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     

Extensional rheology of dilute polymer solutions in oscillatory cross-slot flow: the transient behaviour of birefringent strands

Birefringent strands are key to understanding polymeric non-Newtonian flows, especially in extension. Utilising microfluidic extensional flow oscillatory rheometry coupled with microvelocimetry (μ-PIV), we report experiments on the genesis, steady state and decay of such strands, together with rheological consequences. For closely monodisperse atactic polystyrene, we report massive effects of the polymer on flow even at low concentrations. The often observed startup “overshoot” in stress and birefringence is observed at unprecedented dilution and discussed in terms of the local strain rate. Strand decay shows pronounced hysteresis. These factors are most important in modelling real flows such as cyclic and capillary entrance flows. Even with the closely monodisperse and well-characterised samples used, residual polydispersity plays a vital role in flow behaviour.     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

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.     

Gradient and vorticity banding

“Banded structures” of macroscopic dimensions can be induced by simple shear flow in many different types of soft matter systems. Depending on whether these bands extend along the gradient or vorticity direction, the banding transition is referred to as “gradient banding” or “vorticity banding,” respectively. The main features of gradient banding can be understood on the basis of a relatively simple constitutive equation. This minimal model for gradient banding will be discussed in some detail, and its predictions are shown to explain many of the experimentally observed features. The minimal model assumes a decrease of the shear stress of the homogeneously sheared system with increasing shear rate within a certain shear-rate interval. The possible microscopic origin of the severe shear-thinning behaviour that is necessary for the resulting nonmonotonic flow curves is discussed for a few particular systems. Deviations between experimental observations and predictions by the minimal model are due to obvious simplifications within the scope of the minimal model. The most serious simplifications are the neglect of concentration dependence of the shear stress (or on other degrees of freedom) and of the elastic contributions to the stress, normal stresses, and the possibility of shear-induced phase transitions. The consequences of coupling of stress and concentration will be analyzed in some detail. In contrast to predictions of the minimal model, when coupling to concentration is important, a flow instability can occur that does not require strong shear thinning. Gradient banding is sometimes also observed in glassy- and gel-like systems, as well as in shear-thickening systems. Possible mechanisms that could be at the origin of gradient-band formation in such systems are discussed. Gradient banding can also occur in strongly entangled polymeric systems. Banding in these systems is discussed on the basis of computer simulations. Vorticity banding is less well understood and less extensively investigated experimentally as compared to gradient banding. Possible scenarios that are at the origin of vorticity banding will be discussed. Among other systems, the observed vorticity-banding transition in rod-like colloids is discussed in some detail. It is argued, on the basis of experimental observations for these colloidal systems, that the vorticity-banding instability for such colloidal suspensions is probably related to an elastic instability, reminiscent of the Weissenberg effect in polymeric systems. This mechanism might explain vorticity banding in discontinuously shear-thickening systems and could be at work in other vorticity-banding systems as well. This overview does not include time-dependent phenomena like oscillations and chaotic behaviour.     

Specific rheology – morphology relationships for some blends containing LCPs

For the blend melts of isotropic polysulfone (PSF) and LC polyester (PES), differing in viscosity more than 10 times, the flow curves with maxima were observed in cone and plate geometry. The low shear rate branch is located near the PSF flow curve, and the high shear rate branch is close to the PES flow curve. At high strains, the formation of the ring-like morphology of the blend sample, accompanied by appearance of maximum on flow curve, was registered. The scaling analysis of the reasons for the ring morphology formation was based on stretching of the large, low-viscous LC droplet, embedded to the high-viscous polymer matrix, in a homogeneous shear field. It was shown that, if the critical Taylor radius is not exceeded, the droplet may form the closed torus. Under strong flows, the PSF melt manifests the “spurt effect”, consisting of a drastic increase of the shear rate when the critical value of the shear stress is reached. The pattern of the blend flow curves with maxima may be explained by a vanishing PSF input to the total shear stress, inherent for blends, while the PES melt continues to be in a liquid state and, consequently, is responsible for the blend viscosity at the high shear rates. The presence of regular heterogeneities in the blend in the form of LC rings may initiate the rupture of the entanglements network of the matrix PSF (close to LC rings) under strong shear flows. The appearance of the low-viscous “cracks” at the critical shear stress will diminish the contribution of the PSF to the blends rheological response. Received: 20 April 1999 Accepted: 28 January 2000     

Measurements of mean and fluctuating wall shear stress beneath spanwise-invariant separation bubbles

 Pulsed-wire measurements of wall shear stress have been made beneath two separation bubbles. In one a cross flow was generated by means of a (25°) swept separation line. Fluctuating stresses in orthogonal “streamwise” and cross-flow directions are very nearly equal and independent of at least moderate cross flow velocity. These fluctuations are largely determined by large-scale motions in the outer flow, whereas the mean shear stresses are not. The pdf of the “streamwise” fluctuations is unchanged by the cross flow. When a cross flow is present the pdf of the cross-flow stresses is similar to the “streamwise” pdf. Dependence on Reynolds number is the same in both flows. Received: 10 April 1998/Accepted: 17 July 1998     

A convenient way of interpreting steady shear rheo-optical data of semi-dilute polymer solutions

Rheo-mechanical and rheo-optical investigations were carried out with the aim of determining the influence of deformation and orientation or disentangling of polymer coils on the flow behavior in the non-Newtonian region of the flow curve, for a moderately concentrated network solution. To avoid the influence of polydispersity this was done on a series of narrowly distributed polystyrene standards (dissolved in toluene). By using steady state shear flow measurements it was possible to detect qualitatively a reduction in the entanglement density within the non-Newtonian flow region. Birefringence experiments were able to show that deformation of the polymer coils also occurs in the Newtonian flow region, which has no effect on the flow behavior in this range, whereas in the non-Newtonian flow region the increase in deformation is lower than in the Newtonian range. The flow birefringence and its orientation can be described over the whole range of the flow curve with a newly developed equation system (Eq. 8 and 14) derived from the stress states of a sheared solution using the stress-optical rule. Starting from these equations, it could be shown, that in the Newtonian flow region a mastercurve in form of a reduced birefringence Δn′/η₀=f(γ˙) and a reduced orientation φ= f(γ˙/γ˙ _crit) can be plotted, independent from concentration and molar mass. A comparison of the experimentally determined orientation angle and birefringence curve form with theoretical deformations and orientations of polymer coils in a solution state, without intermolecular interactions, was able to demonstrate that the flow behavior of a moderately concentrated network solution is determined decisively (approximately to 85%) by the disentanglement. Received: 8 May 2000 Accepted: 12 September 2000     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

Experimental and conceptual problems in the rheological characterization of wheat flour doughs

Wheat flour dough is an industrially important material and a better understanding of its rheological behavior could have long ranging impact on the agricultural and the food processing industries. However, rheological characterization of dough is proving to be difficult due to a range of testing issues and anomalies in flow behavior. In a cone-and-plate rheometer wheat flour doughs “roll-out” of the gap before steady state viscosities can be established, as discussed by Bloksma and Nieman (1975). However, the mirror image of the transient viscosity-time plot obtained using a cone-and-plate viscometer has been used to obtain an estimate of steady shear viscosity behavior (Gleissle, 1975). To check this transient methodology for doughs, a second method, in addition to cone-and-plate transient flow, for determination of the shear viscosity, was needed. For this, capillary extrusion was chosen. Both a piston-driven and pressure driven capillary rheometer were employed. End corrections were determined to provide information on both the shear viscosity and, following Binding (1988), the extensional viscosity of the doughs. There are few data available on end corrections for doughs, though published data by Kieffer indicate that the corrections are unexpectedly very high. In this present work it was found that the end correction experiments were very difficult and imprecise in part due to the time-dependent nature of the doughs and difficulties in preparing replicate batches required to compare dies of differing L/R values. Further it was unexpectedly found that the samples, though prepared by normal mixing procedures to the “optimum” level, were so heterogeneous that large fluctuations in the pressure at constant output rate (in the piston-driven rheometer) and in output rate at constant pressure (in the pressure-driven instrument) were observed. These fluctuations could be eliminated by overmixing of the doughs, but overmixed doughs are of little practical interest. Although the problems encountered in this work were significant, it was encouraging that even these preliminary studies indicate that rheological measurements are effective in differentiating between spring and winter wheats. Defining a constitutive model for dough rheology still remains a major challenge, as results from one type of testing do not corroborate the findings from a different type of testing. Received: 19 May 1998 Accepted: 27 July 1998     

Unsteady flow through screens

Summary Reflected and transmitted pressure waves take place when an incident pressure wave reaches a screen placed in a duct and crossed by a previous steady flow. The thickness of the screen is assumed to be negligible and essure losses through it in the quasi steady flow, which takes place after the passage of the incident pressure wave, are assumed be steady flow ones. The strength of reflected and transmitted pressure waves is related to the strength of the incident wave, to the porosity of the screen and to the previous flow through it. Subsonic and just supersonic quasi steady flow inside the screen are considered. Five commercial screens have been tested with incident shock waves and the experimental results have been compared with the theoretical ones: the agreement is quite satisfactory.

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Sommario L'interazione di un'onda di pressione con una rete intubata attraversata da un flusso preesistente produce un'onda riflessa ed una trasmessa. Si presenta un modello teorico basato essenzialmente sulle potesi che lo spessore della rete sia trascurabile e che le perdite carico attraverso di essa, nel flusso quasi stazionario indotto dalla suddetta interazione, coincidano con quelle proprie di un oto permanente. Si analizza il caso di flusso indotto nella rete ia subsonico che sonico. Si eseguono prove sperimentali su cinque reti di diverse porosità differenti condizioni di flusso, e si verifica l'attendibilità del odello teorico proposto.

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The present work has been supported by the “Consiglio Nazionale delle Ricerche” (C.N.R.) under Contract no. 115.4075.69.01220.

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The suffixes “u” and “d” pertain to the conditions upstream and downstream of the screen with reference to the flow through it. The suffix “j” refers to the region “j” of the wave diagrams in Fig. 1.     

Pressure effects on viscosity and flow stability of polyethylene melts during extrusion

In the present work, the effects of pressure on the viscosity and flow stability of four commercial grade polyethylenes (PEs) have been studied: linear-low-density polyethylene copolymer, high-density polyethylene, metallocene polyethylenes with short-chain branches (mPE-SCB), and metallocene polyethylenes with long chain branching (mPE-LCB). The range of shear rates considered covers both stable and unstable flow regimes. “Enhanced exit-pressure” experiments have been performed attaining pressures of the order of 500×10⁵ Pa at the die exit. The necessary experimental conditions have been clearly defined so that dissipative heating can be neglected and pressure effects isolated. The results obtained show an exponential increase in both shear and entrance-flow pressure drop with mean pressure when shear rate is fixed and as long as flow is stable. These pressure effects are described by two pressure coefficients, β_S under shear and, β_E under elongation, that are calculated using time–pressure superposition and that are independent of mean pressure and flow rate. For three out of four PE, pressure coefficient values can be considered equal under shear and under elongation. However, for the mPE-LCB, the pressure coefficient under elongation is found to be about 30% lower than under shear. Flow instabilities in the form of oscillating flows or of upstream instabilities appear at lower shear rates as mean pressure increases. Nevertheless, the critical shear stress at which they are triggered remains independent of mean pressure. Moreover, it is found that the β_S values obtained for stable flows do not differ much from the values obtained during upstream instability regimes, and differ really from pressure effects observed under oscillating flow and slip conditions.     

On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

Wall slip of mayonnaises in viscometers

The shear flow of mayonnaise is generally characterized by an apparent yield stress, shear thinning in steady flow, stress overshoots upon inception of flow and other time-dependent effects. These observations are usually understood to be the result of structural rearrangement within the material. Additionally and separately, the possibility that emulsions may exhibit apparent wall slip on a microscopic scale at a solid-liquid boundary has been reported by some researchers. Thus, observed rheological behavior is likely to be the result of the interplay between these two phenomena. In the present work, it is demonstrated that when measurements are sought to be made on mayonnaise using rotational viscometers visible wall slip occurs, rendering such instruments ineffective for the purpose of making viscosity measurements even at shear rates as low as 10^–3s^–1. The factors that influence the onset and extent of slip are investigated with the help of parallel plate viscometers, and it is concluded that the observed “yielding” of mayonnaise is actually an artifact of the onset of macroscopic slip. Slip effects are also found in capillary flow but are ameliorated with increasing shear rate. To circumvent these problems, it is proposed that extensional viscometry be employed for determining the flow behavior of mayonnaises. Received: 18 August 1997 Accepted: 1 April 1998     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.     

Flow pattern transition,pressure gradient,hold-up predictions in gas/non-Newtonian power-law fluid stratified flow

This work focuses on gas/non-Newtonian power-law fluid stratified pipe flow. Two different theoretical approaches to obtain pressure gradient and hold-up predictions are presented: the steady fully developed two-fluid model and the pre-integrated model. The theoretical predictions are compared with experimental data available for horizontal and for slightly downward inclined air/shear thinning fluid stratified flow taken from literature. The predictions of the pre-integrated model are validated showing a good agreement when compared with experimental data. The criteria for the transition from the stratified flow pattern are applied to gas/non-Newtonian stratified flow. The neutral stability analysis (smooth/wavy stratified flow) and the well-posedness (existence region of stratified flow) of governing equations are carry out. The predicted transition boundaries are obtained using the steady fully developed two-fluid model and the pre-integrated model, where the shape factors and their derivatives are accounted for. A comparison between the predicted boundaries and experimental flow pattern maps is presented and shows a good agreement. A comment on the shear stress modeling by the pre-integrated model is provided.     

Prediction of multiple overshoots in shear stress during fast flows of bidisperse polymer melts

We present a differential constitutive model of stress relaxation in polydisperse linear polymer melts and solutions that contains contributions from reptation, contour-length fluctuations, and chain stretching. The predictions of the model during fast start-up and steady shear flows of polymer melts are in accord with experimental observations. Moreover, in accordance with reported experimental literature (Osaki et al. in J Polym Sci B Polym Phys 38:2043–2050, 2000), the model predicts, for a range of shear rates, two overshoots in shear stress during start-up of steady shear flows of bidisperse polymer melts having components with widely separated molar masses. Two overshoots result only when the stretch or Rouse relaxation time of the higher molar mass component is longer than the terminal relaxation time of the lower molar mass component. The “first overshoot” is the first to appear with increasing shear rate and occurs as a result of the stretching of longer chains. Transient stretching of the short chains is responsible for the early time second overshoot. The model predictions in steady and transitional extensional flows are also remarkable for both monodisperse and bidisperse polymer solutions. The computationally efficient differential model can be used to predict rheology of commercial polydisperse polymer melts and solutions.     

Materials with Internal Variables and Relaxation to Conservation Laws

.The theory of materials with internal state variables of Coleman & Gurtin [CG] provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics. After reviewing the requirements imposed on constitutive theories by the principle of consistency with the Clausius‐Duhem inequality, we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy. They each lead to a relaxation framework for the theory of thermoelastic non‐conductors of heat, equipped with globally defined “entropy” functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and compactness for solutions of the relaxation system in the zero‐relaxation limit, and establish convergence to the system of isothermal elastodynamics by using compensated compactness. Finally, we prove a strong dissipation estimate for the relaxation approximations proposed in Jin & Xin [JX] when the limit system is equipped with a strictly convex entropy. (Accepted June 17, 1998)     

Chaos in Acoustic Subspace Raised by the Sommerfeld–Kononenko Effect

This paper deals with vibrations of an infinite plate in contact with an acoustic medium where the plate is subjected to a point excitation by an electric motor of limited power-supply. The whole system is divided into two “exciter - foundation” and “foundation-plate-medium”. In the system “motor-foundation” three classes of steady state regimes are determined: stationary, periodic and chaotic. The vibrations of the plate and the pressure in the acoustic fluid are described for each of these regimes of excitation. For the first class they are periodic functions of time, for the second they are modulated periodic functions, in general with an infinite number of carrying frequencies, the difference between which is constant. For the last class they correspond to chaotic functions. In another mathematical model where the exciter stands directly on an infinite plate (without foundation) it was shown that chaos might occur in the system due to the feedback influence of waves in the infinite hydro-elastic subsystem in the regime of motor shaft rotation. In this case the process of rotation can be approximately described as a solution of the fourth order nonlinear differential equation and may have the same three classes of steady state regimes as the first model. That is the electric motor may generate periodic acoustic waves, modulated waves with an infinite number of frequencies or chaotic acoustic waves in a fluid.