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     

Numerical flow simulation of viscoplastic fluids in annuli

This paper describes numerical solutions for the laminar flow of non-Newtonian fluids in vertical annuli using the Herschel-Bulkley model to describe the rheological behaviour of such materials. Numerical solutions have been obtained when there is both axial and tangential flows in either a concentric or eccentric annulus. The tangential flow arises from the rotation of the inner cylinder of the annulus and the axial flow from a constant axial pressure gradient. The flow is analysed by solving the momentum and continuity equation numerically using the finite element method. The dimensionless velocity, deformation and stress profiles with other quantities such as the apparent viscosity and pressure distribution have been calculated for various eccentricities, radius ratios, fluid properties and flow parameters; the results give insights into the flow behaviour in the annuli. It is shown that the inclusion of rotational effects, for a fixed pressure gradient, is likely to increase the axial volumetric flowrate over non-rotating situations in concentric geometries. New results reveal that, in eccentric annuli, the situation is reversed and the flowrate gradually decreases as the rotation rate is increased.     

压力驱动微流道内流动电势及壁面滑移效应

This study investigated the streaming potential and wall slip effects on pressure-driven liquid flow in hydrophobic microchannels.The Poisson-Boltzmann equation for the electrical double layer(EDL)and Navier-Stokes equation for incompressible viscous fluid were established.For those microchannels with high wall zeta potential, the traditional Debye-H點kel linear approximation for solving the potential distributions of EDL would produce big error, therefore, analytical expression for potential distributions and Navier slip boundary condition were introduced to solve the N-S equation analytically, then analytical solution of streaming potential could be obtained by using the electrical current balancing condition.The influences of electrokinetic parameter(K), wall zeta potential and slip coefficient on streaming potential and velocity distributions were discussed in detail.The results showed that streaming potential decreased with increasing electrokinetic parameter, while increased significantly with increasing slip coefficient.It also tended to reach a maximum value at a certain zeta potential and then decreased rapidly with increasing zeta potential.Streaming potential and wall slip both affected fluid flow in microchannels, the former retained the development of liquid flow, but the latter accelerated flow velocity.Wall slip effect played a major role at lower zeta potentials, that is, flow velocity increased at lower zeta potentials when the combined effects of streaming potential and hydrodynamic slippage appeared in microchannels.Wall slip velocity gradually reduced to zero at higher zeta potential, then wall slip effect on pressure-driven flow in microchannels could be ignored.     

Viscosity corrections for concentrated suspension in capillary flow with wall slip

Corrections for viscosity measurements of concentrated suspension with capillary rheometer experiments were investigated. These corrections include end effects, Rabinowitsch effect, and wall slip. The effects of temperature, particle concentration, and contraction ratio on the end effects were studied and their effects were accounted for using an entrance and exit losses model. The non‐Newtonian effect and the nonlinearity of slip velocity against wall shear stress were described using a slip model. The true viscosity of a concentrated suspension with glass powder suspended in a non‐Newtonian binder system was calculated as a function of shear rate and effective particle concentration, taking into consideration particle migration, which is calculated by a diffusive numerical model. Particle size was found to affect significantly the viscosity of the suspension with viscosity decreasing with increasing particle size, which can be reflected by a decrease in the value of the power‐law index in the Krieger model. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

The onset of wall slip and sharkskin melt fracture in capillary flow

The capillary die flow of high density and linear low density polyethylenes is simulated under slip conditions to investigate the origin of sharkskin melt fracture. As suggested in the literature, it is shown that sharkskin originates at the exit of the die and is due to the acceleration (high stretching rate) of the melt as it exits the die. It is also shown that both adhesion and slip promoters eliminate surface defects by decreasing the stretching rate of the polymer melt at the exit region of the die. The effect of length-to-diameter ratio of the die on the sharkskin melt fracture is also examined. It is found that sharkskin is more pronounced in short dies which is in accord with experimental observations. Finally, it is suggested that applied pressure at the capillary exit suppresses surface defects.     

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.     

On creeping drag flow of a viscoplastic fluid past a circular cylinder: wall effects

The creeping drag flow of a Bingham plastic past a circular cylinder kept symmetrically between parallel plates was analyzed via numerical simulations with the finite element method. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous analyses for creeping drag flow past a cylinder in an infinite medium based on variational principles and provide calculations of the drag coefficient around a cylinder in the case of wall effects.     

Entrance laminar flows of viscoplastic fluids in concentric annuli

Developing flows of generalized Bingham (Herschel-Bulkley) fluids in concentric annuli were studied numerically. A control volume approach based upon an upwinding finite difference technique was used to solve the equation of motion. The results in terms of velocity and pressure drop profiles are shown graphically. Radius ratios of 0.02, 0.2, 0.4 and 0.6; power-law indices (n) of 0.7, 1.0 and 1.2; generalized Bingham numbers of 5, 10 and 15 were investigated. At present, there are no experimental results with which to make comparisons. However, there are results for fully developed flows and comparison has been made with these. In all cases the agreement was good.     

Rayleigh-Jeffreys stability of thermohaline stratified fluids subject to vertical flows

The stability of an infinite horizontal layer of fluid with a density stratification due to both temperature and solute gradients, subject to an initial vertical flow field, is studied for different sets of homogeneous boundary conditions. Sufficient conditions for the maintenance of the original stratification, i.e., for stability, are obtained as a relation between R_aT, the thermal Rayleigh number, and R_as, the solute Rayleigh number. The relation $\text{R}{}_{\mathrm{as}}\text{R}{}_{\mathrm{aT}}$<-($\text{P}{}_{\mathrm{s}}\text{P}{}_{\mathrm{T}}$) is obtained as a sufficient condition for stability for all sets of boundary conditions, where P_T and P_S are the thermal and solute Prandtl number, respectively. This condition may thus be applied to cases in which the physical boundary conditions are not well defined; e.g., to practical engineering use of solar ponds.In addition, in conjunction with a similar result for initial horizontal flow fields, the result gives a sufficient condition for stability of arbitrary initial flows under all combinations of boundary conditions.     

Rayleigh-Jeffreys stability of thermohaline stratified fluids subject to horizontal flows

The stability of a horizontal layer of a fluid with density stratification due to both temperature and solute gradients, subject to horizontal flow, is studied for different sets of homogeneous boundary conditions. Sufficient conditions for stability, i.e., the maintenance of the original stratification are obtained as the relation between the thermal Rayleigh number, R_aT, and the solute Rayleigh number, R_aS. For stabilizing solute gradients —<-R_aS/R_aT<(P_rS/P_rT)² is obtained as a sufficient condition for stability for all sets of boundary conditions, where P_rT and P_rS are the thermal and the solute Prandtl numbers, respectively. This condition is, therefore, also applicable in cases where the physical boundary conditions are not well defined, e.g., in practical engineering use of solar ponds.     

Effect of volume fraction and particle size on wall slip in flow of polymeric suspensions

For especially highly concentrated suspensions, slip at the wall is the controlling phenomenon of their rheological behavior. Upon correction for slip at the wall, concentrated suspensions were observed to have non‐Newtonian behavior. In this study, to determine the true rheological behavior of model concentrated suspensions, “multiple gap separation method” was applied using a parallel‐disk rheometer. The model suspensions studied were polymethyl methacrylate particles having average particle sizes, in the range of 37–231 μm, in hydroxyl terminated polybutadiene. The effects of particle size and solid particle volume fraction on the wall slip and the true viscosity of model concentrated suspensions were investigated. It is observed that, as the volume fraction of particles increased, the wall slip velocity and the viscosity corrected for slip effects also increased. In addition, for model suspensions in which the solid volume fraction was ≥81% of the maximum packing fraction, non‐Newtonian behavior was observed upon wall slip correction. On the other hand, as the particle size increased, the wall slip velocity was observed to increase and the true viscosity was observed to decrease. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 98: 439–448, 2005     

A novel unified wall slip model for Poiseuille flow of polymer melt in the circular tube

In this study, to better reflect the slip effect of Poiseuille flow for polymer melt extruded through a circular tube, a novel unified wall slip model and flow equation based on two phase fluid system were proposed via a purely phenomenological approach. According to the different combinations of boundary conditions and flow parameters, the novel slip model was transformed into other models, such as adsorption–desorption model, entanglement–disentanglement model, lubrication layer model, Z–W model, and no‐slip model. The numerical simulation based on computed fluid dynamics was performed to verify the feasibility of the novel slip model. In the simulations, the radial flow velocity profile, shear rate, and viscosity distribution were obtained for six different models. Moreover, the effect of different slip coefficient combinations for the novel slip model on the radial flow velocity, slip velocity, volumetric flow rate error, and viscosity distribution of melt were also investigated and discussed. Results showed that the novel unified slip model not only incorporated the characteristics of other five models above mentioned, but also well interpreted the reason of simultaneously occurring the sharkskin surface defect and gross melt fracture phenomenon when flow rate of melt was extremely large. POLYM. ENG. SCI., 56:328–341, 2016. © 2015 Society of Plastics Engineers     

Effect of the surface roughness and construction material on wall slip in the flow of concentrated suspensions

Recent studies on polyethylene, elastomers, and thermoplastics have revealed that the construction material and surface roughness are two important factors affecting wall slip. In this study, to determine the true rheological behavior of model concentrated suspensions, a multiple‐gap separation method was used in a parallel disk rheometer. The model suspensions studied were poly (methyl methacrylate) particles with an average particle size of 121.2 μm in hydroxyl‐terminated polybutadiene. The aim of this study was to investigate the effect of disk R_a in the range of 0.49–1.51 μm and disk construction material on the wall slip and the true viscosity of the model concentrated suspensions. The wall slip velocity and the viscosity were found to be independent of R_a for particle size‐to‐disk R_a ratios of 80–247. Also, the true viscosity was found not to be affected by the rheometer surface construction material. Glass surfaces resulted in the highest slip velocity, whereas aluminum surfaces resulted in the lowest slip velocity. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 3341–3347, 2007     

Melt fracture and wall slip of thermoplastic vulcanizates

Thermoplastic vulcanizates (TPVs) are blends of polypropylene (PP) (thermoplastic phase) and ethylene propylene diene monomer (EPDM) rubber (rubber phase) in which a high content of rubber EPDM is cross-linked and dispersed in a thermoplastic matrix (PP) in the presence of oil (lubricant) and filler. Depending on the molecular characteristics of the constituent polymers, the level of curing and the amount of cross-linked rubber, their processing (extrusion) exhibits various difficulties such as melt fracture (extrudate distortions). In this study, a number of different TPVs with various characteristics, including the degree of curing and amount of cross-linked rubber are examined in capillary extrusion at two different temperatures (190°C and 205°C) relevant to real processing. First, the effect of the temperature on the yield stress is investigated using rheological measurements. Consequently, the flow behavior of the TPVs in capillary flow is studied concluding that TPVs slip massively (nearly plug flow) due to the presence of lubricant and the vulcanized rubber phase. Although there is little slip observed in PP samples, EPDMs themselves exhibit severe slip and melt fracture. As a consequence, the TPV samples essentially follow the slip behavior of EPDMs. Finally, the melt fracture analysis of several TPVs has shown that with increase of temperature and amount of cross-linked rubber, the severity of TPVs' surface defects increases accordingly.     

Rayleigh-Jeffreys stability of thermohaline stratified fluids subject to arbitrary temperature and salinity gradients

The stability of a horizontal layer of fluid of infinite extent and having a vertical density stratification due to arbitrary temperature and salinity gradients is studied for various sets of homogeneous boundary conditions. A sufficient condition for stability, i.e., for the maintenance of the original stratification is obtained in the form of a relation between the thermal Rayleigh number, Ra_T and the solute Rayleigh number, Ra_s. For no initial flow, the relation

is obtained as a sufficient condition for all sets of boundary conditions, where P_T and P_s are the thermal and solute Prandtl numbers, respectively, and $\text{df}$_T and $\text{df}$_s are mean values of the gradients of the initial temperature and solute distributions. This condition is a natural modification of that previously obtained for the case of constant thermal and salinity gradients [1, 10, 16], namely

The condition is applicable to situations in which the physical boundary conditions are not well-defined, e.g., in practical engineering development and operation of solar ponds.     

Effect of wall slip on blown film thickness distribution

One of the most important materials for blown film is high‐density polyethylene (HDPE) with wide molecular weight distribution. First, we computed a wall stress at the entrance of a spiral groove in a particular die during blown film processing on a particular condition, to which a similar condition is widely utilized in a film works. The computed value is about 170 kPa, while the HDPE melt slips at die wall at stresses above approximately 50 kPa. The stress of 170 kPa is sufficiently large for the slip occurrence of the melt. Then, we investigated the effects of wall slip and melt visosity on film thickness distribution in the circumferential direction; the distribution tends to decrease with decreasing wall slip and melt viscosity. This tendency is explained by considering flow distribution in a spiral mandrel die and polymer melt flow characteristics.