Three-dimensional viscoelastic simulation of coextrusion process: comparison with experimental data

 Three-dimensional numerical simulation of viscoelastic coextrusion process has been performed and numerical results were compared with the experimental data of Karagiannis et al. (1990). By varying the magnitude of the second normal stress difference and its ratio of Fluid I and Fluid II, we were able to control the interface profile and the degree of encapsulation along the downstream direction. By increasing the parameter α (α_Fluid I=α_Fluid II) from 0.1 to 0.4 in the Giesekus model and increasing the α ratio (α_Fluid Iα_Fluid II) between Fluid I and Fluid II from 2.0 to 4.0 in the permissible range of realistic polymeric systems, the interface profile and the degree of encapsulation along the downstream direction were fitted with the experimental results. There was little difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction when the α ratio was set to 3.0 (0.3:0.1). Fluid I with larger magnitude of the second normal stress difference protrudes into Fluid II with smaller magnitude of the second normal stress difference around the symmetric plane, while Fluid II wraps around Fluid I near the side walls. As the ξ ₁ ratio (ξ _{1 ,Fluid I}ξ _{1 ,Fluid II}) increases from 1.0 to 3.0 for the two-mode Phan-Thien and Tanner model, it was found that the curvature of the interface profile increased, and the difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction was almost negligible when the ξ ₁ ratio was set to 3.0 (0.54:0.18). Although the parameters of viscoelastic models were fitted by using the shear viscosity data only, quantitative agreements between the numerical results and the experimental coextrusion data were quite satisfactory. Received: 24 April 2001 Accepted: 5 June 2001     

Numerical simulation of coextrusion and film casting

In the first part of this paper a numerical strategy is developed for the numerical simulation of the coextrusion process. Coextrusion consists of extruding many polymers in the same die in order to combine their respective properties. The die is generally flat and quite large and consequently a two-dimensional approximation is sufficient. The main difficulty is to accurately predict the interfaces between the different layers of polymers. A finite element method based on a pseudoconcentration function is developed to calculate these fluid interfaces. Numerical results are presented for the coextrusion of up to five fluids. In the second part of the paper the above strategy is slightly modified to simulate the film-casting process. In this case a polymer is stretched (with a draw velocity U_L) at the exit of the die in order to produce a very thin layer of polymer that is cooled in contact with a chill roll. Only one polymer-air interface has to be computed. The draw ratio is defined as Dr = U_L/U , where U is the mean velocity of the polymer at the exit of the die. As the draw ratio is increased, instabilities appear and numerical results put in evidence the draw resonance phenomenon.     

Shear and normal stress relaxation in polymer blends: stress predictions from interface velocity and Laplace pressure terms

We derived for the first time the relationships among shear stress and normal stress differences for ellipsoidal interfaces under large step shear strains considering interface velocity term and Laplace pressure term in the expression of the stress tensor for mixtures of two Newtonian fluids. In the derivation, orientation angle of the interface is assumed to be given by the affine deformation assumption and is independent of time based on experimental results for blends with 0.048 ≤ K ≤ 0.54 where K is the ratio of droplet viscosity to matrix viscosity. For ellipsoidal droplets, the shear stress is only proportional to the first normal stress difference. On the other hand, for spheroidal droplets, proportionality among the shear stress, the first and the second normal stress differences was derived, and the ratio of the second normal stress difference to the first normal stress difference was given as a function of step strain. The shear stress and the first normal stress difference obtained experimentally satisfy the derived relationship, indicating applicability of the stress expression for polymer blends.     

Three-dimensional studies on bicomponent extrusion

The present work is concerned with the mathematical modelling and numerical simulation of three-dimensional (3-D) bicomponent extrusion. The objective is to provide an understanding of the flow phenomena involved and to investigate their impact on the free surface shape and interface configuration of the extruded article. A finite element algorithm for the 3-D numerical simulation of bicomponent stratified free surface flows is described. The presence of multiple free surfaces (layer interface and external free surfaces) requires special free surface update schemes. The pressure and viscous stress discontinuity due to viscosity mismatch at the interface between the two stratified components is handled with both a double node (u–v–w–P ₁ –P ₂ –h ₁ –h ₂) formulation and a penalty function (u–v–w–P–h ₁ –h ₂) formulation.The experimentally observed tendency of the less viscous layer to encapsulate the more viscous layer in stratified bicomponent flows of side-by-side configuration is established with the aid of a fully 3-D analysis in agreement with experimental evidence. The direction and degree of encapsulation depend directly on the viscosity ratio of the two melts. For shear thinning melts exhibiting a viscosity crossover point, it is demonstrated that interface curvature reversal may occur if the shearing level is such that the crossover point is exceeded. Extrudate bending and distortion of the bicomponent system because of the viscosity mismatch is shown. For flows in a sheath-core configuration it is shown that the viscosity ratio may have a severe effect on the swelling ratio of the bicomponent system.Modelling of the die section showed that the boundary condition imposed at the fluid/fluid/wall contact point is critical to the accuracy of the overall solution.     

MHD flow and heat transfer of a UCM fluid over a stretching surface with variable thermophysical properties

In this paper we investigate the effects of temperature-dependent viscosity, thermal conductivity and internal heat generation/absorption on the MHD flow and heat transfer of a non-Newtonian UCM fluid over a stretching sheet. The governing partial differential equations are first transformed into coupled non-linear ordinary differential equation using a similarity transformation. The resulting intricate coupled non-linear boundary value problem is solved numerically by a second order finite difference scheme known as Keller-Box method for various values of the pertinent parameters. Numerical computations are performed for two different cases namely, zero and non-zero values of the fluid viscosity parameter. That is, 1/?? _r ??0 and 1/?? _r ??0 to get the effects of the magnetic field and the Maxwell parameter on the velocity and temperature fields, for several physical situations. Comparisons with previously published works are presented as special cases. Numerical results for the skin-friction co-efficient and the Nusselt number with changes in the Maxwell parameter and the fluid viscosity parameter are tabulated for different values of the pertinent parameters. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid phenomena, especially the UCM fluid phenomena. Maxwell fluid reduces the wall-shear stress.     

Characterization of micellar structure dynamics for a drag-reducing surfactant solution under shear: normal stress studies and flow geometry effects

Some surfactant solutions have been observed to exhibit a strong drag reduction behavior in turbulent flow. This effect is generally believed to result from the formation of large cylindrical micelles or micellar structures. To characterize and understand better these fluids, we have studied the transient rheological properties of an efficient drag-reducing aqueous solution: tris (2-hydroxyethyl) tallowalkyl ammonium acetate (TTAA) with added sodium salicylate (NaSal) as counter ion. For a 5/5 mM equimolar TTAA/NaSal solution, there is no measurable first normal stress difference (N ₁) immediately after the inception of shear, but N ₁ begins to increase after a well-defined induction time — presumably as shear-induced structures (SIS) are formed — and it finally reaches a fluctuating plateau region where its average value is two orders of magnitude larger than that of the shear stress. The SIS buildup times obtained by first normal stress measurements were approximately inversely proportional to the shear rate, which is consistent with a kinetic process during which individual micelles are incorporated through shear into large micellar structures. The SIS buildup after a strong preshear and the relaxation processes after flow cessation were also studied and quantified with first normal stress difference measurements. The SIS buildup times and final state were also found to be highly dependent on flow geometry. With an increase in gap between parallel plates, for example, the SIS buildup times decreased, whereas the plateau viscosity increased.     

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.     

Normal stress differences in large-amplitude oscillatory shear flow for the corotational ��ANSR�� model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant ?? ₀ controlling the ratio ??₂/??₁ for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.     

Investigation of liquid–liquid drop coalescence using tomographic PIV

High-speed tomographic PIV was used to investigate the coalescence of drops placed on a liquid/liquid interface; the coalescence of a single drop and of a drop in the presence of an adjacent drop (side-by-side drops) was investigated. The viscosity ratio between the drop and surrounding fluids was 0.14, the Ohnesorge number (Oh = μ_d/(ρ_dσD)^1/2) was 0.011, and Bond numbers (Bo = (ρ _d  − ρ _s )gD ²/σ) were 3.1–7.5. Evolving volumetric velocity fields of the full coalescence process allowed for quantification of the velocity scales occurring over different time scales. For both single and side-by-side drops, the coalescence initiates with an off-axis film rupture and film retraction speeds an order of magnitude larger than the collapse speed of the drop fluid. This is followed by the formation and propagation of an outward surface wave along the coalescing interface with wavelength of approximately 2D. For side-by-side drops, the collapse of the first drop is asymmetric due to the presence of the second drop and associated interface deformation. Overall, tomographic PIV provides insight into the flow physics and inherent three-dimensionalities in the coalescence process that would not be achievable with flow visualization or planar PIV only.     

Stress optical measurement of the third normal stress difference in polymer melts under oscillatory shear

Results are reported for the dynamic moduli,G andG, measured mechanically, and the dynamic third normal stress difference, measured optically, of a series bidisperse linear polymer melts under oscillatory shear. Nearly monodisperse hydrogenated polyisoprenes of molecular weights 53000 and 370000 were used to prepare blends with a volume fraction of long polymer, _L, of 0.10, 0.20, 0.30, 0.50, and 0.75. The results demonstrate the applicability of birefringence measurements to solve the longstanding problem of measuring the third normal stress difference in oscillatory flow. The relationship between the third normal stress difference and the shear stress observed for these entangled polymer melts is in agreement with a widely predicted constitutive relationship: the relationship between the first normal stress difference and the shear stress is that of a simple fluid, and the second normal stress difference is proportional to the first. These results demonstrate the potential use of 1,3-birefringence to measure the third normal stress difference in oscillatory flow. Further, the general constitutive equation supported by the present results may be used to determine the dynamic moduli from the measured third normal stress difference in small amplitude oscillatory shear. Directions for future research, including the use of birefringence measurements to determineN ₂/N ₁ in oscillatory shear, are described.     

The rheology of suspensions of vinylon fibers in polymer liquids. II. Suspensions in polymer solutions

Summary The rheological properties of vinylon fiber suspensions in polymer solutions were studied in steady shear flow. Shear viscosity, first normal-stress difference, yield stress, relative viscosity, and other properties were discussed. Three kinds of flexible vinylon fibers of uniform length and three kinds of polymer solutions as mediums which exhibited remarkable non-Newtonian behaviors were employed. The shear viscosity and relative viscosity ( _r ) increased with the fiber content and the aspect ratio, and depended upon the shear rate. Shear rate dependence of _r was found only in the low shear rate region. This result was different from that of vinylon fiber suspensions in Newtonian fluids. The first normal-stress difference increased at first slightly with increasing fiber content but rather decreased and showed lower values for high content suspensions than that of the medium. A yield stress could be determined by using a modified equation of Casson type. The flow properties of the fiber suspensions depended on the viscosity of the medium in the suspensions under consideration.With 16 figures and 1 table     

Instability investigation of creeping viscoelastic flow in a curved duct with rectangular cross-section

In this paper, instability in the creeping viscoelastic flow inside a curved rectangular duct is investigated numerically for the first time. Using the Criminale–Eriksen–Filbey (CEF) model as the constitutive equation, the governing equations are solved by a second order of finite difference method based on the artificial compressibility algorithm in a staggered mesh. The effects of normal stress differences on the flow stability are investigated. The numerical results obtained indicate that the increase of the negative second normal stress difference of viscoelastic fluid causes stability in the creeping flow in curved ducts, however, the increase of the first normal stress difference intensifies the instability. Furthermore, at the special value of Ψ₂/Ψ₁=?0.5, the interaction of the two normal stress differences results in a stable flow field.     

Droplet deformation of a strongly shear thinning dense suspension of polymeric micelles

We investigated the deformation of a strong shear thinning droplet undergoing simple shear flow in a Newtonian liquid. The droplet was an aqueous solution of poly(ethylene oxide) end capped with an alkyl group that forms spherical micelles in aqueous solution. At high concentrations and below a critical temperature, the jammed micelles showed strong shear thinning behaviour, and neither a yield stress nor a Newtonian viscosity was observed. At small shear rates, the droplet rotated and aligned in the flow, but did not deform or only very weakly. At high shear rates, the droplet deformation increased with increasing shear rate. The deformed droplet did not relax after the shear was stopped except for a modest rounding of the edges. For each shear rate, an apparent viscosity, η _ad, of the equivalent Newtonian droplet was calculated assuming affine deformation. η _ad showed a power law dependence on the capillary number Ca with an exponent of − 1.8 and was larger than the shear viscosity of the micelle suspension at the same shear rates. The results were explained by the existence of a strong gradient of the viscosity inside the droplet leading to a very low viscosity fluid layer near the droplet/matrix interface.     

A shear rheometer for measuring shear stress and both normal stress differences in polymer melts simultaneously: the MTR 25

The MTR 25 is a multitask rheometer (for shear and squeeze flow) with 25 kg of normal force and a partitioned plate. Torque and normal force are measured at both, the inner disk and the outer ring of the plate. The first and second normal stress differences can be determined from a single test. The axial stiffness is high (10⁷ N/m) by using rigid springs and strain gauges for the load cell. Monodisperse polystyrene (M _w = 206 kg/mol, 180°C) has been sheared in the range from 0.05 to 47 s^− 1. The viscosity and first normal stress difference are highly reproducible. The second normal stress difference scatters and mirrors the instability at the rim. A critical comparison is made between the MTR 25 method and the single transducer evaluation method (RMS 800 method, Schweizer, Rheol Acta 41:337–344, 2002): Both yield excellent and coinciding viscosity and first normal stress difference data. The RMS 800 method gives more stable second normal stress difference data, since the normal force from the outer ring, which is influenced by edge fracture, is not used. Data for the RMS 800 method can be acquired on the MTR 25. The high normal force capacity permits larger samples and higher shear rates than on the RMS 800.

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

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

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

A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele‐Shaw cell

In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele‐Shaw cell. Fingering instabilities initiated at the interface between a low‐viscosity fluid and a high‐viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model. The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO₂ sequestration but does not pretend to analyse this complex problem. Instead, we present a detailed study of the analogous problem occurring in a Hele‐Shaw cell, giving indications of possible plume patterns that can develop during the CO₂ injection. The numerical scheme utilises a boundary element method in which the normal velocity at the interface of the two fluids is directly computed through the evaluation of a hypersingular integral. The boundary integral equation is solved using a Neumann convergent series with cubic B‐Spline boundary discretisation, exhibiting sixth‐order spatial convergence. The convergent series allows the long‐term nonlinear dynamics of growing viscous fingers to be explored accurately and efficiently. Simulations in low‐mobility ratio regimes reveal large differences in fingering patterns compared with those predicted by previous high‐mobility ratio models. Most significantly, classical finger shielding between competing fingers is inhibited. Secondary fingers can possess significant velocity, allowing greater interaction with primary fingers compared with high‐mobility ratio flows. Eventually, this interaction can lead to base thinning and the breaking of fingers into separate bubbles. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of element order and interface reconstruction in FEM/volume‐of‐fluid incompressible flow simulation

In previous studies, the moment‐of‐fluid interface reconstruction method showed dramatic accuracy improvements in static and pure advection tests over existing methods, but this did not translate into an equivalent improvement in volume‐tracked multimaterial incompressible flow simulation using low‐order finite elements. In this work, the combined effects of the spatial discretization and interface reconstruction in flow simulation are examined. The mixed finite element pairs, Q₁Q₀ (with pressure stabilization) and Q₂P _{? 1} are compared. Material order‐dependent and material order‐independent first and second‐order accurate interface reconstruction methods are used. The Q₂P _{? 1} elements show significant improvements in computed flow solution accuracy for single material flows but show reduced convergence using element‐average piecewise constant density and viscosity in volume‐tracked simulations. In general, a refined Q₁Q₀ grid, with better material interface resolution, provided an accuracy similar to the Q₂P _{? 1} element grid with a comparable number of degrees of freedom. Moment‐of‐fluid shows more benefit from the higher‐order accurate flow simulation than the LVIRA, Youngs', and power diagram interface reconstruction methods, especially on unstructured grids, but does not recover the dramatic accuracy improvements it has shown in advection tests. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Purely elastic interfacial instabilities in superposed flow of polymeric fluids

Purely elastic interfacial stability of superposed plane Poiseuille flow of polymeric liquids has been investigated utilizing both asymptotic and numerical techniques. It is shown that these instabilities are caused by an unfavorable jump in the first normal stress difference across the fluid interface. To determine the significance of these instabilities in finite experimental geometries, a comparison between the maximum growth rates of purely elastic instabilities with instabilities driven primarily by a viscosity or a combined viscosity and elasticity difference is made. Based on this comparison, it is shown that purely elastic interfacial instabilities can play a major role in superposed flow of polymeric liquids in finite experimental geometries.