable uncertainty in choosing the appropriate measure of the fluid relaxation time for consistently modeling the critical condition for the elastic instability, the decay of normal stresses, and the migration of polymer species.     

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value. (Accepted April 22, 1996)     

Three-dimensional disturbances of planar viscometric flow

This paper examines three-dimensional disturbances of a plane steady shear flow of simple fluids with short memory. Under the assumption of nearly-viscometric flow, constitutive equations are derived and then a general form of the Reynolds-Orr energy equation is obtained. With the aid of this derived energy formula, sufficient conditions are generated for the stability of three-dimensional disturbances of the planar viscometric flow. These conditions are analysed and a comparison is made with the corresponding two-dimensional stability problem. There is a strong indication that the basic flow is less stable against three-dimensional disturbances than against two-dimensional ones.     

Selection of kernel functions used in a new constitutive equation for viscoelastic fluids

A selection of kernel functions is given to be used in a new integral constitutive equation proposed by Piau whereby the deviatoric stress is calculated from the integral of the history of the past intrinsic rate of rotation and rate of deformation tensors through a representation theorem. Piau has demonstrated the objectivity of a frame moving with a given particle whose axis are directed along the eigenvectors of the rate of deformation tensor. The use of such a framework provides a new approach in the attempt to reduce the computational difficulties associated with conventional constitutive equations written in co-deformational or co-rotational reference frames.The shear and primary normal-stress material functions and the extensional (elongational) stress growth function are defined for the proposed integral constitutive equation. These material functions are used to calculate the kernel functions using steady state, stress relaxation and stress growth data of Attané in simple shear flow for monodisperse polystyrene solutions. The shear and extensional stress growth data of Meissner for a polyethylene melt are also used to show the flexibility of the rheological model.The material functions are first written in terms of five monotonically decreasing functions of the time lag between the past and the present time. Then kernel functions are chosen such that when substituted in the new integral constitutive equation they yield the functions used to describe the data. A further condition imposed on the normalized kernel functions is that they be decreasing functions of time lag.     

The phenomenological description of the thixotropic behaviour of fresh cement pastes

The present work reports an investigation on the rheological behaviour of fresh cement pastes. Three types of Portland cement were used. The water/cement ratio was varied in the range 0.35 ÷ 0.40. The rheological tests were performed using the coaxial-cylinder viscometer Rotovisko-Haake RV 11. The material to be tested was subjected immediately after mixing to a constant shear rate until a steady value of shear stress (equilibrium value) was attained. All the pastes tested exhibited a flow behaviour of the partially thixotropic type. A rheological model consisting of a sole constitutive equation \(\tau = \tau (\dot \gamma ,t)\) was defined according to the Cheng and Evans approach. The parameters of the constitutive equation were correlated with the cement specific surface and the water/cement ratio.     

Strain localization in ductile single crystals

This paper is concerned with an analysis of strain localization in ductile crystals deforming by single slip. The plastic flow is modelled as rate-insensitive, and localization, viewed as a bifurcation from a homogeneous deformation mode to one which is concentrated in a narrow ‘shear band’, is found to be possible only when the plastic hardening modulus for the slip system has fallen to a certain critical value h_cr, sensitive to the precise form of the constitutive law governing incremental shear. We develop the general form of this constitutive law, incorporating within it the possibility of deviations from the Schmid rule of a critical resolved shear stress, and we show that h_cr may in fact be positive when there are deviations from the Schmid rule. It is suggested that micromechanical processes such as ‘cross-slip’ in crystals provide specific cases for which stresses other than the Schmid stress may influence plastic response and, further, there is an experimental association of localization with the onset of large amounts of cross-slip. Thus, we give the specific form of h_cr for a constitutive model that corresponds to non-Schmid effects in cross-slip, and we develop a dislocation model of the process from which we estimate the magnitude of the parameters involved. The work supports the notion that localization can occur with positive strain-hardening, h_cr > 0, and the often invoked notions of the attainment of an ideally-plastic or strain-softening state for localization may be unnecessary.     

Convergence analysis of a finite element method based on different moduli in tension and compression

When analyzing materials that exhibit different mechanical behaviors in tension and compression, an iterative approach is required due to material nonlinearities. Because of this iterative strategy, numerical instabilities may occur in the computational procedure. In this paper, we analyze the reason why iterative computation sometimes does not converge. We also present a method to accelerate convergence. This method is the introduction of a new pattern of shear modulus that was strictly derived according to the constitutive model based on the bimodular elasticity theory presented by Ambartsumyan. We test this procedure with a numerical example concerning a plane stress problem. Results obtained from this example show that the proposed method reduces the cost of computation and accelerates the convergence of the solution.     

Material functions for constant-strain-rate elongational flows

The problem of determination of invariant material functions for elongational flows in which two components of the constant strain-rate tensor are equal is briefly discussed, and a method of its solution described. The method is based on simultaneous modeling of the elongational viscosity as measured in uniform uniaxial elongational flow, and the shear viscosity as measured in steady viscometric flow. A single integral model with a strain-rate dependent memory is used to correlate both viscosities over a given experimental range of strain rates. The procedure has been applied to a set of experimental data obtained for a low-density polyethylene melt by Laun and Münstedt.     

Dynamic properties of shear thickening colloidal suspensions

The transient shear rheology (i.e., frequency and strain dependence) is compared to the steady rheology for a model colloidal dispersion through the shear thickening transition. Reversible shear thickening is observed and the transition stress compares well to theoretical predictions. Steady and transient shear thickening are observed to occur at the same value of the average stress. The critical strain for shear thickening is found to depend inversely on the frequency at fixed applied stress for low frequencies (high strains), but is limited to an apparent minimum critical strain at higher frequencies. This minimum critical strain is shown to be an artifact of slip. Lissajous plots illustrate the transition in material properties through the shear thickening transition, and the energy dissipated by a shear thickening suspension is analyzed as a function of strain amplitude.     

A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported “steady-state” elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J.L. White, A. Kondo, Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation, J. Non-Newtonian Fluid Mech. 3 (1977) 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements.     

The interface migration in shear-banded micellar solutions

In this work, we study the diffusion of the interface between bands in wormlike micellar solutions that exhibit shear banding flow regimes, namely, systems undergoing coexistence of states of different shear rates along a constant stress plateau. The migration of the interface between bands possessing different birefringence levels is predicted by the BMP (Bautista-Manero-Puig) model in which a structural parameter (the fluidity) presents two states with differing order separated by an interface. The mechanical potential derived from the constitutive equations and a diffusion term for the structure evolution equation predict various time scales of interface migration at the inception of shear flow and under shear-rate changes along the plateau stress. It is shown that the extremes of the plateau (binodals) correspond to the minima in the mechanical potential as a function of fluidity or shear rate. We also predict the dependence of the diffusive length scale on the applied shear rate.