Range Of Weissenberg Numbers Research Articles

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries

The accurate and stable simulation of viscoelastic flows remains a significant computational challenge, exacerbated for flows in non-trivial and practical geometries. Here we present a new high-order meshless approach with variable resolution for the solution of viscoelastic flows across a range of Weissenberg numbers. Based on the Local Anisotropic Basis Function Method (LABFM) of King et al. (2020), highly accurate viscoelastic flow solutions are found using Oldroyd B and PPT models for a range of two dimensional problems — including Kolmogorov flow, planar Poiseulle flow, and flow in a representative porous media geometry. Convergence rates up to 9th order are shown. Three treatments for the conformation tensor evolution are investigated for use in this new high-order meshless context (direct integration, Cholesky decomposition, and log-conformation), with log-conformation providing consistently stable solutions across test cases, and direct integration yielding better accuracy for simpler unidirectional flows. The final test considers symmetry breaking in the porous media flow at moderate Weissenberg number, as a precursor to a future study of fully 3D high-fidelity simulations of elastic flow instabilities in complex geometries. The results herein demonstrate the potential of a viscoelastic flow solver that is both high-order (for accuracy) and meshless (for straightforward discretisation of non-trivial geometries including variable resolution). In the near-term, extension of this approach to three dimensional solutions promises to yield important insights into a range of viscoelastic flow problems, and especially the fundamental challenge of understanding elastic instabilities in practical settings.

Particle migration in channel flow of an elastoviscoplastic fluid

We study the dynamics of a neutrally buoyant rigid sphere carried by an elastoviscoplastic fluid in a pressure-driven channel flow numerically. The yielding to flow is marked by the yield stress which splits the flow into two main regions: the core unyielded region and two sheared yielded regions close to the walls. The particles which are initially in the plug region are observed to translate with the same velocity as the plug without any rotation/migration. Keeping the Reynolds number fixed, we study the effect of elasticity (Weissenberg number) and plasticity (Bingham number) of the fluid on the particle migration inside the sheared regions. In the viscoelastic limit, in the range of studied parameters (low elasticity), inertia is dominant and the particle finds its equilibrium position between the centreline and the wall. The same happens in the viscoplastic limit, yet the yield surface plays the role of centreline. However, the combination of elasticity and plasticity of the suspending fluid (elastoviscoplasticity) trigger particle-focusing: in the elastoviscoplastic flow, for a certain range of Weissenberg numbers (≈0.5), isolated particles migrate all the way to the centreline by entering into the core plug region. This behaviour suggests a particle-focusing process for inertial regimes which was not previously found in a viscoelastic or viscoplastic carrying fluid.

Observation of lock-in for viscoelastic fluid–structure interactions

Under the right conditions, viscoelastic fluids can exhibit elastic flow instabilities in the absence of inertia. If the fluid is in contact with a flexible or flexibly-mounted structure, these elastic flow instabilities lead to time-dependent forces on the structure, which can cause the structure to oscillate. This constitutes a new class of Viscoelastic Fluid–Structure Interactions (VFSI). Up until now, the VFSI reported in the literature demonstrated a one-way coupling between the fluid and the structure where little feedback from the structure to the flow was observed. Here, we report for the very first time, the presence of a lock-in behavior in VFSI. We have designed and conducted a set of experiments in which the frequency of the elastic flow instabilities and the natural frequency of the flexible structure become equal as the flow velocity is increased, and therefore lock-in is observed. We present amplitude and frequency responses of two sets of flexibly-mounted cylinders over a range of Weissenberg numbers and reduced velocities to show the lock-in range. The time-varying flow fields inside and outside the lock-in range were studied through streak-line imaging and particle image velocimetry. The viscoelastic lock-in range was observed for a Weissenberg number range of 5≤Wi≤15 for one of the cylinders tested and for a range of 20≤Wi≤30 for the other. For both cylinders, the lock-in started at a reduced velocity of Vr≈1.5. In Cylinder 1, the lock-in ended at Vr≈4, while for Cylinder 2 the end of the lock-in was not observed, because the data points did not cover a wide enough range. Similar to the lock-in observed in the typical VIV response in a Newtonian fluid, the frequency of the elastic flow instability was found to increase with increasing flow velocity before plateauing at the natural frequency of the cylinder in the lock-in range. Unlike the VIV response of a Newtonian fluid, where the lock-in range corresponds to the maximum observed amplitude of oscillations, in VFSI the amplitude of oscillations reached a plateau while in lock-in, but increased with Weissenberg number and reduced velocity before and after the lock-in range.

Polymer extrudate-swell: From monodisperse melts to polydispersity and flow-induced reduction in monomer friction

This paper describes finite-element simulations and associated experimental studies of extrudate swell for near-monodisperse and polydisperse polystyrenes. The tube-model based Rolie-Poly constitutive model, when extended to include a reduction of monomeric friction at high extension rates, makes much-improved predictions of extrudate swell at high Weissenberg number. This is especially significant for near-monodisperse polymers where rheological features are unchanged by the effects of polydispersity. Extension of this molecular rheology scheme to a polydisperse constitutive model addresses extrusion experiments on polydisperse polystyrenes inside a multipass rheometer, accounting for experimental data up to Rouse Weissenberg numbers of 50. We, therefore, show that from a measurement of polymer molecular weight distribution, it is possible to predict extrudate swell over a broad range of processing conditions for polydisperse polymers and realistic extrusion processes. Small changes in the capillary length to diameter ratio have little effect on extrudate swell in this range of Weissenberg number. This is because the capillary residence time is sufficiently long for a steady state to be reached within the polymer stretch relaxation time, which controls the most decisive physics responsible for extrudate swell.

Proper Orthogonal Decomposition (POD) of the flow dynamics for a viscoelastic fluid in a four-roll mill geometry at the Stokes limit

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by the decomposition, indicates that the motion of the stagnation points drives the different flow transitions. In particular, a transition to an asymmetric state occurs when the extensional stagnation points lose their pinning to the background forcing. A further transition to higher frequency modal dynamics occurs when the stagnation points that were initially tied by the forcing to the centers of the rolls, begin to move. The relative frequencies of the motion of these stagnation points is a critical factor in determining the complexity of the flow, measured by the number of modes needed to capture most of the energy in the system. Even when the flows are more complex a small number of modes is sufficient to capture the time evolution of these flows, demonstrating the usefulness of the POD applied to viscoelastic fluids at zero Reynolds number.

Dynamics of a single buoyant plume in a FENE-P fluid

The dynamics of a single laminar buoyant plume in a FENE-P (finitely extensible nonlinear elastic-Peterlin) fluid has been investigated by performing a series of direct simulations over a range of Weissenberg numbers. Examination of this flow has given insight into the heat transfer reduction phenomenon observed recently in more complex Rayleigh-Benard turbulence. The simulations, which were performed with a Rayleigh number of 2.53×106 and a maximal polymeric extension of L = 100, show that the wall heat flux is reduced by about 28% at a Weissenberg number of 20, which is in reasonable agreement with the results obtained in Rayleigh-Benard turbulence. In addition, the global flow kinetic energy was reduced by about one order of magnitude.

Experimental investigation of the impact of elastic turbulence on heat transfer in a serpentine channel

The characteristics of convective heat transfer and fluid flow within a square cross-section serpentine channel are experimentally studied for two groups of polymeric viscoelastic fluids, shear-thinning and constant-viscosity Boger solutions. The elastic turbulence can be created by the non-linear interaction between elastic stresses generated within the flowing high-molecular-weight polymer solutions and the streamline curvature. In order to confirm elastic turbulence in this geometry, pressure drop across the serpentine channel was measured. The findings indicate that the measurements of non-dimensional pressure-drop increase approximately from 1.48 to 4.82 for viscoelastic solutions compared with the Newtonian fluid over a range of Weissenberg number from 4 to 211. The convective heat transfer enhances due to elastic turbulence by up to 200% for low polymer concentration (dilute) solutions and reaches up to 380% for higher polymer concentration (semi-dilute) solutions under creeping-flow conditions in comparison to that achieved by the equivalent Newtonian fluid flow at low Graetz number (up to 14.6). We propose a modified Weissenberg number which is able to approximately collapse the mean Nusselt number data for each solution group.

Approximation of the two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched finite element shape functions

The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we present a finite element method to solve the two immiscible fluid flow problems using the level set method. When the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The method is tested on two problems: the first example consists of a sloshing case that involves the interaction of a Giesekus and a Newtonian fluid. This example shows that the enriched pressure functions permit the exact resolution of the hydrostatic rest state. The second example is the classical jet buckling problem used to validate our method. To permit the use of equal interpolation between the variables, we use a variational multiscale formulation proposed recently by Castillo and Codina (2014) [21], that has shown very good stability properties, permitting also the resolution of the jet buckling flow problem in the the range of Weissenberg number 0 < We < 100, using the Oldroyd-B model without any sign of numerical instability. Additional features of the work are the inclusion of a discontinuity capturing technique for the constitutive equation and some comparisons between a monolithic resolution and a fractional step approach to solve the viscoelastic fluid flow problem from the point of view of computational requirements.

Modelling polymeric flows in reverse roll coating processes with dynamic wetting lines and air-entrainment: FENE and PTT solutions

Viscoelastic FENE and PTT constitutive models are considered in the simulation of high-speed reverse roll coating flow with polymeric paint solutions, where wetting lines and air-entrainment are incorporated. Both FENE-CR and FENE-P models are utilised to illustrate the influence of shear-thinning in the context of strain-hardening properties. Strain-softening is introduced via PTT solutions in contrast. Steady solutions are derived numerically through a time-stepping hybrid finite element-finite volume sub-cell algorithm with dynamic free-surface location, drawing upon a fractional staged, predictor-corrector, and semi-implicit time-stepping procedure. A systematic computational and rheological study allows for parametric variation in elasticity (Wi-variation), level of strain-hardening (AL), shear-thinning and solvent fraction (β). Various problem aspects are investigated to reveal the influence of viscoelasticity on vortex developments, pressure and lift profiles, shear and strain-rates, and critical stress states. Specific features of viscoelastic flow response are observed by analysing stress and flow structures over a range of Weissenberg numbers. The novel aspects of the work lie in the application of the algorithm to the reverse roll coating process under such viscoelastic fluid models.

Quantitative characterization of high molecular weight polymer solutions in microfluidic hyperbolic contraction flow

Nonlinear flows of polyacrylamide (PAAm) ( $$M_{\text{w}} = 5.7 \times 10^{6}\,{\text{g}}/{\text{mol}}$$ ) aqueous solutions through a micro-fabricated, hyperbolic contraction geometry with high Hencky strain ( $$\varepsilon_{{\text{H}}} = 3.7$$ ) have been characterized by micro-particle image velocimetry ( $$\mu $$ -PIV). Various flow dynamics regimes in a range of Weissenberg number (Wi) and Reynolds number (Re) are presented in a Wi–Re diagram. The symmetric corner vortices are only observed in the flow of low concentration PAAm solution ( $$c/c^{*}=3.3$$ ). In a higher concentration ( $$c/c^{*}=8.3$$ ), PAAm solution exhibits chaotic-like flow patterns in the strong nonlinear flow regime ( $$Wi>350$$ ). Extensional deformation in nonlinear flows of Wi up to 860 has been analyzed. Furthermore, the local stretch experienced by the polymer chain in complex flow is systematically quantified and linked to the corresponding velocity vector fields, which are valuable for understanding the highly nonlinear flow phenomena.

Viscoelastic computations for reverse roll coating with dynamic wetting lines and the Phan-Thien-Tanner models

The computational modelling of reverse roll coating with dynamic wetting line has been analysed for various non-Newtonian viscoelastic materials appealing to the Phan-Thien-Tanner (PTT) network class of models suitable for typical polymer solutions, with properties of shear thinning and strain hardening/softening. The numerical technique utilizes a hybrid finite element-sub-cell finite volume algorithm with a dynamic free-surface location, drawing upon a fractional-staged predictor-corrector semi-implicit time-stepping procedure of an incremental pressure-correction form. The numerical solution is investigated following a systematic study which allows for parametric variation in elasticity (We-variation), extensional hardening-softening (e), and solvent fraction (β). Under incompressible flow conditions, linear PTT (LPTT) and exponential PTT (EPTT) models were used to solve the paint strip coatings, under reverse roll-coating configuration. This involves two-dimensional planar reverse roll-coating domains, considering a range of Weissenberg numbers (We) up to critical levels, addressing velocity fields and vortex development, pressure and lift profiles, shear rate, and stress fields. Various differences are observed when comparing solutions for these constitutive models. Concerning the effects of elasticity, increase in We stimulates vortex structures, which are visible at both the downstream meniscus and upstream narrowest nip region, whilst decreasing the peak pressure and lift values at the nip constriction. At low values (e > 0.5, β = 0.1) of extensional viscosity, the LPTT flow fields were much easier to extract, attaining critical We levels up to unity, in contrast to critical We levels of 0.4 for EPTT solutions. This finding is reversed at higher extensional viscosity levels (e < 0.5). This trend reveals qualitative agreement with theoretical studies. Noting flow behaviour under EPTT solution, increasing the peak level of strain hardening/softening is found to stimulate vortex activity around the nip region, with a corresponding increase in peak pressure and lift values.

Settling of spherical particles in unbounded and confined surfactant-based shear thinning viscoelastic fluids: An experimental study

An experimental study is performed to understand and quantify settling velocity of spherical particles in unbounded and confined surfactant-based shear thinning viscoelastic fluids. Experimental data is presented to show that elastic effects can increase or decrease the settling velocity of particles, even in the creeping flow regime. Experimental data shows that a significant drag reduction occurs with increase in Weissenberg number. This is followed by a transition to increasing drag at higher Weissenberg numbers. A new correlation is presented for the sphere settling velocity in unbounded viscoelastic fluids as a function of the fluid rheology and the proppant properties.The wall factors for sphere settling velocities in viscoelastic fluids confined between solid parallel plates are calculated from experimental measurements made on these fluids over a range of Weissenberg numbers. Results indicate that elasticity reduces the effect of the confining walls and this reduction is more pronounced at higher ratios of the particle diameter to spacing between the walls. Shear thinning behavior of fluids is observed to reduce the retardation effect of the confining walls. A new empirical correlation for wall factors for spheres settling in a viscoelastic fluid confined between two parallel walls is presented.

Peristaltic pumping and irreversibility of a Stokesian viscoelastic fluid

Peristaltic pumping by wavelike contractions is a fundamental biomechanical mechanism for fluid and material transport and is used in the esophagus, intestine, oviduct, and ureter. While peristaltic pumping of a Newtonian fluid is well understood, in many important settings, as in the fluid dynamics of reproduction, the fluids have non-Newtonian responses. Here, we present a numerical method for simulating an Oldroyd-B fluid coupled to contractile, moving walls. A marker and cell grid-based projection method is used for the fluid equations and an immersed boundary method is used for coupling to a Lagrangian representation of the deforming walls. We examine numerically the peristaltic transport of a highly viscous Oldroyd-B fluid over a range of Weissenberg numbers and peristalsis wavelengths and amplitudes.

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.

A spectral domain decomposition method for the planar non-Newtonian stick-slip problem

A spectral domain decomposition method for solving the planar stick-slip problem for an Oldroyd-B fluid is presented. The flow domain is decomposed into two semi-infinite rectangular subdomains. A comparison of the treatment of these subdomains by domain truncation and algebraic mapping is given. A fixed point method is used to linearize the constitutive equation. The linearized systems are solved in a manner sympathetic towards the structure of the coefficient matrices at each iterative step. The algorithm employed in solving the stick-slip problem is shown to be efficient in its demand upon array storage and CPU time. The numerical results demonstrate the necessity of adequate entry and exit lengths for the non-Newtonian problem and the adverse effect that the singularity has upon the range of Weissenberg numbers over which converged solutions are possible. Results generated from a regularized stick-slip geometry where the boundary singularity is smoothed out provide evidence that it is possible to extend this range by solving a slightly modified problem in which the effect of the singularity is weakened.