Flow of an inelastic Non-Newtonian fluid through an abrupt contraction: A recalculation

The axi-symmetric flows of Newtonian and Non-Newtonian fluids through a sudden one-to-four contraction are numerically simulated. As the Non-Newtonian constitutive equation the inelastic form of the four-constant Oldroyd implicit rheological model is used. Because of the Non-Newtonian viscosity, a single non-dimensional parameter cannot be found to characterize the equations of motion. Therefore in this work the equations of motion are solved using dimensional values, that is, with actual increase of the mass flow.     

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

Evolution of time constants during sol-gel transition

In developing a rheological constitutive equation for crosslinking polymers the determination and explanation of time constants are of great importance. For stoichiometrically balanced and imbalanced reactions of polydimethylsiloxane and polyurethane model polymers with appropriate cross-linkers, it is shown that the experimental data for the storage and loss moduli are well fitted within a large frequency range by a linear rheological constitutive equation with a truncated power law relaxation function.Special attention is given to the problem of estimating the different time constants. A method is proposed for the determination of the terminal relaxation time (in the case below the gel point) and a characteristic retardation time (in the case above the gel point) from the intersection points of the high and low frequency asymptotes of the storage modulus.     

Elastic recovery and polymer-polymer interactions

i) Elastic recovery in polymeric liquids is a cooperative phenomenon in the sense that individual polymer molecules undergoing retraction must interact with one another in order to generate recovery. Stress generated by polymer molecules under an externally imposed flow field may or may not be a cooperative phenomenon. We suggest that the ability to describe the large elastic recovery exhibited by many polymeric liquids furnishes a crucial test of the validity of methods used to model the interaction of a given polymer molecule with its neighbors. Temporary-junction network models appear to be capable of explaining observed recoveries. Elastic recovery cannot be explained by single-molecule-in-a-mean-field theories which involve no calculation of the effect of the single molecule on the mean field. ii) A Gaussian network theory equation for the change of volume with elongation for a cross-linked elastomer is generalized in order to allow the bulk compliance to depend on elongation. iii) It is proved that two classes of flow history, namely shear-free and shear, are constitutively independent in the sense that, for a given viscoelastic liquid of unknown constitutive equation, the behavior in one class cannot be predicted from rheological measurements (however extensive) made solely in the other class.Dedicated to Prof. Dr. J. Meissner on the occasion of his 60th birthday.     

Generalized Cole-Cole behavior and its rheological relevance

Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.     

The time-dependent stress-optical behavior of polycarbonate in the glass transition region

The mechanical and stress-optical behavior of Bisphenol-A polycarbonate was investigated in the glass-transition region. For this purpose, optical creep experiments were carried out in shear and elongation on a tensile tester specially designed for use on a microscope state. A Kohlrausch Williams Watts equation (KWW) with a temperature-independent parameter could successfully be applied to the curves describing the time-dependent values of the stress-optical coefficient for several temperatures. The temperature dependence of the corresponding retardation time could be established and described by the WLF equation. For variable stresses the time-dependent birefringence is obtained from a generalized linear stress-optical rule as modeled according to linear superposition. The time-temperature superposition principle was applied to all measurements. With the dynamic moduli some deviations were observed at the transition from the rubbery plateau to the relaxation. The strain-optical coefficient was found to decrease with increasing time and strain. The strain dependence was found to be independent of temperature at constant stress.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

Taylor-Couette stability analysis for a Doi-Edwards fluid

The stability of Taylor-Couette flow of entangled polymeric solutions to small axisymmetric stationary disturbances is analyzed using the Doi-Edwards constitutive equation in the small gap limit. A previous analysis of Karlsson, Sokolov, and Tanner for the general K-BKZ equation, of which the Doi-Edwards equation is a special case, reduces the problem to one of numerically evaluating seven viscoelastic functions of the shear rate in the gap. Of these seven, only three — two of which are related to the second normal stress difference, and one of them to shear thinning — significantly affect the flow stability. The negative second normal stress difference of the Doi-Edwards fluid stabilizes the flow at low values of the Weissenberg number ₁ , while shear thinning produces strong destabilization at moderate Weissenberg number. Here ₁ is the longest relaxation time. Non-monotonic effects of viscoelasticity on Taylor-Couette stability analogous to those predicted here have been observed in experiments of Giesekus. The extreme shear thinning of the Doi-Edwards fluid is also predicted to produce a large growth in the height of the Taylor cells, a phenomenon that has been seen experimentally by Beavers and Joseph.     

Rheological properties of lightly crosslinked carboxy copolymers in aqueous solutions

The rheological properties of a series of lightly crosslinked carboxy copolymers in aqueous solutions have been evaluated in steady shear and dynamic oscillatory modes. Viscosity profiles and the behavior of storage modulus are related to the chemical composition of the copolymers and their crosslinking density. A maximum in viscosity and in storage modulus which depends on the type of crosslinking agent used is explained by a combination of a chain entanglement mechanism and a closely-packed spheres model. The recovery of viscosity and storage modulus after shearing is very fast and is related to the very fast rearrangement of the microgel structure as a function of time.     

Linear viscoelastic behavior of narrow molecular weight distribution 1,4 polybutadiene: Comparison with Doi-Edwards theory of reptation

Linear viscoelastic behavior of narrow molecular weight distribution 1,4 polybutadiene samples with molecular weights between 42500 and 779000 has been correlated with molecular structure using a simple modification of the Doi-Edwards theory of reptation. The entire GPC curve is required for the calculations of viscoelastic behavior.The plateau modulus obtained from the experimental data is comparable to literature values, while the equilibrium compliance (which is indicative of polydispersity) is greater than values reported in the literature for nearly monodisperse polybutadienes. Reasonable agreement between theory and experiment is obtained over the entire molecular weight range. The agreement between theory and experiment using the GPC curve is better than that obtained by assuming the polymer to be monodisperse or by using the Doi fluctuation model. The model appears to break down for a more polydisperse sample . This study indicates that it may be possible to use the Doi-Edwards theory to explain the viscoelastic behavior of narrow MWD polybutadienes without introducing any new concepts into the theory (fluctuations, constraints release, etc.).     

Consistency tests for start-up and cessation deformations of viscoelastic fluids

The time-dependent shear stress and first normal stress difference were measured for a polystyrene solution for start-up and cessation-flow experiments over a relatively wide range of shear rate. Consistency tests for the K-BKZ model were applied to the data, and it was concluded that the K-BKZ equation generally does not satisfactorily describe the start-up and cessation data. Modified consistency tests were developed using a strain-coupling constitutive equation, and the evidence suggests that most of the differences between the predictions of the K-BKZ theory and experiment can be explained by including a strain-coupling effect in the rheological constitutive equation.     

Rheological characteristics of glass-fibre-filled polypropylene melts

The rheological properties of glass fibre-filled polypropylene melts have been investigated. A high pressure capillary rheometer has been used for the experimental study. The effect of shear rate, temperature, and fibre concentration on the melt viscosity and viscoelastic properties have been studied. An equation has been proposed to correlate the melt viscosity with shear rate, temperature and fibre content. A master curve relation on this basis has been brought out using the shift factora _T . a _T shift factor (=/ _r ) - A _i coefficients of the polynomical of eq. (1) (i = 0, 1, 2, ,n) - B constant in the AFE equation (eq. (2)) (Pa s) - B constant in eq. (3) - D extrudate diameter - d capillary diameter - activation energy at constant shear rate (kcal/mole) - E activation energy at constant shear stress (kcal/mole) - T melt temperature (K) - X fraction glass fibre by weight - shear rate (s^–1) - shear viscosity (Pa s) - normal stress coefficient (Pa s²) - ₁ – ₂ first normal-stress difference (Pa) - shear stress (Pa) - r at reference temperature     

Flow birefringence and rheological measurements on shear induced micellar structures

Aqueous solutions of cationic surfactant systems with strongly binding counterions show the striking phenomenon of shear induced phase transitions. At low shear rates or angular frequencies, the solutions exhibit Newtonian flow. At high rates of shear, however, the rheological properties change dramatically. Above a well defined threshold value of the velocity gradient, a supermolecular structure can be formed from micellar aggregates. This shear induced structure (SIS) behaves like a gel and exhibits strong flow birefringence. The formation of the shear induced structure is very complicated and depends on the specific conditions of the surfactant system. In this paper we discuss new results which have been obtained from rheological measurements and from flow birefringence data. We examine the stability of the shear induced state as a function of temperature, surfactant concentration and salt concentration and we analyse the effect of solubilisation of alcohols and hydrocarbons. The results are interpreted in terms of a kinetic model which accounts for the observed behavior.Dedicated to the 60. birthday of Prof. H. Harnisch, Hoechst AGPartly presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Finite difference solutions have been obtained by the perturbation method to investigate the influence of shear thinning and elasticity on the flow around an inclined circular cylinder of finite length in a uniform flow. In this numerical analysis a generalized upper-convected Maxwell model, in which the viscosity changes according to the Cross model, has been used.The local flow over the cylinder is only slightly deflected. However, in the wake flow behind the cylinder the particle path is remarkably influenced by the axial flow and rapidly flows up parallel to the cylinder's axis. Then it gradually rejoins direction of the incoming flow. It is found that viscoelastic fluids are prone to flow axially in the vicinity of the cylinder. The numerical predictions generally agree with the flow visualization results.The numerical solutions also demonstrate that elasticity has a strong effect on the velocity profile especially around both ends of the cylinder; elasticity increases the asymmetric profiles of both circumferential velocity and axial velocity with respect to equal to 90° and decreases a difference in the circumferential velocity between the windward end and the leeward end.For non-Newtonian fluids, the length of the wake flow is influenced by not only the Reynolds number but also the cylinder diameter and it is larger for the cylinder with the smaller diameter at the same Reynolds number.Partly presented at the 9th Australasian Fluid Mechanics Conference, University of Auckland, New Zealand, 8–12 December, 1986     

On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.