Showing papers in "Journal of Non-newtonian Fluid Mechanics in 2003"
TL;DR: In this article, the authors developed a theory for fast flows of entangled polymer melts which includes the processes of reptation, convective and reptation-driven constraint release, chain stretch and contour length fluctuations.
Abstract: Recently we developed a theory for fast flows of entangled polymer melts which includes the processes of reptation, convective and reptation-driven constraint release, chain stretch and contour length fluctuations The theory is derived from a stochastic microscopic equation of motion of the chain inside the tube and of the tube itself As a result we obtain a partial differential equation for the tube tangent correlation function, the solution of which requires quite intensive calculations At the same time the application of this theory to realistic flows (which is anything other than the laboratory rheometer) requires a simple and less computationally intensive set of equations for the stress tensor similar to the Giesekus, PTT, Larson or Pom–Pom equations In particular, the last was derived from molecular theory for a generic type of branched polymer In this paper we demonstrate that molecular tube theory can also provide a route to constructing a family of very simple differential constitutive equations for linear polymers They capture the full model quite well and therefore can be used in flow solving software to model spatially inhomogeneous flows We present a comparison of the proposed equations with our full model and with experimental data
314 citations
TL;DR: In this paper, the authors present very accurate numerical results for the vortex size, the vortex intensity and the Couette correction, in planar contraction flows of Oldroyd-B and PTT fluids, with both the linear and the exponential stress function, and with a solvent viscosity ratio equal to 1/9.
Abstract: The paper presents very accurate numerical results for the vortex size, the vortex intensity and the Couette correction, in planar contraction flows of Oldroyd-B and PTT fluids ( e = 0.25) with both the linear and the exponential stress function, and with a solvent viscosity ratio equal to 1/9. The accuracy of these results is quantified, being generally below 1% (0.3% for most results), and the finest mesh employed had over 1 million degrees of freedom. Such degree of mesh fineness is shown to be required for accurate results with the Oldroyd-B fluid, especially at high Deborah numbers, but the shear-thinning PTT fluid in general does not require the finest meshes. In terms of level of elasticity, steady results for the PTT fluid could be obtained for values of the Deborah number in excess of 100 (linear PTT) and 10,000 (exponential PTT). © 2003 Elsevier Science B.V. All rights reserved.
238 citations
Abstract: A charged polymer jet may be accelerated and stretched by an external electric field, and this process is relevant to electrospinning for making nanofibers. The stretching of an electrified jet is governed by the interplay among electrostatics, fluid mechanics and rheology, and the role of viscoelasticity has not been systematically explored before. This paper presents a slender-body theory for the stretching of a straight charged jet of Giesekus fluid. Results show strain-hardening as the most influential rheological property. It causes the tensile force to rise at the start, which enhances stretching of the jet. Further downstream, however, the higher elongational viscosity tends to suppress jet stretching. In the end, strain-hardening leads to thicker fibers. This confirms the main result of a previous study using empirical rheological models. The behavior of the electrically driven jet forms an interesting contrast to that in conventional fiber spinning.
203 citations
TL;DR: Large amplitude oscillatory shear behavior of complex fluids, which form microstructures depending on their deformation history, has been investigated by using a network model and it is suggested that the LAOS behavior can be effectively used as a tool for classifying complex fluids.
Abstract: Large amplitude oscillatory shear (LAOS) behavior of complex fluids, which form microstructures depending on their deformation history, has been investigated by using a network model. According to recent experimental observations, the LAOS behavior of complex fluids could be classified by at least four types: type I, strain thinning (G � , G �� decreasing); type II, strain hardening ( G � , G �� increasing); type III, weak strain overshoot ( Gdecreasing, G �� increasing followed by decreasing); type IV, strong strain overshoot ( G � , G �� increasing followed by decreasing). To understand such complex behavior, we have applied a general network model. As there is little information available on the form of creation and loss rates of network junctions, we have modeled the creation and loss rates as exponential functions of shear stress. By adjusting the model parameters that define the creation and loss rates, the types of LAOS behavior observed in the experiments could be reproduced. Despite highly simplistic modeling, the model reproduced the types of LAOS behavior observed in the experiments, which means that the behavior can be explained in terms of the model parameters, that is, the creation and loss rates of network junctions. It is also suggested that the LAOS behavior can be effectively used as a tool for classifying complex fluids. © 2003 Elsevier B.V. All rights reserved.
169 citations
TL;DR: In this article, a method to include full hydrodynamic interactions (HI) using the Rotne-Prager tensor into the bead-spring model without excluded-volume interactions was described.
Abstract: We describe a method to include full hydrodynamic interactions (HI) using the Rotne–Prager tensor into the bead–spring model without excluded-volume interactions in a way suitable for Brownian dynamics (BD) simulations of the transient nonlinear rheological properties of dilute polymer solutions. First, we develop a scheme to determine the HI parameter h∗ and bead radius “a” that keeps the number of beads N modest at high molecular weight and yet matches the bead–spring model to the “real” polymer both in its longest relaxation time or diffusivity near equilibrium and in the drag on the chain at full extension, the latter being estimated by a formula of Batchelor. Second, we compare three different numerical integration methods, namely an explicit Euler’s method, a semi-implicit Newton’s method and a semi-implicit predictor–corrector method, and conclude that the predictor–corrector method is the best one available now, because of its ability to use larger time-steps and the relatively low computational expense for each step. Third, we perform simulations for two different macromolecules, namely λ-phage DNA and high molecular weight polystyrene (PS). We find that we can model λ-phage DNA with full HI with only 10 beads, and find that HI has negligible effect on extensional-flow behavior because of DNA’s expanded configuration even at rest, and therefore, its small value of h ∗ =0.03 . For PS in a theta solvent, however, molecular configurations are much more compact, and to avoid bead overlap we must keep h ∗ . This requires the use of more beads, at least N=20 for a molecular weight of 2 million, even if we drop the requirement that we match the longest relaxation time. Surprisingly, despite our inability to match the experimental longest relaxation time with a limited number of beads, excellent agreement with experimental filament-stretching strain–stress data is obtained except for the plateau Trouton ratio for PS of molecular weight of 2 million at various Weissenberg numbers. The inclusion of HI eliminates the “lag”, the delay in growth of the Trouton ratio relative to experimental data, seen in earlier simulations.
150 citations
TL;DR: In this article, the authors present a numerical investigation of instability in non-Newtonian flows through a sudden expansion, which is manifested by a symmetry breaking of the flow separation and the onset of the instability depends on the specific parameters involved in each model's constitutive equation.
Abstract: The paper presents a numerical investigation of instabilities occurring in non-Newtonian flows through a sudden expansion. Three non-Newtonian models, used in the literature for simulating the rheological behaviour of blood, are employed, namely the Casson, Power-Law, and Quemada models. The computations reveal that similar to Newtonian flow through a suddenly expanded channel, an instability also occurs in non-Newtonian flows. The instability is manifested by a symmetry breaking of the flow separation. The onset of the instability depends on the specific parameters involved in each model’s constitutive equation. The investigation encompasses a parametric study for each model, specifically the critical values at which transition from stable to unstable flow occurs. Due to the fact that for each of the Casson and Quemada models, two characteristic flow parameters exist, the relation between the critical values for each of these parameters is also examined.
116 citations
TL;DR: In this article, an entangled worm-like micelle solution of 0.05% cetyletrimethylammonium bromide and 0.5% sodium salicylate dissolved in water was experimentally studied.
Abstract: With the increasing use of wormlike micelle solutions as rheological modifiers in many consumer products, the prediction of the behavior of these fluids has grown increasingly important in recent years. In this paper, the flow past a sphere falling at its terminal velocity through a column of a wormlike micelle solution is experimentally studied. The working fluid is an entangled wormlike micelle solution of 0.05 mol/l cetyletrimethylammonium bromide and 0.05 mol/l sodium salicylate dissolved in water. The rheology of the fluid is characterized in both shear and transient homogeneous uniaxial extension. Sphere-to-tube ratios of a / R =0.0625 and a / R =0.125 are investigated over a wide range of Deborah numbers. The drag on the sphere is initially found to decrease with increasing Deborah number because of shear thinning effects. As the Deborah number is increased, the establishment of a strong extensional flow in the wake of the sphere causes the drag to increase to a value larger than that of a Newtonian fluid with the same viscosity. At a critical Deborah number, the flow becomes unstable and fluctuations in the sedimentation velocity of the sphere are observed. Particle image velocity measurements are used to analyze the flow fields around the falling spheres. These measurements show the presence of a strong negative wake. For the unstable flows, the velocity flow field is observed to fluctuate between a negative and extended wake. Pointwise and full-field flow-induced birefringence measurements are used to track the evolution in the deformation of the wormlike micelle chains. Strong evidence is found that suggests that the flow instability is the result of a breakdown of the wormlike micelle network structure in the wake of the sphere. This breakdown is related to the filament rupture observed in the extensional rheology experiments.
108 citations
TL;DR: In this article, the authors studied the flow of viscoelastic liquids with constant shear viscosity through symmetric sudden expansions, and the geometry considered is planar and the constitutive model follows the modified FENE-CR equation, valid for relative dilute solutions of polymeric fluids.
Abstract: The flow of viscoelastic liquids with constant shear viscosity through symmetric sudden expansions is studied by numerical means. The geometry considered is planar and the constitutive model follows the modified FENE-CR equation, valid for relative dilute solutions of polymeric fluids. For Newtonian liquids in a 1:3 expansion we predict the result that the flow becomes asymmetric for a Reynolds number (based on upstream mean velocity and channel height) of about 54, in agreement with previously published results. For the non-Newtonian case the transition depends on both the concentration and the extensibility parameters of the model, and the trend is for the pitch-fork bifurcation to occur at higher Reynolds numbers. Detailed simulations are carried out for increasing Reynolds number, at fixed concentration and Weissenberg number, and for increasing concentration at a fixed Reynolds number of 60. The results given comprise size and strength of the recirculation zones, bifurcation diagrams, and streamline plots.
93 citations
TL;DR: In this paper, the authors analyzed how the microstructure develops in dependence of the shear rate and cooling rate during the solidification and it was observed that the average particle size increased with increasing shear-rate and decreasing cooling rate.
Abstract: Semi-solid metal alloys, as used in industrial thixoforming, have a special microstructure of globular grains suspended in a liquid metal matrix. The complex rheological properties are strongly influenced by the local solid fraction, particle shape, particle size and state of agglomeration. It was analysed how the microstructure develops in dependence of the shear rate and cooling rate during the solidification and it was observed that the average particle size increased with increasing shear rate and decreasing cooling rate. In order to account for those phenomena, the rate of crystal growth and the relationship between average particle diameter and viscosity was modelled by applying the Sherwood two-film model for the mass transport. The dependence of the viscosity from the particle size were modelled with a modified Krieger–Dougherty model. Based on the rheological and microstructural observations an evaluation method was elaborated that allows for the construction of objective master curves that are independent of the particle growth during the experimentation. The isothermal experiments for the characterisation of the rheological behaviour consisted of step-change of shear-rate and yield-stress experiments. From the experimental data the steady-state flow curves could be determined as well as the time-dependent relaxation of the shear stress after a change of shear rate. The steady-state rheological behaviour was found to be shear thinning. Nevertheless, immediately after a shear-rate change an overshoot was observed that resulted from a short-time shear-thickening behaviour. The yield stress was found to strongly depend on the microstructure and the degree of agglomeration of the solid phase. With increasing rest time the yield stress was increasing strongly, because of the agglomeration of the solid particles. Based on the step-change of shear-rate experiments a single-phase flow has been developed that consists of a modified Herschel–Bulkley approach and accounts for the thixotropic as well as for the yield-stress behaviour of the alloys.
82 citations
TL;DR: In this article, an experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers.
Abstract: An experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers. The measurements of vortex shedding frequency and formation length (lf) were carried out by laser-Doppler anemometry in Newtonian fluids and in aqueous polymer solutions of CMC and tylose. These were shear thinning and elastic at weight concentrations ranging from 0.1 to 0.6%. The 10 and 20 mm diameter cylinders (D) used in the experiments had aspect ratios of 12 and 6 and blockage ratios of 5 and 10%, respectively. The Reynolds number (Re∗) was based on a characteristic shear rate of U∞/(2D) and ranged from 50 to 9×103 thus encompassing the laminar shedding, the transition and shear-layer transition regimes. Increasing fluid elasticity reduced the various critical Reynolds numbers ( Re etr ∗ , Re lf ∗ , Re bbp ∗ ) and narrowed the extent of the transition regime. For the 0.6% tylose solution the transition regime was even suppressed. On the other end, pseudoplasticity was found to be indirectly responsible for the observed reduction in Re otr ∗ : it increases the Strouhal number which in turn increases the vortex filaments, precursors of the transition regime. Elasticity was better quantified by the elasticity number Re′/We than by the Weissenberg number. This elasticity number involves the calculation of the viscosity at a high characteristic shear rate, typical of the boundary layer, rather than at the average value (U∞/(2D)) used for the Reynolds number, Re∗.
81 citations
TL;DR: In this article, the authors examine the transient displacement of a viscoplastic material from straight or suddenly constricted cylindrical tubes of finite length and develop accurate and efficient numerical methods for the fundamental study of processes in which a gas is displacing a liquid from prototype geometries under various operating conditions.
Abstract: We examine the transient displacement of a viscoplastic material from straight or suddenly constricted cylindrical tubes of finite length. Our general goal is to develop accurate and efficient numerical methods for the fundamental study of processes in which a gas is displacing a liquid from prototype geometries under various operating conditions. Such processes can be part of the Gas Assisted Injection Molding (GAIM) or enhanced oil recovery. To this end, we use the mixed finite element method coupled with a quasi-elliptic mesh generation scheme in order to follow the very large deformations of the fluid volume. The displacing fluid is gas at high pressure, which forms a bubble of increasing length and a shape that depends on the fluid properties, the flow conditions, and the tube geometry. The cross-section of the bubble is always smaller than that of the tube due to adherence of fluid on the tube walls. The thickness of the remaining film depends on the same parameters and for most of its length it behaves as unyielded material. Unyielded material also arises in front of the bubble, around the axis of symmetry of the tube(s) and in the case of a constricted tube near the recirculation corner, but not around the entrance of the secondary tube. The rate of growth of the ‘tip splitting’ instability, that arises at relatively large values of the Reynolds number for Newtonian fluids in straight tubes, decreases as the Bingham number increases and, eventually, the instability disappears. The resistance provided by the constricted tube downstream makes the bubble move at a nearly constant velocity only when the Bingham number is not large. When the bubble approaches the constriction it becomes more pointed, but after entering it, the bubble reassumes its well-developed profile. Depending on parameter values, the bubble in the secondary tube may periodically split, thus forming a train of smaller bubbles directed towards the exit of the tube, a phenomenon for which experimental evidence exists.
TL;DR: In this article, the rheological and transport properties of highly concentrated inverse phase (water-in-oil, w/o) emulsions were studied, and the flow curve of fresh emulsion was described by the Cross equation.
Abstract: The rheological and transport properties of highly concentrated inverse phase (water-in-oil, w/o) emulsions were studied. The concentration of the aqueous (disperse) phase was 94%. The flow curve of fresh emulsion is described by the Cross equation. Aging leads to changes in the rheological behavior of the emulsion: the Newtonian flow domain disappears, the apparent viscosity increases and typical yielding is observed. The flow curve of this “aged” emulsion is described by the Hershel–Bulkley equation. Preliminary deformations lead to a slight dilatant effect. Shearing the emulsion through a transition stress threshold of approximately 300 Pa leads to radical changes in the rheological behavior, including loss of stability in tube flow. Stable invert, highly concentrated emulsions have elastic moduli that do not depend on stress and are not affected by transition through the yield limit. Their storage modulus is stable with respect to the deformation amplitude up to 10% strain, while the loss modulus is very sensitive to deformation amplitude. The rheological properties of samples taken from a pipeline are close to the properties of ‘’aged” samples. Experimental data obtained in a rheometer can be used successfully for the prediction of the transport characteristics of the emulsion in pipelines for pipes of large diameter. The choice of the flow curve parameters is not crucial for the results of calculations of transport characteristics of the emulsion.
TL;DR: In this paper, the authors present the results of non-equilibrium molecular dynamics simulations of planar shear flow and planar elongational flow of melts of model linear chain molecules, in which the number of beads per molecule is varied from N = 4 to 50.
Abstract: We present the results of non-equilibrium molecular dynamics simulations of planar shear flow and planar elongational flow of melts of model linear chain molecules, in which the number of beads per molecule is varied from N = 4 to 50. The shear viscosity η, normal stress coefficients Ψ1 and Ψ2, and the two planar elongational viscosities η1 and η2 have been computed as a function of strain rate. The results are analysed using the third order retarded motion expansion (RME). The limiting zero strain rate values of the viscosity ratios agree with their expected values: η1/η = 4 and η2/η1 = 0.5. At low N, values of the coefficient of the lowest order nonlinear term in the RME obtained independently from the shear flow simulations and the elongational flow simulations also agree. However, the consistency check fails for a higher order retarded motion coefficient. This is attributed to insufficient data at low strain rates for the higher values of N. The N-dependence of the viscosities and normal stress coefficients as well as higher order RME constants is studied. We find that the zero strain rate values of the shear viscosity and both elongational viscosities are approximately proportional to N and the limiting values of the first and second normal stress coefficients Ψ1 and Ψ2 are approximately proportional to N 3 . The higher order RME constants have exponents nearer to 6.
TL;DR: In this article, numerical simulations were carried out to determine the flow characteristics of a Herschel-Bulkley viscoplastic fluid around a cylinder in an infinite medium, and the location, dimension and kinematics of the rigid zones were determined.
Abstract: The purpose of the numerical simulations carried out in this study is to determine the flow characteristics of a Herschel–Bulkley viscoplastic fluid around a cylinder in an infinite medium. Inertia is assumed to be negligible. Two types of boundary conditions are considered: the fluid adheres or slips (zero tangential stress) on the cylinder wall. Finite-element modelling involves regularising the Herschel–Bulkley model, as proposed by Papanastasiou [J. Rheol. 31 (1987) 385]. The effect of the yield stress value and shear-thinning index on the kinematic field and drag exerted on the cylinder were explored systematically. The location, dimension and kinematics of the rigid zones were determined. The results are compared with available theoretical data.
TL;DR: In this article, the analysis of the linear theory of capillary instability of threads of Maxwell fluids of diameter D is carried out for the unapproximated normal mode solution and for a solution based on viscoelastic potential flow.
Abstract: Analysis of the linear theory of capillary instability of threads of Maxwell fluids of diameter D is carried out for the unapproximated normal mode solution and for a solution based on viscoelastic potential flow. The analysis here extends the analysis of viscous potential flow [Int. J. Multiphase Flow 28 (2002) 1459] to viscoelastic fluids of Maxwell type. The analysis is framed in dimensionless variables with a velocity scale based on the natural collapse velocity V=γ/μ (surface tension/liquid viscosity). The collapse is controlled by two dimensionless parameters, a Reynolds number J=VDρ/μ=ργD/μ2=(Oh)2 where Oh is the Ohnesorge number, and a Deborah number Λ1=λ1V/D where λ1 is the relaxation time. The density ratio ρa/ρ and μa/μ are nearly zero and do not have a significant effect on growth rates. The dispersion relation for viscoelastic potential flow is cubic in the growth rate σ and it can be solved explicitly and computed without restrictions on the Deborah number. On the other hand, the iterative procedure used to solve the dispersion relation for fully viscoelastic flow fails to converge at very high Deborah number. The growth rates in both theories increase with Deborah number at each fixed Reynolds number, and all theories collapse to inviscid potential flow (IPF) for any fixed Deborah number as the Reynolds number tends to infinity.
TL;DR: In this paper, the deformation-rate-dependent rheology is described by power law and Carreau models, and the scaling exponents that determine the variation with time remaining to breakup of the bridge radius or radial length scale, axial length scale and axial velocity are evaluated by a combined analytical and numerical approach.
Abstract: Pinch-off dynamics of slender liquid bridges of generalized Newtonian fluids without and with inertia are studied using asymptotic analysis and numerical computation. The deformation-rate-dependent rheology is described by power law and Carreau models. Because the bridges are slender, their dynamics are governed by a pair of spatially one-dimensional (1D), non-linear evolution equations for the bridge shape and axial velocity. A bridge of a power law fluid under creeping flow conditions exhibits self-similar dynamics in the vicinity of the axial location where the bridge radius is a minimum. The scaling exponents that determine the variation with time remaining to breakup of the bridge radius or radial length scale, axial length scale, and axial velocity are evaluated by a combined analytical and numerical approach. Similarity solutions are obtained by collapsing numerically computed profiles of both the bridge shape and the axial velocity in the vicinity of the axial location where the bridge radius is minimum by rescaling of the transient profiles with radial and axial scalings deduced from theory. This scaling behavior is transitory and inertial effects become significant as pinch-off is approached. Thereafter, a new balance is established between viscous, capillary, and inertial forces that leads to a new self-similar regime which persists until pinch-off. The scaling exponents appropriate to this regime are also determined. Moreover, it is shown theoretically that interface shapes in the vicinity of the singularity are non-slender for values of the power law exponent below 2/3. Similarity solutions are once again obtained in the same manner as that used in the creeping flow limit. Low-viscosity bridges of Carreau fluids are known to exhibit a transition from potential flow (PF) scaling to Newtonian scaling. Here it is demonstrated that high-viscosity bridges of Carreau fluids exhibit a transition from power law scaling to Newtonian scaling. The point of transition between the latter two regimes is predicted in terms of parameters of the Carreau model.
TL;DR: In this paper, a detailed experimental study on the flow characteristics of various vortex shedding regimes was carried out for the flow of non-Newtonian fluids around a cylinder, which had varying degrees of shear-thinning and elasticity.
Abstract: A detailed experimental study on the flow characteristics of various vortex shedding regimes was carried out for the flow of non-Newtonian fluids around a cylinder. The fluids were aqueous solutions of carboxymethyl cellulose (CMC) and tylose at weight concentrations ranging from 0.1 to 0.6%, which had varying degrees of shear-thinning and elasticity. Two cylinders of 10 and 20 mm diameter were used in the experiments, defining an aspect ratio of 12 and 6 and producing blockages of 5 and 10%, respectively. The Reynolds number (Re) ranged from 50 to 9 × 10 3 . Shear-thinning gave rise to a decrease of the cylinder boundary-layer thickness and to a reduction of the diffusion length (ld), which raised the Strouhal number, St. In the laminar shedding regime, a modified Strouhal number was successful at overlapping the shedding frequency variation with the Reynolds number for the various solutions. In contrast, fluid elasticity was found to increase the formation length ( lf ), and this contributed to a decrease of the Strouhal number. The overall effect of shear-thinning and elasticity was an increase in the Strouhal number. The increase in polymer concentration and the corresponding increase in fluid elasticity were responsible for the reduction of the critical Reynolds number marking the sudden decrease of the formation length, Relf . In the shear layer transition regime, the formation length and Strouhal number data collapsed onto single curves as function of a Reynolds number difference, which confirmed Coelho and Pinho (J. Non-Newtonian Fluid Mech. (2003), accepted for publication) finding that an important effect of fluid rheology was in changing the demarcations of the various flow regimes. © 2003 Elsevier Science B.V. All rights reserved.
TL;DR: In this paper, the deformation of a Boger fluid under steady-state slow shear flow is investigated by video-enhanced microscopy and image analysis, and an optical determination of the first normal stress difference of the matrix fluid can be obtained.
Abstract: The small deformation of a single drop in a Boger fluid under steady-state slow shear flow is here investigated by video-enhanced microscopy and image analysis. Data are compared to predictions of a recent perturbation analysis, dealing with the drop-in-a-flow-field problem for viscoelastic fluid components [J. Non-Newt. Fluid Mech. 107 (2002) 111]. The main experimental result, in agreement with theory, is that drop orientation towards the flow direction is significantly enhanced as compared to the Newtonian case. In fact, based on this result, an optical determination of the first normal stress difference of the matrix fluid can be obtained. Furthermore, it is shown here that the interfacial tension of the fluid pair can be readily obtained by comparison between the theoretical predictions and the optical data taken in both the “side” and “top” views (i.e. along the vorticity and velocity gradient direction, respectively) of the deformed drop.
TL;DR: In this paper, a modification of the dissipation term in the Leonov model using relaxation time-dependent adjustable parameters is proposed to increase the model capability to represent elongational flow behavior, and the fitting/predicting capabilities of the proposed modified model are compared with the eXtended Pom-Pom model and modified White-Metzner model in steady shear and uniaxial extensional flows of LDPE, mLLDPE and PVB melts in a wide range of deformation rates.
Abstract: A slight modification of the dissipation term in the Leonov model using relaxation time-dependent adjustable parameters is proposed to increase the model capability to represent elongational flow behavior. The fitting/predicting capabilities of the proposed modified Leonov model are compared with the eXtended Pom–Pom model and modified White–Metzner model in steady shear and uniaxial extensional flows of LDPE, mLLDPE and PVB melts in a wide range of deformation rates. The input low-shear-rate viscosity and first normal stress coefficient data was measured on the advanced rheometric expansion system (ARES) Rheometrics parallel-plate rheometer, whereas the RH7-2 capillary rheometer was used for the determination of shear viscosity (capillary), first normal stress coefficient (slit die) and uniaxial extensional viscosity (Cogswell method). A newly developed ‘effective entry length correction’ was applied to deal with all uniaxial extensional viscosity data.
TL;DR: In this paper, the authors present energy estimates for the stresses and velocity components in a general setting, for both inertial and inertialess flows of an Oldroyd B fluid.
Abstract: In this paper, we present energy estimates for the stresses and velocity components in a general setting, for both inertial and inertialess flows of an Oldroyd B fluid. Our results apply to flows in bounded domains in any number of dimensions, subject to Dirichlet and possibly inflow boundary conditions. A novel numerical scheme is introduced and shown to be superior to a conventional Galerkin discretization of the Oldroyd B equations. In particular, the new scheme respects the derived energy estimates and guarantees positive definiteness of the stress tensor τ +((1−β)/We) I at all times, β being a solvent-to-total viscosity ratio and We a Weissenberg number. Numerical results for the planar viscoelastic Poiseuille problem illustrate some differences between the new and conventional schemes and reveal that the conventional scheme may lead to violation of the theoretical energy bounds in certain circumstances.
TL;DR: In this article, a low Reynolds number k-e turbulence model is developed for predicting turbulent wall flows of viscoelastic fluids, and a new damping function is introduced to account for near-wall effects.
Abstract: A low Reynolds number k–e turbulence model is developed for predicting turbulent wall flows of viscoelastic fluids. The model uses a non-linear molecular viscosity that is affected by the turbulent fluctuations and a new damping function is introduced to account for near-wall effects. This new function was made equal to the eddy viscosity damping function which was derived taking into account viscometric and elastic effects. Flow predictions compare favourably with results from experiments with several viscoelastic fluids, especially the friction factor and the mean velocity. Comparisons of the turbulence kinetic energy are less good, but the model is able to capture the shift of the peak turbulence kinetic energy and rate of dissipation away from the wall, the decrease in those peak values and of the production of k, and the Reynolds shear stress deficit across the pipe. An advantage of the present single-point turbulence closure relative to previous attempts at modelling polymer drag reduction, is the fact that here the input is only the mean velocity and fluid properties and no other modifications of the turbulence model are required to deal with different fluids. Further developments of the turbulence model are suggested at the end.
TL;DR: In this paper, a closed model for the time-average viscosity is proposed that takes into account its nonlinearity and dependence on the second and third invariants of the fluctuating rate of deformation tensor.
Abstract: Based on a generalised Newtonian fluid (GNF) model, modified to account for strain-thickening of the extensional viscosity, this paper derives transport equations for mass, momentum, Reynolds stresses, turbulent kinetic energy and its rate of dissipation. An analysis of order of magnitude identifies the relevant new terms and suggestions are made to model those terms needed to ensure closure in the perspective of a low Reynolds number k–e model. Specifically, a closed model for the time-average viscosity is proposed that takes into account its non-linearity and dependence on the second and third invariants of the fluctuating rate of deformation tensor. The turbulence model is qualitatively shown to increase the rate of decay of turbulent kinetic energy in isotropic grid turbulence for certain rheological conditions. The performance of the turbulence model in a pipe flow is assessed in a companion paper by Cruz and Pinho [J. Non-Newtonian Fluid Mech., in press]. © 2003 Elsevier B.V. All rights reserved.
TL;DR: In this paper, a non-monotonic slip equation was proposed for the Poiseuille and extrudate-swell flows of a shear-thinning fluid that obeys the Carreau constitutive model.
Abstract: We solve the time-dependent, compressible Poiseuille and extrudate-swell flows of a shear-thinning fluid that obeys the Carreau constitutive model, using finite elements in space and a fully-implicit scheme in time. Slip is assumed to occur along the die wall following a non-monotonic slip equation that relates the wall shear stress to the slip velocity and is based on experimental measurements with polyethylene melts. Thus, the resulting flow curve is also non-monotonic, and consists of two stable positive-slope branches and a linearly unstable negative-slope branch. The steady-state numerical results compare well with certain analytical solutions for Poiseuille flow. The time-dependent calculations at fixed volumetric flow rates demonstrate the existence of periodic solutions in the unstable regime, due to the combination of compressibility and slip. Self-sustained oscillations of the pressure-drop and of the mass-flow rate are obtained. In the extrudate region, high-frequency, small amplitude waves are generated on the free-surface, which also oscillates radially. The wavelength and the amplitude of the free-surface waves and the amplitude of the oscillations in the radial direction are reduced, as the Reynolds number is decreased and approaches the conditions of the experiments.
TL;DR: In this paper, the rheological behavior of Brownian rigid-rod particles (fibres) in turbulent channel flow is investigated by numerical means and the response of the conformation distribution function of ensembles of passive Brownian fibres along these Lagrangian pathes is computed by a stochastic simulation using Jeffery's equation (1922) and the Rheological theory of dilute suspensions of fibres in Newtonian solvents.
Abstract: The rheological behaviour of Brownian rigid-rod particles (fibres) in turbulent channel flow is investigated by numerical means. From a direct numerical simulation (DNS) of turbulent channel flow, Lagrangian time traces of the velocity derivative tensor, as experienced by small inertia-free particles, are generated. The response of the conformation distribution function of ensembles of passive Brownian fibres along these Lagrangian pathes is computed by a stochastic simulation using Jeffery’s equation (1922) and the rheological theory of dilute suspensions of fibres in Newtonian solvents of [Int. J. Multitphase Flow 1 (2) (1974) 195]. Results are presented for the influence and importance of turbulent fluctuations and Brownian motion on the moments of the distribution function and stresses generated by the presence of the particles. The stresses peak in the buffer layer and it is found that stress levels as well as rms of stresses rise quickly with the square of the fibre aspect ratio. At aspect ratios of more than about r =100, the stress distributions remain qualitative similar. Smaller fibres generate higher average stress levels but lower stress fluctuation levels than larger fibres for which the effect of Brownian motion is weak. Large normal stresses always come in hand with large shear stresses showing that, in such suspensions possible drag reduction will always be connected to a significant stress deficit.
TL;DR: In this article, a wide range of Bingham numbers using the continuous modification of Papanastasiou for the Bingham model is presented, where axisymmetric geometries give smaller unyielded regions than planar ones.
Abstract: Squeeze flow of Bingham plastics shows small unyielded regions confined only near the center of the disks. Previous simulation results on axisymmetric coaxial disks are extended to aspect ratios ranging from 0.01 to 1, and to planar parallel plates of infinite width. New results are given for a wide range of Bingham numbers using the continuous modification of Papanastasiou for the Bingham model. Axisymmetric geometries give smaller unyielded regions than planar ones, while big aspect ratios give larger unyielded regions than small ones for the same Bingham number. Calculations of the squeeze force along disks or plates for different aspect ratios are also given and fitted to easy-to-use equations.
TL;DR: In this paper, it was shown that the Giesekus term in the XPP model has practically no effect at any deformation rate except the generation of a second normal stress difference at small shear rates.
Abstract: It is shown that several (single-mode) models (PTT, XPP) are basically special cases of the general network model. The XPP model [J. Rheol. 45 (2001) 823] is shown to give an identical response to a PTT model at high elongational rates, but the models differ in shear flow at high shear rates. The Giesekus term in the XPP model has practically no effect at any deformation rate except the generation of a second normal stress difference at small shear rates. The possibility of tailoring the shear flow response at high shear rates is explored. The results for the models are also given to show the effect of branching on the response in both transient and steady flows—elongation, shear, planar elongation and biaxial deformation. Finally, the fitting of data with multi-mode models is investigated.
TL;DR: In this article, the interactions of two identical rigid spheres of radius R translating in a Bingham material in creeping flow along their line of centers are calculated at various sphere separations using the finite element method.
Abstract: The interactions of two identical rigid spheres of radius R translating in a Bingham material in creeping flow along their line of centers are calculated at various sphere separations using the finite element method. The yield surfaces are determined by an extrapolation using a regularized constitutive model. Two spheres falling in a line interact at separations greater than that corresponding to the superposition of yield surfaces for isolated spheres falling in an unbounded medium. Three distinct regimes are identified: for L/R>5.5, the spheres move in separate yield envelopes; for 4
TL;DR: In this article, the dynamics of dilute solutions of finitely extensible nonlinear elastic (FENE) bead-spring dumbbell and chain models in start-up and relaxation of uniaxial elongational and shear flow and in the flow kinematics obtained from a DNS database of a turbulent channel flow is investigated for Weissenberg numbers (We) of 1, 10, and 100.
Abstract: The dynamics of dilute solutions of finitely extensible nonlinear elastic (FENE) bead-spring dumbbell and chain models in start-up and relaxation of uniaxial elongational and shear flow and in the flow kinematics obtained from a DNS database of a turbulent channel flow is investigated for Weissenberg numbers (We) of 1, 10, and 100. Five FENE models are considered: FENE chain, FENE dumbbell, FENE-P chain, FENE-P dumbbell, and FENE-PM chain. The FENE chain is used as the standard of reference to evaluate all coarse-grained models. It is shown that in transient elongational flow, pre-averaged models incur large errors while the FENE dumbbell, with appropriate selection of the bead-spring parameters, can provide reasonably accurate predictions once the polymer has unravelled. Faithful prediction of the dynamics prior to polymer unravelling requires a multi-mode model. In transient shear flow, none of the coarse-grained models studied can correctly replicate the dynamics predicted by the FENE chain over the entire range of We considered. The flow in wall-bounded turbulence is shown to be predominantly simple shear within the viscous sublayer and a mixture of simple shear and elongational flow in the buffer layer. The dominant contributions to polymer stress arise from patches of biaxial or uniaxial elongational flow encountered in the buffer layer. The Weissenberg number is the critical parameter which determines polymer stretching, stresses, and drag reduction in a given polymer–solvent system. Effective drag reduction requires a large polymer extensibility and Weτ∼O(100) or higher. Comparison of the predictions of different models reveals that the FENE dumbbell, with appropriate selection of the bead-spring parameters, can provide reasonably accurate predictions of the polymer dynamics in turbulent flow at high Weτ. Pre-averaged models, while in qualitative agreement with the FENE chain, are quantitatively inaccurate and over-estimate the polymer stresses by up to 200–400% in regions of strong polymer stretching. These results suggest that the most promising approach to accurate computation of polymer drag reduction at present is through stochastic simulations.
TL;DR: In this paper, a mean-field model for a reversible network of telechelic chains is proposed, and the stochastic differential equations for this model are solved numerically using a standard Brownian dynamics method.
Abstract: A reversible network of associative, telechelic chains is represented by a mean-field model similar to that proposed recently by Vaccaro and Marrucci [J. Non-Newtonian Fluid Mech. 92 (2000) 261]. The model contains neither the topology of the network nor explicit interactions between different chains. Instead, it consists of two separate ensembles of dumbbells. One of these ensembles represents the ‘active’ chains, which are connected by both ends to other chains and carry most of the stress on the system. The other ensemble represents ‘dangling’ chains, connected to the network by one end only. Association of dangling chains to the network, to become active, and dissociation, the reverse process, are simulated by appropriate transition rules. The stochastic differential equations for this model are solved numerically using a standard Brownian dynamics method. This circumvents the need for (questionable) closure approximations to solve analytically the equivalent Fokker–Planck equations. Under simple shear flow, this system shows the main characteristics of a reversible network of telechelic chains, e.g. a Newtonian plateau, shear thickening and shear thinning. The simulation results confirm some of the predictions of Vaccaro and Marrucci.
TL;DR: In this paper, the authors studied the flow field around air bubbles rising in aqueous polyacrylamide (PAAm) solutions using a particle image velocimetry (PIV) system.
Abstract: The flow field around air bubbles rising in aqueous polyacrylamide (PAAm) solutions was studied using a particle image velocimetry (PIV) system. This flow was analyzed in the vicinity of the critical bubble volume where the discontinuity of the terminal bubble velocity occurs. It was found that the flow configuration changes drastically below and above the critical bubble volume. The flow just below the critical bubble volume shows an upward flow at the front and back of the bubble with a symmetrical vortex around the bubble. The flow field obtained for a volume just above the critical one shows the appearance of the so-called negative wake behind the bubble. Additionally, it was found that the container walls affect significantly the magnitude of the terminal velocity as well as the velocity jump. However, the critical volume at which the velocity jump appears does not change for different container sizes.