Fountain flow of pseudoplastic and viscoplastic fluids

Numerical simulations have been undertaken for the benchmark problem of fountain flow present in injection-mold filling. The Finite Element Method (FEM) is used to provide numerical results for both cases of planar and axisymmetric domains under steady-state conditions. The Herschel–Bulkley model of viscoplasticity is used, which reduces with appropriate modifications to the Bingham, power-law and Newtonian models. The present results extend previous ones regarding the shape of the front, which is essential in correctly capturing the flow field. In particular the centreline front position is found as a function of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress (in the case of viscoplasticity). The pressures from the simulations have been used to compute the excess pressure losses in the system (front pressure correction or exit correction). Both shear-thinning and shear-thickening lead to more extended front positions relative to the Newtonian values, which are 0.895 for the planar case and 0.835 for the axisymmetric one. Viscoplasticity leads also to more extended front positions as the dimensionless yield stress goes from zero (Newtonian behaviour) to higher values of the yield stress. In both cases of non-Newtonian behaviour, the front tends to follow the development of the fully developed Poiseuille velocity profile, which tends towards a plug-like profile at the extreme cases of non-Newtonianness. The front pressure (exit) correction increases monotonically with the decrease in the power-law index and the increase in the dimensionless yield stress.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

Numerical study of the combined effects of inertia,slip, and compressibility in extrusion of yield stress fluids

The axisymmetric extrudate swell flow of a compressible Herschel–Bulkley fluid with wall slip is solved numerically. The Papanastasiou-regularized version of the constitutive equation is employed, together with a linear equation of state relating the density of the fluid to the pressure. Wall slip is assumed to obey Navier’s slip law. The combined effects of yield stress, inertia, slip, and compressibility on the extrudate shape and the extrudate swell ratio are analyzed for representative values of the power-law exponent. When the Reynolds number is zero or low, swelling is reduced with the yield stress and eventually the extrudate contracts so that the extrudate swell ratio reaches a minimum beyond which it starts increasing asymptotically to unity. Slip suppresses both swelling and contraction in this regime. For moderate Reynolds numbers, the extrudate may exhibit necking and the extrudate swell ratio initially increases with yield stress reaching a maximum; then, it decreases till a minimum corresponding to contraction, and finally, it converges asymptotically to unity. In this regime, slip tends to eliminate necking and may initially cause further swelling of the extrudate, which is suppressed if slip becomes stronger. Compressibility was found to slightly increase swelling, this effect being more pronounced for moderate yield stress values and wall slip.     

Computations with viscoplastic and viscoelastoplastic fluids

This numerical study focuses on regularised Bingham-type and viscoelastoplastic fluids, performing simulations for 4:1:4 contraction?Cexpansion flow with a hybrid finite element?Cfinite volume subcell scheme. The work explores the viscoplastic regime, via the Bingham?CPapanastasiou model, and extends this into the viscoelastoplastic regime through the Papanastasiou?COldroyd model. Our findings reveal the significant impact that elevation has in yield stress parameters, and in sharpening of the stress singularity from that of the Oldroyd/Newtonian models to the ideal Bingham form. Such aspects are covered in field response via vortex behaviour, pressure-drops, stress field structures and yielded?Cunyielded zones. With rising yield stress parameters, vortex trends reflect suppression in both upstream and downstream vortices. Viscoelastoplasticity, with its additional elasticity properties, tends to disturb upstream?Cdownstream vortex symmetry balance, with knock-on effects according to solvent-fraction and level of elasticity. Yield fronts are traced with increasing yield stress influences, revealing locations where relatively unyielded material aggregates. Analysis of pressure drop data reveals significant increases in the viscoplastic Bingham?CPapanastasiou case, O (12%) above the equivalent Newtonian fluid, that are reduced to 8% total contribution increase in the viscoelastoplastic Papanastasiou?COldroyd case. This may be argued to be a consequence of strengthening in first normal stress effects.     

Extrudate swell of Boger fluids

Boger fluids are dilute polymer solutions exhibiting high elasticity at low apparent shear rates, which leads to high extrudate swell. Numerical simulations have been undertaken for the flow of three Boger fluids (including benchmark Fluid M1), obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of dilute polymer solutions. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries without or with gravity. The results provide the extrudate swell and the excess pressure losses (exit correction), as well as the shape and extent of the free surface. All these quantities increase rapidly and monotonically with increasing elasticity level measured by the stress ratio, S_R. It was found that the main reason for the high extrudate swelling is high normal stresses exhibited in shear flow (namely, the first normal-stress difference, N₁). Surprisingly, the elongational parameter of the model or a second normal-stress difference N₂ do not affect the results appreciably. Gravity serves to lower the swelling considerably, and makes the simulations easier and in overall agreement with previous experiments.     

A finite difference analysis of the extrudate swell problem

The extrudate swell phenomenon of a purely viscous fluid is analysed by solving simultaneously the Cauchy momentum equations along with the continuity equation by means of a finite difference method. The circular and planar jet flows of Newtonian and power-law fluids are simulated using a control volume finite difference method suggested by Patankar called SIMPLER (semi-implicit method for pressure-linked equations). This method uses the velocity components and pressure as the primitive variables and employs a staggered grid and control volume for each separate variable. The numerical results show good agreement with the analytical solution of the axisymmetric stick-slip problem and exhibit a Newtonian swelling ratio of 13.2% or 19.2% for a capillary or slit die respectively in accordance with previously reported experimental and numerical results. Shear thinning results in a decrease in swelling ratio, as does the introduction of gravity and surface tension.     

Approche analytique de la convection forcée des fluides de Bingham dans un tubeAnalytical approach for forced convection of Bingham fluids in a tube

The asymptotic behaviour of laminar forced convection for Bingham fluid in a circular tube subjected to an axially varying wall heat flux is studied analytically. The effect of viscous dissipation is taken into account while the axial heat conduction in the fluid is considered as negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial wall heat flux distributions which yield a thermally developed region. The results obtained show a diminution in asymptotic Nusselt number when the Brinkman number and the dimensionless radius of the plug flow region increase. Comparisons with the results in the literature for Newtonian fluids show the validity of the present analysis. To cite this article: R. Khatyr et al., C. R. Mecanique 330 (2002) 69–75.     

Stability of a Rayleigh-Bénard Poiseuille flow for yield stress fluids—Comparison between Bingham and regularized models

A linear stability analysis of a Rayleigh-Bénard Poiseuille flow is performed for yield stress fluids whether we use the Bingham or regularized models. A fundamental difference between those models is that the effective viscosity is not defined in the plug zone for the Bingham model, while it is defined in the whole domain for the regularized models. For these models, the viscosity depends highly on a parameter ? near the axis and increases drastically in an intermediate region. The convergence of the critical conditions between the simple and the Bingham models is not obtained. However, we show that the Bercovier and Papanastasiou models can tend to the exact Bingham results.     

Compressible extrudate swell

There are few computations of polymer forming processes which include compressibility. Here we estimate the effect of compressibility in Newtonian and PTT fluids on extrudate swell and stick-flip flow. Changes of the order of a few per cent occur in swelling, which is in accord with expectations.     

Extrusion of a compressible Newtonian fluid with periodic inflow and slip at the wall

We explore a mechanism of extrusion instability, based on the combination of nonlinear slip and compressibility. We consider the time-dependent compressible Newtonian extrudate swell problem with slip at the wall. Steady-state solutions are unstable in regimes where the shear stress is a decreasing function of the velocity at the wall. Compressibility provides the means for the alternate storage and release of elastic energy, and, consequently, gives rise to periodic solutions. The added novelty in the present work is the assumption of periodic volumetric flow rate at the inlet of the die. This leads to more involved periodic responses and to free surface oscillations similar to those observed experimentally with the stick-slip instability. To numerically simulate the flow, we use finite elements in space and a fully-implicit scheme in time.Dedicated to the memory of Prof. Tasos Papanastasiou     

Shallow Water equations for Non-Newtonian fluids

The purpose of this paper is to provide a consistent thin layer theory for some Non-Newtonian fluids that are incompressible and flowing down an inclined plane under the effect of gravity. We shall provide a better understanding of the derivation of Shallow Water models in the case of power-law fluids and Bingham fluids. The method is based on asymptotic expansions of solutions of the Cauchy Momentum equations in the Shallow Water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q. Such a method has been first introduced in the case of Newtonian fluids where the computations are proved to be rigorous (Vila, in preparation [20]; Bresch and Noble, 2007 [9]) whereas the more complex case of arbitrary topography has been treated formally (Boutounet et al., 2008 [5]). The well posedness of the free surface Cauchy Momentum equations for these Non-Newtonian fluids is still an open problem: the computations carried out here are only formal.     

Extensional dynamics of viscoplastic filaments: I. Long-wave approximation and the Rayleigh instability

We derive an asymptotic reduced model for the extensional dynamics of long, slender, axisymmetric threads of incompressible Herschel–Bulkley fluids. The model describes the competition between viscoplasticity, gravity, surface tension and inertia, and is used to explore the viscoplastic Rayleigh instability. A finite-amplitude initial perturbation is required to yield the fluid and initiate capillary-induced thinning. The critical amplitude necessary for thinning depends on both the wavelength of the perturbation and on the yield stress. We also numerically examine the inertialess growth of the instability and the progression towards pinch-off. The final self-similar form of inertialess pinch-off is similar to that for a power-law fluid.     

Free (open) boundary condition: some experiences with viscous flow simulations

The free (or open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. (A new outflow boundary condition, International Journal for Numerical Methods in Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been tested in several benchmark problems of viscous flow in fluid mechanics. The FEM is used to provide numerical results for both cases of planar and axisymmetric domains under laminar, isothermal or non‐isothermal, steady‐state conditions, for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually in the extrudate‐swell benchmark problem for a wide range of the relevant parameters. The present results extend previous ones regarding the applicability of the FBC and show cases where the FBC is inappropriate, namely in the extrudate‐swell problem with gravity or surface‐tension effects. Particular emphasis has been given to the pressure at the outflow, which is the most sensitive quantity of the computations. In all cases where FBC is appropriate, excellent agreement has been found in comparisons with results from very long domains. The formulation for Picard‐type iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects. Copyright © 2011 John Wiley & Sons, Ltd.     

Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times

We solve the one-dimensional cessation Couette and Poiseuille flows of a Bingham plastic using the regularized constitutive equation proposed by Papanastasiou and employing finite elements in space and a fully implicit scheme in time. The numerical calculations confirm previous theoretical findings that the stopping times are finite when the yield stress is nonzero. The decay of the volumetric flow rate, which is exponential in the Newtonian case, is accelerated and eventually becomes linear as the yield stress is increased. In all flows studied, the calculated stopping times are just below the theoretical upper bounds, which indicates that the latter are tight.     

Transient laminar forced convection to power-law fluids inside ducts resulting from a sudden change in wall temperature

The transient laminar forced convection to power-law fluids in thermally developing, hydrodynamically developed flow inside parallel-plate ducts and circular tubes resulting from a sudden change in wall temperature is studied. The generalized integral transform and the Laplace transform techniques are employed to develop approximate analytic solutions. The local Nusselt number and average fluid temperature are presented over the range of the dimensionless axial coordinate Z varying from 10^?4 to 10^?1 for several dimensionless times. Three different values of the power-law index are considered in the study includedn=1/3,n=1 andn=3 corresponding to, respectively, the pseudoplastic, Newtonian and dilatant fluids.     

Taylor–Dean instability of yield-stress fluids at large gaps

Centrifugal instability of Bingham fluids is investigated in Taylor–Dean flow when the gap size is large compared to the cylinders radii. To determine conditions for the onset of instability, an infinitesimal axisymmetric disturbance was introduced to the basic flow and its evolution in time was monitored using a normal-mode linear stability analysis. To avoid the problem with the stress discontinuity at the location of the yield surface(s), use was made of the Papanastasiou’s regularized variation of the Bingham model in order to obtain the basic flow velocity profiles. An eigenvalue problem was obtained for the exact Bingham fluid which was solved numerically using the collocation method. A plot of the neutral instability curve at different Bingham numbers suggests that the yield stress can have a stabilizing or destabilizing effect on Taylor–Dean flow depending on the sign and magnitude of the pressure gradient, and also on the sense of rotation of the two cylinders with respect to each other.     

A modern look on yield stress fluids

A concept of viscoplasticity advanced exactly one century ago by Bingham appears very fruitful because there are many natural and artificial materials that demonstrate viscoplastic behavior, i.e., they are able to pass from a solid to a liquid state under the influence of applied stress. However, although this transition was originally considered as a jump-like phenomenon occurring at a certain stress—the yield stress—numerous subsequent studies have shown that the real situation is more complicated. A long-term discussion about the possibility of flow at low stresses less than the yield stress came to today’s conclusion denying this possibility as being opposite to the existence of the maximal Newtonian viscosity in viscoelastic polymeric fluids. So, there is a contradiction between the central dogma of rheology which says that “everything flows” and the alleged impossibility for flow at a solid-like state of viscoplastic fluids. Then, the concept of the fragile destruction of an inner structure responsible for a solid-like state at the definite (yield) stress was replaced by an understanding of the yielding as a transition extending over some stress range and occurring in time. So, instead of the yield stress, yielding is characterized by the dependence of durability (or time-to-break) on the applied stress. In this review, experimental facts and the new understanding of yielding as a kinetic process are discussed. Besides, some other alternative methods for measuring the yield stress are considered.     

聚合物熔体三维挤出胀大的数值模拟

采用有限元方法分析K-BKZ本构方程描述的聚合物熔体的三维挤出胀大.对于本构方程中偏应力张量的计算,首先给出质点的运动轨迹,分段求出局部的变形梯度张量,再求出整体的变形梯度、Cauchy-Green应变张量和 Finger应变张量,沿轨迹采用分段高斯积分计算应力.把应力作为方程的右端项,给出迭代方法,求解非线性方程组.并根据自由面处的边界条件,迭代得出出口处自由面的最终位置.对轴对称流道和矩形流道进行分析计算,并把结果与二维分析和实验结果进行了比较,显示方法是可行的.     

Extrudate swell in some dilute elastic solutions

An experimental programme has been carried out to obtain data for the isothermal swelling of dilute viscoelastic fluids based on the polyisobutylene-polybutene system. The fluid was thus a constant viscosity one and so the observed swell was predominantly due to elastic effects alone. To establish the experimental rig and procedure, preliminary results for the creeping flow of a Newtonian fluid, based on a polybutene, were acquired. The Newtonian results, showing a creeping flow swell of 13.8%, demonstrated that the rig and procedure were satisfactory for the formation of free, isothermal jets. The viscoelastic results were quite unexpected, showing that dilute elastic solutions can exhibit significant swelling despite the fact that the recoverable shear is small (S _R < 0.1). These results qualitatively confirm a recent numerical study [1].     

Numerical solution of a casson fluid flow in the entrance of annular tubes

The solution of the momentum equation for a Casson fluid flowing in the entrance region of an annular tube has been obtained. The results have been presented for a large range of radii ratio and dimensionless yield stress. The mathematical accuracy of the numerical procedure is demonstrated by comparing the asymptotic velocity profiles at large axial distance with fully developed solution [1]. In addition, the results of the numerical solution for the case of yield stress equal to zero are compared with the entrance flow solution for a Newtonian fluid [2].