Squeezing flow of viscoplastic fluids subject to wall slip

Polymer processing operations such as compression molding, sheet forming and injection molding can be modeled by squeezing flows between two approaching parallel surfaces in relative motion. Squeezing flows also find applications in the modeling of lubrication systems, and in the determination of rheological properties. Here, analytical solutions are developed for the constant-speed squeezing flow of viscoplastic fluids. It is assumed that the fluid is purely viscous, and hence viscoelastic effects unimportant. The rheological behavior of the viscoplastic fluids is represented by the Herschel-Bulkley viscosity function. The deformation behavior of commonly encountered viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which is a function of the wall shear stress. The slip coefficient that relates the slip velocity to the shear stress is affected by the material of construction and also the roughness of the solid surfaces, leading to the possibility of different slip coefficients at various solid surfaces. The model developed in this study accommodates the use of different slip coefficients at different solid surfaces. The accuracy of the solutions is established, and the effects of various parameters such as slip coefficient and apparent yield stress are examined. The solutions provide useful design expressions that can be utilized for squeezing flows of viscoplastic fluids, with or without wall slip at the solid boundaries.     

Single screw extrusion of viscoplastic fluids subject to different slip coefficients at screw and barrel surfaces

Commonly encountered viscoplastic fluids including concentrated suspensions of polymeric and ceramic composites, foams, gels, concrete, food products, and energetic compounds exhibit wall slip during their flow and processing. For some viscoplastics fluids, especially highly filled suspensions, wall slip may dominate the flow and deformation and hence the processing behavior of the suspension. The wall slip velocity is generally a function of the wall shear stress and temperature. Various factors including the materials of construction i.e., chemical nature and the roughness of the wall surface affect the wall slip behavior of viscoplastic fluids. In this study an analytical model of the extrusion of viscoplastic fluids under isothermal and fully-developed conditions in shallow channels is developed. The model accommodates the use of different slip coefficients at barrel and screw surfaces. It thus permits the investigation of effects of introducing different materials of construction for the barrel and screw surfaces and development of design expressions.     

An analytical model for steady coextrusion of viscoplastic fluids in thin slit dies with wall slip

Coextrusion is widely used to fabricate multilayered products with each layer providing a separate functionality, including barrier resistance to gases, strength, and printability. Here an analytical model of the coextrusion die flow of two incompressible, viscoplastic fluids in a slit die, subject to nonlinear wall slip and under fully developed and isothermal conditions, is developed to allow the prediction of the steady‐state velocity and shear stress distributions and the flow rate versus pressure gradient relationship. The resulting model is applied to the coextrusion of two layers of viscoplastic fluids in a thin rectangular slit die (slit gap, h ? slit width, W). The analytical solution recognizes a number of distinct flow conditions (eleven cases) that need to be treated separately. The solutions for all eleven cases are provided along with an apriori identification methodology for the determination of the applicable case, given the shear viscosity and wall slip parameters of the two viscoplastic fluids, the slit geometry and the flow conditions. Simplifications of the model would provide the solutions for the fully developed and isothermal coextrusion flows of any combination of Hershel‐Bulkley, Bingham, power‐law and Newtonian fluids with or without wall slip at one or both walls of the slit die. POLYM. ENG. SCI., 2010. © 2009 Society of Plastics Engineers     

Turbulence structure for plane Poiseuille–Couette flow and implications for drag reduction over surfaces with slip

Direct numerical simulations were used to simulate plane channel and plane Poiseuille–Couette flows. For Poiseuille–Couette flow, the walls of the channel were moving with a specified velocity. This is equivalent to forcing a slip velocity at the wall of the channel, and such flow behaviour can be viewed as the effect due to an ultra‐hydrophobic wall. It was found that the location of the zero Reynolds stress value shifted towards the wall moving in the streamwise direction. The near‐wall eddies were found to be longer and weaker than for the plane‐Poiseuille channel flow. It appears that such an eddy structure can lead to turbulence drag reduction.     

FLOW OF VISCOPLASTIC FLUIDS ON AN INCLINED PLANE: EVALUATION OF YIELD STRESS

Within the general category of so called generalized Newtonian fluids, there exists a class of materials which do not deform until the applied shear stress exceeds a critical value called yield stress. Conversely, such a material behaves like a solid as long as the shear stress is less than the yield stress. This class of materials is called viscoplastic fluids. The question whether the true yield stress exists or not is indeed far from settled (1), but the notion of a yield stress has proved to be quite useful in practice in describing the steady shear rheological behaviour of a range of materials especially of particulate suspensions. Consequently, yield stress appears as a parameter in all constitutive relations (which purport to describe the steady shear behaviour of viscoplastic materials), and its evaluation is important before an engineering flow problem can be solved. Conversely, there are some simple hydrodynamic situations which allow the value of the yield stress to be extracted from macroscopic quantities such as flow rate-pressure drop data. In this paper, we examine this possibility using gravity driven flow of a viscoplastic material on an inclined plane.     

Effects of wall slip on the rheological measurement and extrusion die design of a filled rubber compound

We investigate the rheological properties of a filler-reinforced rubber compound using a rotorless shear rheometer specially designed for rubber and a rotational rheometer with parallel-plate geometry. Our aim is to evaluate the effects of wall slip on both the oscillatory and steady shear modes of the rotational rheometer. The rheological measurements show that the slip generally does not affect the oscillatory shear but does exist in the steady shear flow and tends to make the measured shear viscosity lower than the true value. Besides, we extend the investigation into engineering applications. The extrusion die for a given extrudate profile is obtained using a finite-element model in which the wall slip is considered as a boundary condition. To validate the die design, an extrusion experiment is carried out and the results confirm that, for filled elastomers, it is necessary to consider the effect of wall slip in the extrusion die design.     

Effect of wall slip on the uniformity of flow from sheeting extrusion dies

A theoretical study for analyzing the uniformity of flow from sheeting extrusion dies is presented. In this study it is assume that a slip condition exists at the wall of the die, the magnitude of slip velocity is proportional to the shear stress at the wall, the flow is isothermal and steady state, and a power law model is valid for viscosity. Two extrusion dies, T-dies and coat-hanger dies, are examined. The flow uniformity at the exit of the die is calculated and compared with that for a nonslip analysis. The discrepancies between the slip and nonslip models imply that the wall slip condition induces a significant nonuniform flow distribution. Traditional design criticism based on the nonslip model are invalid for flow with the wall slip condition, and it is necessary to increase the length of the die land to even the flow distribution at the exit of the die.     

Rheological behavior of zirconia feedstock flowing through various channels considering wall-slip

The existence of wall slip for ZrO₂ feedstock flow in micro powder injection molding was investigated based on capillary rheometer experiments using dies of three dimensions. A power law function was derived by data fitting to determine the wall slip velocity based on which numerical simulation was carried out to explore the influence of wall slip on micro injection molding. Experimental results indicate that the feedstock is less sensitive to temperature fluctuation at higher shear rates. Power-law model can provide higher accuracy than the modified Cross model to depict the rheological behavior of the feedstock in capillary flows with different channels. Numerical simulation results show that in case of steady flow higher dynamic viscosity of the feedstock and higher pressure losses of the flow appeared when the wall slip boundary was included as compared to no-slip assumption in micro powder injection molding. This is because that when the wall slip boundary was included the shear rate distribution of the feedstock was lower than that of the feedstock assuming no-slip boundary.     

A mechanism for extrusion instabilities in polymer melts

A mechanism for explaining some of the instabilities observed during the extrusion of polymer melts is further explored. This is based on the combination of non-monotonic slip and elasticity, which permits the existence of periodic solutions in viscometric flows. The time-dependent, incompressible, one-dimensional plane Poiseuille flow of an Oldroyd-B fluid with slip along the wall is studied using a non-monotonic slip equation relating the shear stress to the velocity at the wall. The stability of the steady-state solutions to one-dimensional perturbations at fixed volumetric flow rateis analyzed by means of a linear stability analysis and finite element calculations. Self-sustained periodic oscillations of the pressure gradient are obtained when an unstable steady-state is perturbed, in direct analogy with experimental observations.     

Dynamic slip and nonlinear viscoelasticity

Several significant problems arise when film is fabricated on a large scale. One of these is the appearance of irregularities on the extrudate surfaces when the polymer melt is extruded at high rates. These irregularities vary in intensity and form and are generally known as sharkskin melt fracture. This phenomenon, which occurs when the wall shear stress exceeds a critical value, is a limiting factor for production rates in many industrial extrusion operations such as film blowing of polyethylene. We used a sliding plate rheometer incorporating a shear stress transducer to study slip in both steady and unsteady flows. By combining a dynamic slip model with a nonlinear viscoelastic constitutive model, we determined the slip model parameters for LLDPE film resin with and without a fluoropolymer sharkskin suppressant. The models give good prediction of our slip data in steady shear but show insufficient gap dependence in exponential shear. Our own film blowing studies demonstrated the efficiency of the sharkskin suppressant; it has more than doubled the throughput in our laboratory setup. The fluoropolymer additive was found to profoundly affect both the steady and dynamic slip parameters. Hence, the sharkskin suppressant alters how the LLDPE remembers its past slipping motions.     

Process viscometry of complex fluids and suspensions with helical ribbon agitators

The viscosity of highly inelastic shear thinning fluids and aqueous suspensions of kaolin clay particles has been investigated using a helical ribbon impeller fitted to a rheometer. Viscosity data for the single phase non-Newtonian fluids adequately processed with a generalized Reynolds number based on the impeller tip speed are shown to superimpose very well to the results obtained with a cone and plate rheometer. In the case of the two-phase system, it is shown that the data treatment for single phase system can be extended. The helical ribbon impeller yields more stable viscosity values than with the traditional geometries and no spurious flow phenomena (i.e., sedimentation, slip at the wall, etc.) was observed, making this system a superior device for suspension rheology over cone and plate and Couette flow rheometers.     

EFFECT OF COMBINED SHEAR,SECONDARY AND AXIAL FLOW OF BLOOD ON OXYGEN UPTAKE

The oxygen uptake of red blood cells (RBC) has been investigated in a blood flowing model in which defined shear and secondary flows are produced with blood flow direction perpendicular to the shear plane. The model consists of a Couette flow realized by means of two coaxial cylinders with gas permeable walls. The “venous” blood flows in the axial direction. In the annual gap shear and secondary flows (Taylor-vortices) are produced by rotation of the inner cylinder which determine the amount of oxygen uptake. Different blood Row regions and conditions (Taylor-, Reynolds-number, and shear rates) in regard to non-Newtonian behaviour of blood have been reported. An increase of Taylor-number from 100 to 300 results in a greater than four-fold increase in oxygen uptake. For Taylor-numbers lower than 100 the oxygen uptake is not significantly enhanced due to flow instabilities. This can be explained by non-Newtonian flow behaviour of blood and appearance of different flow regions. The results for normal whole blood are compared to those for water and for artificially regidified red cell membranes.     

管壁厚度对微挤出成型的影响分析

基于Bird-Carreau黏度模型,运用有限元方法对三维等温微管挤出成型流动模型进行了数值分析,主要研究了管壁厚度对微管挤出成型过程中挤出胀大、速度分布、剪切速率和口模压降等重要指标的影响。结果表明,当熔体入口体积流率相等时,随着管壁厚度的增大,挤出物挤出胀大率和横截面尺寸变化量增大;口模出口端面上熔体的二次流动增强,但挤出速度和剪切速率减小;熔体在口模内的压力降明显下降;适当增加管壁厚度,有利于提高微管挤出质量。     

Wall slip heating

When molten plastic is extruded, the upper limiting throughput is often dictated by fine irregular distortions of the extrudate surface. Called sharkskin melt fracture, plastics engineers spike plastics formulations with processing aids to suppress these distortions. Sharkskin melt fracture is not to be confused with gross melt fracture, a larger scale distortion arising at throughputs higher than the critical throughput for sharkskin melt fracture. Sharkskin melt fracture has been attributed to a breakdown of the no slip boundary condition in the extrusion die, that is, adhesive failure at the die walls, where the fluid moves with respect to the wall. In this article, we account for the frictional heating at the wall, which we call slip heating. We focus on slit flow, which is used in film casting, sheet extrusion, curtain coating, and when curvature can be neglected, slit flow is easily extended to pipe extrusion and film blowing. In slit flow, the magnitude of the heat flux from the slipping interface is the product of the shear stress and the slip speed. We present the solutions for the temperature rise in pressure‐driven slit flow and simple shearing flow, each subject to constant heat generation at the adhesive slip interface, with and without viscous dissipation in the bulk fluid. We solve the energy equation in Cartesian coordinates for the temperature rise, for steady temperature profiles. For this simplest relevant nonisothermal model, we neglect convective heat transfer in the melt and use a constant viscosity. We arrive at a necessary dimensionless condition for the accurate use of our results: Pé?1. We find that slip heating can raise the melt temperature significantly, as can viscous dissipation in the bulk. We conclude with two worked examples showing the relevance of slip heating in determining wall temperature rise, and we show how to correct wall slip data for this temperature rise. POLYM. ENG. SCI., 55:2042–2049, 2015. © 2014 Society of Plastics Engineers     

Slow and intermediate flow of a frictional bulk powder in the Couette geometry

Most current research in the field of dry, non-aerated powder flows is directed toward rapid granular flows of large particles. Slow, frictional, dense flows of powders in the so-called quasi-static regime were also studied extensively using Soil Mechanics principles. The present paper describes the rheological behavior of powders in the “intermediate” regime lying between the slow and rapid flow regimes. Flows in this regime have direct industrial relevance. Such flows occur when powders move relative to solid walls in hoppers, bins and around inserts or are mixed in high and low shear mixers using moving paddles. A simple geometry that of a Couette device is used as a benchmark of more complicated flows.The constitutive equations derived by Schaeffer [J. Differ. Equ. 66 (1987) 19] for slow, incompressible powder flows were used in a new approach proposed by Savage [J. Fluid Mech. 377 (1998) 1] to describe flows in the intermediate regime. The theory is based on the assumption that both stress and strain-rate fluctuations are present in the powder. Using Savage's approach, we derive an expression for the average stress that reduces to the quasi-static flow limit when fluctuations go to zero while, in the limit of large fluctuations, a “liquid-like”, “viscous” character is manifested by the bulk powder.An analytical solution of the averaged equations for the specific geometry of the Couette device is presented. We calculate both the velocity profile in the powder and the shear stress in the sheared layer and compare these results to experimental data. We show that normal stresses in the sheared layer depend linearly on depth (somewhat like in a fluid) and that the shear stress in the powder is shear rate dependent. We also find that the velocity of the powder in the vicinity of a rough, moving boundary, decays exponentially so that the flow is restricted to a small area adjacent to the wall. The width of this area is of the order of 10-13 particle diameters. In the limit of very small particles, this is tantamount to a shear band-type behavior near the wall.     

Velocity measurements on a polypropylene melt during extrusion through a flat coat‐hanger die

An experimental investigation of various flow regimes observed during the extrusion of a polypropylene melt through a flat coat‐hanger die by laser‐Doppler velocimetry (LDV) is presented. LDV measurements of the velocity profiles across the gap of the die at various locations along the die reveal three different extrusion regimes. At small wall shear stresses, the velocity profiles can be fitted by symmetrical curves with the velocities becoming zero at the die walls. These profiles are not uniformly distributed along the die. An increase of the wall shear stress reveals a second flow regime characterized by a uniform distribution of the velocity profiles along the die. As the wall shear stress is increased even further, a third flow regime characterized by wall slip on the glass windows is observed. This flow regime is systematically characterized by measurements of the slip velocities at various temperatures and throughputs. The maximum velocities along the die are taken to assess the uniformity of flow which decisively influences the thickness of the extruded film. By measuring velocity profiles, at different throughput, and temperatures, the conditions for constant velocities along the die were determined. POLYM. ENG. SCI., 2012. © 2011 Society of Plastics Engineers     

THE FLOW OF MICROEMULSIONS THROUGH PACKED BEDS AND CAPILLARY TUBES

The flow behavior of water-in-oil microemulsions through beds packed with glass spheres was studied experimentally. The microemulsions used in this study exhibited a shear thinning viscosity described by a simple power law model. The flow of the microemulsions was accompanied by significant apparent slip effects, quantified by an effective slip velocity. The effective slip velocity increased with increasing surfactant concentration, but it differed in magnitude for the packed bed and the capillary tube flows.

In the absence of apparent slip effects, the capillary-power law (CPL) model predicted the superficial velocities in the packed beds with an average error of less than 6%. This provides a direct verification of the applicability of the capillary-power law model to inelastic shear thinning fluids in the absence of “anomalous” wall effects.     

STEADY SHEAR FLOW PROPERTIES OF HIGH SOLIDS SOFTWOOD KRAFT BLACK LIQUORS: EFFECTS OF TEMPERATURE, SOLIDS CONCENTRATIONS, LIGNIN MOLECULAR WEIGHT AND SHEAR RATE

The steady shear flow properties of several softwood kraft black liquors (slash pine) from a two level, four variable factorially designed pulping experiment were determined for solids concentrations from 50% to 85%, temperatures from 40°C to 140°C and shear rates up to 10,000 s-1 by using Instron capillary and Haake coaxial cylinder rheometers. It was shown that the slip velocity at the wall of the capillary is insignificant and that a two capillary method can be used to determine the viscosity of the samples. At high solids, black liquor can exhibit non-Newtonian behavior dependent upon temperature, solids concentrations, solids composition and shear rate. In general, the liquors behave as pseudoplastic fluids. The exact level of viscosity at any given condition is dependent upon the solids composition which will vary from liquor-to-liquor. The flow behavior of the liquors was described using power-law, Cross and Carreau-Yasuda models. Superposition principles developed for polymer melts and concentrated polymer solutions were applied to obtain reduced correlations for viscosity behavior of the liquors. By using a suitable reference temperature, related to the glass transition temperature of black liquors, a generalized WLF type shift factor was obtained for the liquors used in this study and can be used to obtain a reduced plot of viscosity behavior of other black liquors.