Increasing Weissenberg Number Research Articles

Swirling Flow of a Viscoelastic Fluid in a Cylindrical Casing

The swirling flow of a viscoelastic fluid in a cylindrical casing is investigated experimentally, using aqueous solutions of 0.05–1.0wt.% polyacrylamide as the working fluid. The velocity measurements are made using laser Doppler anemometer. The aspect ratios H∕R (H: axial length of cylindrical casing; R: radius of rotating disk) investigated are 2.0, 1.0, and 0.3. The Reynolds numbers Re0 based on the zero shear viscosity and the disk-tip velocity are between 0.36 and 50. The velocity measurements are mainly conducted for the circumferential velocity component. The experimental velocity data are compared to the velocity profiles obtained by numerical simulations using Giesekus model and power-law model. It is revealed that at any aspect ratios tested the dimensionless circumferential velocity component Vθ′ decreases with increasing Weissenberg number We. Both the Giesekus and power-law models could predict the retardation of circumferential velocity fairly well at small We. The extent of the inverse flow region, where the fluid rotates in the direction opposite to the rotating disk, is clarified in detail.

Purely tangential flow of a PTT-viscoelastic fluid within a concentric annulus

An analytical solution is presented for the steady state, purely tangential flow of a viscoelastic fluid obeying the Phan-Thien–Tanner (PTT) constitutive equation in a concentric annulus with relative rotation of the inner and outer cylinders. The influence on the velocity distribution within the annulus and on fRe of the Weissenberg number, aspect ratio and an elongational parameter are investigated. The results show that the differences between the radial location of the minimum velocity and of the critical angular velocity compared with their Newtonian counterparts increase as the fluid elasticity increases. The results also show that fRe decreases with increasing Weissenberg number, radius ratio and the elongational parameter in the case of inner-cylinder rotation. In contrast, fRe increases with increasing radius ratio when the outer cylinder is rotating while the inner cylinder is at rest.

Subcritical Finite-Amplitude Solutions for Plane Couette Flow of Viscoelastic Fluids

Plane Couette flow of viscoelastic fluids is shown to exhibit a purely elastic subcritical instability at a very small-Reynolds number in spite of being linearly stable. The mechanism of this instability is proposed and the nonlinear stability analysis of plane Couette flow of the Upper-Convected Maxwell fluid is presented. Above a critical Weissenberg number, a small finite-size perturbation is sufficient to create a secondary flow, and the threshold value for the amplitude of the perturbation decreases as the Weissenberg number increases. The results suggest a scenario for weakly turbulent viscoelastic flow which is similar to the one for Newtonian fluids as a function of Reynolds number.

DNA Molecules in Microfluidic Oscillatory Flow

The conformation and dynamics of a single DNA molecule undergoing oscillatory pressure-driven flow in microfluidic channels is studied using Brownian dynamics simulations, accounting for hydrodynamic interactions between segments in the bulk and between the chain and the walls. Oscillatory flow provides a scenario under which the polymers may remain in the channel for an indefinite amount of time as they are stretched and migrate away from the channel walls. We show that by controlling the chain length, flow rate and oscillatory flow frequency, we are able to manipulate the chain extension and the chain migration from the channel walls. The chain stretch and the chain depletion layer thickness near the wall are found to increase as the Weissenberg number increases and as the oscillatory frequency decreases.

The mechanism of the self-assembly of associating DNA molecules under shear flow: Brownian dynamics simulation

A shear flow induces the assembly of DNAs with the sticky spots. In order to strictly interpret the mechanism of shear-induced DNA assembly, Brownian dynamics simulations with the bead-spring model were carried out for these molecules at various ranges of the Weissenberg numbers (We). We calculate a formation time and analyze the radial distribution function of end beads and the probability distribution of fractional extension at the formation time to understand the mechanism of shear-induced assembly. At low Weissenberg number the formation time, which is defined as an elapsed time until a multimer forms for the first time, decreases rapidly, reaching a plateau at We = 1000. A shear flow changes the radial distribution of end beads, which is almost the same regardless of the Weissenberg number. A shear flow deforms and stretches the molecules and generates different distributions between end beads with a stickly spot. The fractional extension progresses rapidly in shear flow from a Gaussian-like distribution to a uniform distribution. The progress of the distribution of fractional extension increases the possibility of meeting of end beads. In shear flow, the inducement of the assembly mainly results from the progress of the probability distribution of fractional extension. We also calculate properties such as the radius of gyration, stretch, and so on. As the Weissenberg number increases, the radius of gyration at the formation time also increases rapidly, reaching a plateau at We = 1000.

Computation of incompressible and weakly-compressible viscoelastic liquids flow: finite element/volume schemes

In this study, we analyse viscoelastic numerical solution for an Oldroyd-B model under incompressible and weakly-compressible liquid flow conditions. We consider flow through a planar four-to-one contraction, as a standard benchmark, throughout a range of Weissenberg numbers up to critical levels. At the same time, inertial and creeping flow settings are also addressed. Within our scheme, we compare and contrast, two forms of stress discretisation, both embedded within a high-order pressure-correction time-marching formulation based on triangles. This encompasses a parent-cell finite element/SUPG scheme, with quadratic stress interpolation and recovery of velocity gradients. The second scheme involves a sub-cell finite volume implementation, a hybrid fe/fv-scheme for the full system. A new feature of this study is that both numerical configurations are able to accommodate incompressible, and low to vanishing Mach number compressible liquid flows. This is of some interest within industrial application areas. We are able to provide parity between the numerical solutions across schemes for any given flow setting. Close examination of flow patterns and vortex trends indicates the broad differences anticipated between incompressible and weakly-compressible solutions. Vortex reduction with increasing Weissenberg number is a common feature throughout. Compressible solutions provide larger vortices (salient and lip) than their incompressible counterparts, and larger stress patterns in the re-entrant corner neighbourhood. Inertia tends to reduce such phenomena in all instances. The hybrid fe/fv-scheme proves more robust, in that it captures the stress singularity more tightly than the fe-form at comparable Weissenberg numbers, reaching higher critical levels. The sub-cell structure, the handling of cross-stream numerical diffusion, and corner discontinuity capturing features of the hybrid fe/fv-scheme, are all perceived as attractive additional benefits that give preference to this choice of scheme.

DNA configurations and concentration in shearing flow near a glass surface in a microchannel

We characterize the configurations and concentration of λ-phage deoxyribose nucleic acid (DNA) molecules in shear flow near a glass surface in a microchannel through epifluorescence microscopy. We observe that in Poiseuille flow, over a region extending from a glass surface up to about one third of the contour length of the DNA molecule, the average stretch of a λ-phage DNA molecule is significantly lower than in the bulk, in agreement with results obtained in a steady torsional shear flow by Li et al. (2004). We also find that the concentration of DNA molecules in this same region is notably lower than in the bulk, to a degree that increases with increasing Weissenberg number. A simplified explanation is proposed for the behavior of DNA molecules near the glass surface based on wall influences on hydrodynamic interaction within the chain, motivated by the recent theoretical work of Jendrejack et al. (2003, 2004).

Flow-induced nonequilibrium thermodynamics of lamellar semicrystalline polymers

In this work we present a first attempt to quantify the effect of flow deformation on the microstructure of semicrystalline polymers. This necessitates bridging the macroscopic flow length scale with the microscopic (segment) length scale of the semicrystalline structure. To achieve this connection we developed a hierarchical approach where a thermodynamically consistent macroscopic constitutive equation is interfaced with a microscopic lattice-based Monte Carlo (MC) simulation of the polymer chain conformation. We first illustrate this approach in a two-dimensional (2D) “toy” application where the 2D equivalent of a macroscopic constitutive equation based on reptation theory is applied to describe the chain deformation and extended free energy in the amorphous bulk phase. The values for the derivative of the free energy with respect to the mean segment orientation tensor, calculated for a planar extensional flow, are then used as an extended nonequilibrium thermodynamic forcing term. This is added in a traditional Metropolis Monte Carlo scheme, developed for a 2D lattice representation of a lamellar semicrystalline polymer, to drive the flow-induced microstructure. Significant flow-induced changes are calculated, steadily increasing as the Weissenberg number increases. We subsequently extend these ideas further in a much more realistic three-dimensional (3D) application where the information for the thermodynamics of the bulk amorphous phase under a uniaxial extensional flow is extracted from a macroscopic network model, such as that of Phan-Thien and Tanner (PTT), connecting the free energy to the second moment of the end-to-end distance of a multisegment chain. Through a series of 3D nonequilibrium Monte Carlo simulations of both the amorphous and the semicrystalline microscopic morphologies, it is shown that the interaction of the flow-induced deformation with the semicrystalline microstructure is nonlinear: the amorphous interlamellar structure changes significantly from its corresponding homogeneous bulk amorphous state, even far away from the crystalline interface. Our approach allows for a quantitative estimation of this effect on both thermodynamic quantities, like the extended microscopic free energy, as well as various statistics of the chain conformations.

Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow

It is well known that the Poiseuille flow of a visco-elastic polymer fluid between plates or through a tube is linearly stable in the zero Reynolds number limit, although the stability is weak for large Weissenberg numbers ( Wi ). In this paper, we argue that recent experimental and theoretical work on the instability of visco-elastic fluids in Taylor–Couette cells and numerical work on channel flows suggest a scenario in which Poiseuille flow of visco-elastic polymer fluids exhibits a nonlinear “subcritical” instability due to normal stress effects, with a threshold which decreases for increasing Weissenberg number. This proposal is confirmed by an explicit weakly nonlinear stability analysis for Poiseuille flow of an UCM fluid. Our analysis yields explicit predictions for the critical amplitude of velocity perturbations beyond which the flow is nonlinearly unstable, and for the wavelength of the mode whose critical amplitude is smallest. The nonlinear instability sets in quite abruptly at Weissenberg numbers around 4 in the planar case and about 5.2 in the cylindrical case, so that for Weissenberg numbers somewhat larger than these values perturbations of the order of a few percentage in the wall shear stress suffice to make the flow unstable. We have suggested elsewhere that this nonlinear instability could be an important intrinsic route to melt fracture and that preliminary experiments are both qualitatively and quantitatively in good agreement with these predictions.

Maximum drag reduction in a turbulent channel flow by polymer additives

Maximum drag reduction (MDR) in a turbulent channel flow by polymer additives is studied using direct numerical simulation. An Oldroyd-B model is adopted to express the polymer stress because MDR is closely related to the elasticity of the polymer solution. The Reynolds number considered is 4000, based on the bulk velocity and the channel height, and the amount of MDR from the present study is 44%, which is in good agreement with Virk's asymptote at this Reynolds number. For ‘large drag reduction’, the variations of turbulence statistics such as the mean streamwise velocity and r.m.s. velocity fluctuations are quite different from those of ‘small drag reduction’. For example, for small drag reduction, the r.m.s. streamwise velocity fluctuations decrease in the sublayer but increase in the buffer and log layers with increasing Weissenberg number, but they decrease in the whole channel for large drag reduction. As the flow approaches the MDR limit, the significant decrease in the production of turbulent kinetic energy is compensated by the increase in energy transfer from the polymer elastic energy to the turbulent kinetic energy. This is why turbulence inside the channel does not disappear but survives in the MDR state.

Visualization of Molecular Fluctuations near the Critical Point of the Coil−Stretch Transition in Polymer Elongation

The coil−stretch transition in the extension of polymers in two-dimensional flows was investigated near a critical boundary defined by simple shear flow. Visualization of individual molecules revealed a sharp coil−stretch transition in the steady-state length of the polymer with increasing Weissenberg number (Wi) in flows where the magnitude of the elongational component (||E||) slightly exceeded the rotational component (||Ω||). However, unlike in pure elongational flow, large fluctuations in the length of the polymer were observed near the critical point of the transition. These fluctuations result in a “softening” of the phase transition between coiled and extended states of the polymer. In flows where ||Ω|| is slightly greater than ||E||, significant transient polymer deformation was observed. However, the average length of the polymer as a function of Wi increased much more slowly than in simple shear flow.

Brownian dynamics simulations of isolated polymer molecules in shear flow near adsorbing and nonadsorbing surfaces

We use Brownian dynamics methods to simulate the interactions of polymers with nonadsorbing and adsorbing solid surfaces at equilibrium and during flow under very dilute conditions. Near a nonadsorbing surface, under uniform shearing flow, we find there is a depletion layer near the wall whose thickness decreases with increasing Weissenberg number, because of the compression of the chain in the shear gradient direction due to finite extensibility, in qualitative agreement with a scaling law derived by Hur et al. (2000). We also demonstrate how the presence of the wall interferes with the molecular tumbling in shear flow. For an irreversibly adsorbing wall, the adsorbed molecule is more stretched than an equivalent unadsorbed molecule in the bulk at the same shear rate. This results from a propensity for the “beads” representing the polymer conformation to affix to the surface sequentially starting at one end, due in part to the tendency in a random-walk polymer for a free end to lie at the periphery, and in part to an increased likelihood of initial contact between polymer and wall during fluctuations that expand the coil in the direction perpendicular to the wall.

Finite-amplitude solitary states in viscoelastic shear flow: computation and mechanism

Starting from stationary bifurcations in Couette–Dean flow, we compute stationary nontrivial solutions in the circular Couette geometry for an inertialess finitely extensible nonlinear elastic (FENE-P) dumbbell fluid. These solutions are isolated from the Couette flow branch arising at finite amplitude in saddle–node bifurcations as the Weissenberg number increases. Spatially, they are strongly localized axisymmetric vortex pairs embedded in an arbitrarily long ‘far field’ of pure Couette flow, and are thus qualitatively, and to some extent quantitatively, similar to the ‘diwhirl’ (Groisman & Steinberg 1997) and ‘flame’ patterns (Baumert & Muller 1999) observed experimentally. For computationally accessible parameter values, these solutions appear only above the linear instability limit of the Couette base flow, in contrast to the experimental observations. Correspondingly, they are themselves linearly unstable. Nevertheless, extrapolation of the trend in the bifurcation points with increasing polymer extensibility suggests that for sufficiently high extensibility the diwhirls will come into existence before the linear instability, as seen experimentally.Based on the computed stress and velocity fields, we propose a fully nonlinear self-sustaining mechanism for these flows. The mechanism is related to that for viscoelastic Dean flow vortices and arises from a finite-amplitude perturbation giving rise to a locally unstable profile of the azimuthal normal stress near the outer cylinder at the symmetry plane of the vortex pair. The unstable stress profile, in combination with a ‘tubeless siphon’ effect, nonlinearly sustains the patterns. We propose that these solitary, strongly nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.

Simulation for viscoelastic flow by a finite volume/element method

Stability of a second-order finite element/finite volume (FE/FV) hybrid scheme is investigated on the basis of flows with increasing Weissenberg number. FEs are used to discretise the balances of mass and momentum. For the stress equation a FV method is used, based on the recent development with fluctuation distribution schemes for pure convection problems. Examples considered include a start-up channel flow, flow past a cylinder and the non-smooth 4:1 contraction flow for an Oldroyd-B fluid. A considerable gain in efficiency per time step can be obtained compared to an alternative pure FE implementation. A distribution based on the flux terms is unstable for higher Weissenberg numbers, and this is also true for a distribution based on source terms alone. The instability is identified as being caused by the interaction of the balance equations and stress equation. A combination of distribution schemes based on flux and source terms, however, gives a considerable improvement to the hybrid FE/FV implementation. With respect to limiting Weissenberg number attenuation, the hybrid scheme is more stable than the pure FE alternative for the smooth flow past a cylinder, but less so for the non-smooth contraction flow. The influence of additional strain-rate stabilisation techniques is also analysed and found to be beneficial.

Effect of die gap width on annular extrudates by the annular extrudate swell simulation in steady‐states

AbstractWe calculated the steady‐state annular extrudate swell of polymer melts through flow geometries encountered in processes used to control parison thickness. A streamline‐upwinding finite element method with an under‐relaxation for the rate of deformation tensor was used. The Giesekus model was employed as the constitutive equation. An operation that widens the die gap is appropriate for the control of parison thickness corresponding to the change of die gap width. However, a control process that decreases the die gap width is not useful, because the parison thickness does not correspond to the die gap width. Furthermore, thickness swells change strikingly with the Weissenberg number. It is difficult to control the parison outer diameter in the case of a converging die, because the change of the outer diameter swell becomes large with increasing Weissenberg number. In the case of a diverging die, the changing value of the outer diameter swell is smaller than that in the case of a converging die.

Entry Flow of Polymer Solutions through Multi-Hole Abrupt Contractions.

Flow visualizations and velocity measurements for entry flows of shear-thinning aqueous solutions of polyacrylamide through three kinds of multi-hole contractions have been carried out using the tracer method and LDV technique. In these experiments effects of the number of contractions, distance between contractions and rheology of solutions on the flow structures have been studied. For the multi-hole contractions square arrays of 9 holes and 25 holes were used. In the steady flow regime dimensionless lip vortex length is almost the same for all multi-hole contractions. Swirling-like vortices are also observed for both three-by-three and five-by-five contractions with dimensionless pitch 4, while two-dimensional lip vortices are observed for the three-by-three contraction with dimensionless pitch 8. It is found from LDV measurements that periodic oscillations of the main flow occur at the onset of unsteady flow for all multi-hole contractions and the frequency of the oscillation increases as the Weissenberg number increases. For multi-hole contractions, flow interaction through each contraction is very important factor which decides both steady and unsteady flow structures.

Numerical Calculation of Viscoelastic Flows through Eccentric Abrupt Contraction.

Numerical simulations of viscoelastic flows through an eccentric four-to-one abrupt contraction are carried out using the Giesekus model. The SMAC (Simplified-Marker-and-Cell) method is used to analyze the three-dimensional flows. The velocity profiles along the path line passing through the center of the exit exhibit an overshoot near the entry section, and at high Weissenberg numbers an undershoot follows the overshoot. The magnitude of the stress along the same path line has a peak near the entry section, and its slow relaxation process indicates that a large downstream length is necessary for fully developed stress conditions to exist. The peak is lower than that for the flow through the concentric four-to-one abrupt contraction ; the decrease in the peak amplitude is understood to be due to the distortion of the path line in the eccentric geometry. A corner vortex, the height of which is a maximum at the widest corner, grows as the Weissenberg number increases. Furthermore, the tangential flow toward the widest section inside the vortex is determined.

The time-dependent flow of a viscoelastic fluid around a sphere

We apply the finite element method to calculate the time-dependent velocity of a sphere falling along the axis of a circular cylinder filled with a viscoelastic fluid. The mesh surrounding the sphere translates along the cylindrical wall. For our calculations, we select an Oldroyd-B fluid with viscometric properties similar to those of the M1 Boger fluid. We then calculate the motion of spheres with variable density as one would in the laboratory. We examine the effect of the Weissenberg number and of the geometry upon the essential features of the time-dependent flow. We find that, for a given sphere and a given fluid, the viscoelastic effects are damped when the radius of the cylinder becomes much larger than the radius of the sphere. The velocity overshoot decreases when the Weissenberg number increases. We also examine the sensitivity of the flow to the magnitude of the retardation time.

A finite volume technique to simulate the flow of a viscoelastic fluid

Hydrodynamically developing flow of Oldroyd B fluid in the planar die entrance region has been investigated numerically using SIMPLER algorithm in a non-uniform staggered grid system. It has been shown that for constant values of the Reynolds number, the entrance length increases as the Weissenberg number increases. For small Reynolds number flows the center line velocity distribution exhibit overshoot near the inlet, which seems to be related to the occurrence of numerical breakdown at small values of the limiting Weissenberg number than those for large Reynolds number flows. The distributions of the first normal stress difference display clearly the development of the flow characteristics from extensional flow to shear flow.

Numerical simulation of the hydrodynamically developing flow of a viscoelastic fluid

Hydrodynamically developing flow of OldroydB fluid in the planar die entrance region has been investigated numerically using SIMPLER algorithm in a non-uniform staggered grid system. It has been shown that for constant values of the Reynolds number. the entrance length increases as the Weissenberg number increases. For small Reynolds number flows the center-line velocity distributions exhibit overshoot near the inlet, which seems to be related to the occurrence of numerical breakdown at small values of the limiting Weissenberg number than those for large Reynolds number flows. The distributions of the first normal stress difference display the development of the flow characteristics from extensional flow to shear flow.