SWANSEA UNIVERSITY

REPORT SERIES

COMPUTATIONAL RHEOLGY

by

M.F. Webster, H.R. Tamaddon-Jahromi, and F. Belblidia

Report # CSR 10-2008

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M.F. Webster, H.R. Tamaddon-Jahromi, and F. Belblidia Institute of non-Newtonian Fluid Mechanics - Wales, Department of Computer Science, Swansea University, Singleton Park, SA2 8PP, Swansea, UK Key Words: Rheology, Viscoelasticity, Inelastic, Constitutive models, Kinetic theory, Contraction flows, Finite elements/volume, Weissenberg number, Compressibility, Transient flows, Vortex activity, Pressure-drop, Extensional flow, Numerical predictions, Experiments Contents 1. Introduction 2. Flow and governing equations 3. Numerical discretisation – generalization and a local scheme 4. The contraction flow benchmark 5. Conclusion Glossary Bibliography Biographical Sketch Summary

This chapter offers a general overview of the present state of the art in the field of computational rheology, emphasizing the various numerical challenges that have emerged through the material modeling and discretisation approaches adopted. There is brief coverage of the alternative levels of modeling involved, from macroscopic to mesoscopic, alongside the different discrete techniques utilized and the challenges posed by prescribed benchmark problems. Progress in the field is described historically through consideration of a particular benchmark problem, contraction flow, detailing advances in experimental, numerical and analytical developments, that illustrate the many branches of computational rheology. Several case studies are also presented that illustrate some selected applications, covering such interests areas as transient flow solutions, kinetic-based models, pressure drop analysis and compressible-viscoelastic computations.

1. Introduction

Computational rheology involves the use of numerical simulation for non-Newtonian fluid flow in complex geometries, and in particular introduces viscoelastic effects in polymeric liquids. For more than thirty years, the subject has been under intense development alongside the tremendous increase in computer power, and has now reached a state of relative maturity with new horizons ahead. To date, many reviews have appeared by an evolving international research community, see for example general reviews by Tanner & Walters, Crochet et al., Keunings, Baaijens, and Walters & Webster.

Industrial application remains a principal driving force behind the development and advance in computational rheology. Fluids of interest include molten plastics and polymer solutions, as well as industrial suspensions, multigrade oils, liquid detergents, printing inks and various food and confectionary products. The processes of relevance include injection molding, extrusion, coating,

COMPUTATIONAL RHEOLGY

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mixing and lubrication. The research field is strongly motivated, broad in its scope and challenging. This introduces some fascinating numerical challenges to computational rheology. Across the broad range of application this article is purposely restricted to viscoelastic liquids and contraction flow geometries.

The challenge for computational rheologists is to develop accurate, stable and robust numerical schemes based on realistic physical-mathematical models. This advancing field offers opportunities at different levels. Theoreticians, and material modelers, are required to reduce the gap between the several levels of description employed to predict polymer dynamics, including the choice of appropriate macroscopic constitutive equation, or through molecular modeling. This aids in improved understanding of some of the macroscopic flow phenomena observed in non-Newtonian fluid mechanics, such as vortex activity and flow instability. Experimentalists wish to improve the design of their equipment and experimental procedures through better interpretation of their data. This may be advanced by identifying and eliminating experimental deficiencies, such as flow heterogeneities. It is the combination of flow experiments and numerical predictions that is sought to characterize rheological behavior in complex flows. With industrial application and production in mind, improved properties of end-of-line products are desired through the combination of computational rheology and computer aided design studies.

Polymeric liquids are considered as viscoelastic materials, with their particular properties responsible for some extravagant flow phenomena. There is need to analyze and predict their flow behavior, through a combination of suitable physical models and numerical techniques. The challenge posed by polymeric materials is the broad range of time and length-scales separating the relevant atomistic and macroscopic processes involved, supplemented by the large number of microstructural entities. Analysis is made possible by appealing to a coupling of theoretical models, spanning from quantum mechanics to continuum mechanics.

At the micro-structural level, the atomistic approach remains restricted to coarse models for polymeric liquid flow in geometries of molecular dimensions, such as considered in the wall-slip problem. This is mainly due to the significant computer resources required in such simulations. The ultimate aim, here is to solve such flow problems on the macroscopic level, yet based on this atomistic approach.

The kinetic theory level is the next level of description of polymeric liquids, where stochastic simulation or Brownian dynamics methods are employed. For example, a dilute solution of linear polymers in a Newtonian solvent may be described at a macroscopic level, by a freely-jointed bead-rod Kramers chain of a number of beads connected linearly by rigid segments or springs. Increasing the number of beads provides model resolution at this level of representation. Kinetic theory models provide a coarse-grained description of molecular configurations, wherein atomistic processes are ignored. Such a model tackles important features that govern the flow-induced evolution of configurations. In recent years, kinetic theory has advanced well beyond the classical reptation tube-model proposed by de Gennes (1971) and further developed by Doi & Edwards (1986). An impressive number of recent Brownian dynamics studies have emerged based on Kramers chains, bead-spring chains and dumbbells. The interest in such theory continues to grow, through which numerical studies have significantly advanced the understanding of polymer dynamics in general.

Under the widely utilized continuum mechanics approach, the fluid microstructure is not explicitly taken into account. The combination is forged between a suitable constitutive equation (differential or integral) which reflects material response under deformation, with the governing equations of motion for viscoelastic fluids; typically for isothermal incompressible flow, those of continuity and momentum equations. The resulting system is solved commonly through space-time partial differential equations,

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in the form of non-linear, mixed-type parabolic-hyperbolic equation systems. Some constitutive equations applied in continuum modeling have originated from molecular models and kinetic theory. This is typified through the pom-pom constitutive equation, which was developed for branched polymer architectures.

It is the coupling of the coarse-grained molecular scale of kinetic theory to the macroscopic scale of continuum mechanics that constitutes the micro-macro approach to computational rheology. The micro-macro approach is much more demanding upon computer resources than more conventional continuum simulations, allowing for the direct exploitation of kinetic theory models, and hence, avoiding potentially harmful closure approximations. Under the micro-macro methodology, appropriate distributions of configuration spaces for molecular orientation must be sought, while under the continuum assumption, an integration of a constitutive equation is applied to evaluate the viscoelastic contribution to the stress tensor.

Recently, alternative approaches have begun to appear in the computer modeling of polymeric liquids covering a pore-scale description of flow. The attraction of such approaches is their inherent particle-based construction, resulting in meshless schemes. Here, only two formulations are cited: dissipative particle dynamics and Lattice Boltzmann modeling. Both such methods are increasing in popularity and beginning to tackle the more complex problems of serious interest to the field.

In the rapidly evolving research area of computational rheology, there is an overarching requirement to quality assess any numerical method proposed. This has led the field to establish a set of benchmark problems, each exposing particular features of interest, and against which the properties of a numerical scheme may be judged. Some of these problems are introduced and discussed in Tanner (Engineering Rheology, 2000). The selected benchmarks outlined with their features are: flow around a sphere in a tube, flow past a cylinder, flow near sharp corners and separation points, entry flow such as contraction flow, and the extrusion problem. The flow around a motionless sphere in a tube is a geometrically smooth problem, yet numerical failure occurs for this problem at relatively low levels of elasticity. This is caused by the emergence of a thin stress boundary layer in the wake region beyond the sphere and the drag exerted on the sphere itself. Some experimental observation (as in Walters & Tanner, 1992) has suggested that drag reduction is caused by shear thinning, while fluid elasticity triggers drag enhancement. In this regard, drag calculation has become accepted as a useful means to assess numerical precision and to distinguish between the performance of a wide variety of predictive techniques. In a similar manner, drag characterization has been utilized in the alternative configuration of flow around a cylinder in a channel. Alternatively, flow near corners and separation points introduces additional numerical difficulties, such as typified in the stick-slip flow problem. This problem exhibits steep stress gradients that necessitate rigorous mesh refinement procedures. Similarly, entry flow problems such as contraction flows, present yet further numerical complexity, where the contraction ratio is a parameter of choice, popularly selected as four. In addition to the presence of the stress singularity at the re-entrant corner for abrupt contractions, this problem offers features of vortex enhancement/inhibition and pressure loss at the entry as further means to evaluate predictive precision. Moreover, the availability of asymptotic analytical solution for velocity and stress near the singularity for certain models, such as Upper Convective Maxwell, Oldroyd-B and Phan-Thien/Tanner models, validates mesh convergence findings. Stress singularity is also observed in the extrusion problem, alongside a free-surface to the extrudate. Here, the singularity is related to the change in the boundary condition at the exit of the die (transition from stick to slip), and further complication arises through the discrete treatment of the free boundary condition on the jet. This necessitates tracking of the free-surface, which is a challenging task even for relatively simplistic material properties.

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In this coverage of the wide subject matter on computational rheology, attention is purposely restricted to focus upon applications of interest and features that arise through one of these benchmark problems, that of contraction flow. The merits and features of this problem are drawn out historically, through the many branches of computational techniques that have developed. In order to introduce the discrete treatment of such problems, there is brief coverage of various aspects involved in material and flow modeling. Several case studies on this benchmark problem illustrate some novel and current topics of interest, including: the prediction of some transient flows, solutions with pom-pom models, pressure drop analysis and viscoelastic-compressibility computations.

2. Flow and governing equations

The formulation of constitutive equations is essential in order to predict the behavior of viscoelastic materials in many complex and industrial flows including those involving polymers, colloids, foams, and gel processes. There is need here to provide a brief overview of this wide topic, so that motivation and background introduces the computational aspects to follow. Being a rapidly growing and industrially important field, rheology plays a significant role in polymer processing, food processing, coating and printing, and many other manufacturing processes. In the early days, the so-called Upper-Convected Maxwell (UCM) and Oldroyd-B models (implicit in stress) were strongly favored. This was partly due to the fact that they assumed the ‘bottom-line’ of acceptable mathematical simplicity, whilst also being able to mimic the complex rheometrical behavior for a class of polymer solutions, known as Boger fluids, which became popular in the late 1970s.

To establish a rheological equation of state, the principle of material objectivity states that such an equation remains unchanged under a change of frame, even if the frame is time dependent. Then, if one assumes that the present response of a material is dependent upon its past history of deformation, the constitutive equation assumes a relationship between stress and history of response of the material for all times in the subsequent past. Based upon continuum modeling, the Cauchy stress tensor ikσ may be represented as

ik ik ikp Tσ = − δ + , ( s ) ( p )

ik ik ikT T T ,= + (1)

where p is an arbitrary isotropic pressure,ikδ the Kronecker delta, and ( )ikT is an extra-stress tensor that may be split into viscous ( )( )sikT and polymeric

( )( )pikT contributions. ( )s

ikT may be described by a

generalized Newtonian model, with a viscosity ( )η dependent on second and third invariants, 2I and 3I , of the deformation-rate tensor

†( ) / 2= ∇ + ∇d u u , as

( ) ( ) ( )12 2 tr2

( s )ik ikT d , , , det .η= Γ = = =

22 2 3Γ,Σ I I d I d (2)

The generalized variable Γ can be identified as a shear-rate ( )ɺγ in a steady simple shear flow. Similarly, 3 /=

3 2Σ I I can be identified as an extension-rate ( )εɺ in a steady uniaxial extensional flow.

The specific form of η ( )Γ,Σ is model dependent. Various forms are commended from constant, to shear dependent, to extensional dependent, and blends of both.

To appreciate some of the properties of non-Newtonian fluids under flow, it is useful to consider classical viscometric flow conditions. So, for example, the velocity field ( )u,v,w=u and deformation-rate tensor d under simple shear deformation are given as:

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( )0

0

u y y

v

w

===

ɺγ and

0 01

0 02

0 0 0

.

=

ɺ

ɺd

γγ (3)

Here, other properties that vary between Newtonian and non-Newtonian fluids are the first and second normal stress difference, 1N and 2N , with respective coefficients, 1 2 and ψ ψ . Their magnitudes are zero for Newtonian fluids, whilst for non-Newtonian fluids, their definitions are:

( ) ( ) 21 1xx yyN = − =ɺ ɺ ɺγ σ σ ψ γ γ ,

( ) ( ) 22 2yy zzN = − =ɺ ɺ ɺγ σ σ ψ γ γ . (4)

This normally leads to relations,1 0N > and 1 2N N>> .

Similarly, the velocity field ( )u,v,w=u and deformation-rate tensor d in uniaxial extensional flow can be expressed as a function of extension-rate ( )εɺ as:

( )( )( )

12

12

x = x

y =- y

z =- z

u ε

v ε

w ε

ɺ

ɺ

ɺ

and 1212

0 0

0 - 0

0 0 -

ε

ε .

ε

=

ɺ

ɺ

ɺ

d (5)

Here, the resistance to flow, or extensional viscosity ( )E ,η is then expressed through the relationship,

( )Eηxx yy xx zz ε ε− = − = ɺ ɺσ σ σ σ . (6) For a Newtonian fluid, Eη is constant under all strain-rates, and is simply three times the shear

viscosity ( )Sη :

E Sη = 3η . (7)

Additionally, all fluids, including those with viscoelastic properties, satisfy relation (7) at low deformation-rates, that is, ( ) ( )E Sη 0 3η 0→ = →ɺ ɺε γ . A further related quantity of interest is the so-called Trouton ratio (Tr ), defined as the quotient of extensional and shear viscosities. In order to relate γɺ and εɺ to evaluate shear and extensional viscosities, one may define the Trouton ratio in the form:

( )( )

E

S

η

η 3Tr .=

=

ɺ

ɺ ɺ

εγ ε

(8)

For inelastic fluids with shear-dependent viscosity, Tr is three for all values of εɺ , and for viscoelastic fluids this ratio is anticipated to satisfy:( )0 3Tr ε → =ɺ . In contraction flows, pure extension takes place along the centerline. In axisymmetric configurations this elongation is uniaxial, whilst in 2D planar flows, such deformation is that of planar extension, where the fluid sample is stretched in one direction and compressed in another, leaving one dimension unchanged. Expressions for extensional viscosity and Trouton ratio in planar extension are similar to those for uniaxial flow,

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E Sη = 4η (Newtonian),

( ) ( )E Sη 0 4η 0→ = →ɺ ɺε γ (Viscoelastic),

( )( )

E

S

η

η 2Tr =

=ɺ

ɺ ɺ

εγ ε

, (9)

( )0 4Tr → =ɺε (Viscoelastic). Assuming isothermal condition for simplicity, the governing differential equations for viscoelastic flow may be represented through those for mass-conservation and momentum-transport, in conjunction with equations of state for stress and density (compressible flow). As such, the flow equations governing continuity and momentum balance may be expressed as:

( ) 0t

∂ + ∇ ⋅ =∂

uρ ρ ,

( )223s s s

pt

∂ = ∇ ⋅ + − ∇ ⋅ + − ∇ ∂

ud u I

κρ η ηη

ττττ (10)

where, ,ρ ,u and ττττ represent density, velocity, and extra-stress, respectively. Viscosity material parameters of η , pη and sη , represent factors of total, polymeric-fraction and solvent-fraction, respectively, where p s= +η η η . κ is a generalized factor that mimics the role of bulk viscosity. Bulk viscosity arises as a consequence of active rotational and vibrational modes at the polyatomic molecular level, relevant in compressible gas or granular matter flow. For convenience, (s)τ can be introduced as being the augmented solvent stress referenced within the momentum equation,

( )( ) 22 .3

ss s

s

= + − ∇⋅

d u I

κη ηη

ττττ (11)

Clearly, considerable simplification results under incompressible flow conditions (constant density). To complete the set of governing equations for compressible flow, it is necessary to introduce an equation of state to relate density to pressure. Here, for example, one may employ amongst others, the modified Tait equation of state a well-established formulation for liquids,

0 0

mp B

p B

+ = +

ɶ

ɶ

ρρ

, with augmented pressure1

tr 23

sp p . = − +

ɶ τ dηη

(12)

Parameters B and m are constants, and 0pɶ , 0ρ denote reference scales for pressure and density. Note that, this state law is often approximated to a linear form with index m set to unity.

To extend incompressible algorithms to deal with compressible flows, it is important to appreciate the key role that pressure plays in a compressible flow. In the low Mach number ( )Ma limit, where density is almost constant, the role of pressure is to influence velocity through the continuity equation, so that conservation of mass is satisfied. Indeed, in this instance, density and pressure are only weakly-linked variables. To recast an incompressible scheme into one appropriate for weakly-compressible highly-viscous flow, one may replace the temporal derivative of density in the continuity equation with its

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equivalent in pressure, appealing to an equation of state. Assuming isentropic conditions, and employing the differential chain rule, one gathers,

2(X,t)

ρ ρ p 1 p= =

t p t c t

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

,

2(X,t)

p m(p+B)= =cρ ρ

∂∂ ∂

ɶ, (13)

where, (X,t)

c introduces the speed of sound, a field variable distributed in space and time, often

interpreted as a ratio formed with fluid velocity and expressed through the Mach number Ma u / c= . This switch of representation between density and pressure time derivatives has a major impact in pressure-correction algorithms governing the continuity-pressure equation (see discussion below).

Using general physical principles, constitutive equations have been derived to relate the state-of-stress to the history-of-deformation in a material. Yet, in order to obtain quantitative agreement with experimental data, realistic models are required that are capable of well representing the rheometrical response of actual fluids. Typical examples would include: Phan-Thien/Tanner (PTT), Giesekus, and the Kaye-Bernstein-Kearsley-Zapas (K-BKZ) models, which are phenomenological constitutive equations commonly used to model polymer melt behavior. Characteristically, the Giesekus model in extensional deformation sustains hardening with increase in strain-rates, yet ultimately, reaches a plateau. A modified multi-mode K-BKZ model has also been developed that successfully predicts contraction flows of LDPE melts. Moreover, PTT models can reproduce a variety of rheological responses in both planar and uniaxial extension, though the parameter controlling the degree of extension,PTTε , also affects shear-viscosity properties. In addition, eXtended Pom-Pom (XPP) models

are capable of reproducing the response of polymeric systems in rheometrical flows. These versions are derived from the kinetic-based pom-pom model introduced by McLeish & Larson, based on reptation dynamics of an idealized linear molecule with an equal number of branched arms at both ends.

Here to supply a concrete example, one may employ a generalized formulation to express the relevant constitutive law. The notation encompasses the Single eXtended Pom-Pom (SXPP) model, which requires a set of additional parameters( ), , ,q λ αε , selected to represent system entanglement ( )ε , number of side-branch arms to the backbone segment ( )q , stretch of the backbone segment ( )λ , and

anisotropy of the polymeric network ( )α . With / ss ss p

= =+ηβ η η

η η, such a generalized constitutive

equation statement covers a series of models, taking the form:

( ) ( )s ss

(1 )f , g , 2(1 ) .

(1 )

WiWi

Wi

∇ − α λ + + λ + ⋅ = − −

β ββ

I dτ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ τ (14)

Here, I is the unit tensor, ∇ττττ represents the upper-convected material derivative of ττττ , and the

convected derivative is expressed as

†. ( . .( ).t

∂= + ∇ − ∇ + ∇∂

u u) u∇∇∇∇ τττττ τ τ ττ τ τ ττ τ τ ττ τ τ τ (15)

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The generalized functions f ( , )λ ττττ and g( , )λ ττττ categorizing the choice over models. Table 1 provides the classification through functions { }( )f ,g ,λ ττττ and setting of ( )α -parameter to distinguish between Oldroyd-B, PTT and SXPP fluids.

Model f ( , )λ ττττ g( , )λ ττττ α

Oldroyd B( )− ττττ 1 0 0

LPTT( )ττττ

s

1 tr( )(1 )

PTTε Wi+− β

ττττ 0 0

EPTT( )ττττ exp[ tr( )]

(1 )PTT

s

ε Wi

− βττττ

0 0

SXPP( )ττττ f ( , )λ ττττ f ( , ) 1λ −ττττ α

Table 1. Classification of functions; f ( , )λ ττττ , g( , )λ ττττ

The definition of function f ( , )λ ττττ is all important to this class of models, being

( ) ( ) ( )2

2

12 1 1f , 1 e 1 tr .

(1 ) 3s

Wiν λ − α λ = − + − ⋅ λ λ − ε βτ τ ττ τ ττ τ ττ τ τ (16)

In the expression for ( )f ,λ ττττ , the free parameter ν is estimated by data-fitting and found to be inversely proportional to the number of side-branch arms to the molecular chain-segment ( )2 / qν = . With the single-equation form of the pom-pom model, the backbone stretch parameter λ is identified through an algebraic expression,

( )11 tr3 (1 )s

Wiλ = +− β

ττττ . (17)

Essentially, this formulation collapses the double-equation version of the pom-pom model (DXPP), where a differential equation dictates the change in (t)λ . The pom-pom model has been derived to characterize the rheological behavior of polymer melts with long side-branches, such as applicable for low density polyethylenes. In addition, this model possesses features of the Giesekus model since a non-zero second normal stress-difference is predicted when anisotropy parameter 0α ≠ . The Oldroyd-B model corresponds to setting f ( , ) 1λ =ττττ and, 0α = . Then, the conventional definitions of group numbers, in Reynolds (Re) and Weissenberg (Wi) numbers, with pom-pom parameters sβ and ε , are defined as:

scaleU L

Re= ρη

, b

scale

UWi

L= τ , ss

s b

=+ηβ

η τ0G , s

b

= τετ

. (18)

In this notational form,bτ and sτ represent the backbone orientation and stretch relaxation time-scales, respectively. 0

G is the linear relaxation modulus and the parameter ε is the ratio of stretch to

orientation relaxation times. Typically, re choice of scales and for contraction flows in particular, the

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characteristic velocity (U) and length (Lscale) scales are frequently taken as the downstream mean velocity and channel half-height, respectively. For simplicity in the case of pom-pom models, pη may be defined as 0 bGpη τ= to preserve similarity between the versions of Oldroyd-B and SXPP models.

In contrast to the above instantaneous differential view, integral constitutive models provide an alternative description of non-Newtonian elastico-viscous response, that incorporate the past history of deformation. In this formulation, the Cauchy stress ikσ may be represented as,

( ') ( ( ', )) 't

ik ik ikp h t t g d t t dtσ δ−∞

= − + −∫ (19)

where t and 't are present and past times of the fluid element flowing along its trajectory, h is the memory function of linear viscoelsticity, and g is a model-dependent non-linear strain measure. Such an approach gives rise to integro-differential systems to solve, necessitating integral evaluation along fluid element trajectories throughout the past history of deformation. Each of these approaches has its merits and drawbacks – preference is given henceforth to the differential view, that is pursued below.

3. Numerical discretisation – generalization and a local scheme

Commencing from a generalized standpoint, a number of options arise in the discretisation of viscoelastic flow problems. In particular and relevant for differential constitutive models, mixed-type space-time coupled partial differential systems must be accommodated with solution-dependent source terms. Temporal discretisation introduces some form of time-stepping between solution phases, variants being of implicit, explicit or semi-implicit cross-reference. Any form of implicitness requires matrix-bound solvers, whilst explicit forms may be solved in an independent pointwise fashion. Under spatial discretisation, classical techniques that have emerged would include finite difference (fd), finite element (fe) and finite volume (fv) schemes, though may be extended to high-order spectral schemes or meshless methods (particle-based). In addition, the classical choice of dependent variables has varied between velocity-pressure (primitive) and stream function-vorticity. The full system may be treated directly in a coupled form, or split into subsystems (decoupled) via fractional-staging or operator-splitting. A popular choice of the latter class is pressure-correction (pc), applied successfully for large-scale viscous incompressible flows due to its problem-size reduction property (suitable for three dimensions and transients), splitting computation for velocity and pressure over three staged equations. For current illustration purposes this is the favored option reported through case studies below. In the late 80s/early 90s, pressure-correction algorithms were advanced from fd and fv-base forms through to an fe-framework for application to viscoelastic problems. This generated semi-implicit time-stepping Taylor-Galerkin/pc schemes for Navier-Stokes problems (drawing upon two-step Lax-Wendroff and Crank-Nicolson procedures) and, more generally, Petrov-Taylor-Galerkin/pc schemes (with consistent variational streamline upwind weighting – SUPG). Beyond such issues other aspects of relevance that have lead to branches of study in their own right are: consistency in choice of variable function spaces (velocity/pressure/stress and stress/velocity-gradient); reformulation of equations about type (explicit elliptic momentum equation EEME); alternative splittings of stress (elastico-viscous stress-split EVSS and its discrete-form DEVSS). In addition, velocity gradient treatment may appear in various forms, from discontinuous to continuous (through global/localized procedures), alongside alternative stabilization techniques.

From a general stance, Galerkin fe-schemes are known to be optimal for self-adjoint problems and ideal for elliptic operator discretisation, as under Stokes flow. Nevertheless, viscoelastic problems disclose

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hyperbolic conservation laws (for stress) that demand appropriate treatment, for which conservative finite volume schemes are well-suited. In the past, cell-vertex fv-schemes have been shown to be superior to cell-centered forms (less susceptible to spurious error modes, accurate on unstructured and structured meshes), for advection, Euler and compressible Navier-Stokes equations. This position has suggested a hybrid and local formulation, considering a cell-vertex fv-scheme for stress with a fe-discretisation (semi-implicit, Taylor-Galerkin/pc) for the mass-momentum balance sub-system. With greater detail on alternative schemes and approaches presented under applications on a benchmark problem below, it is in order here to outline the specific features of this local scheme (see case studies). The resulting cell-vertex hybrid fe/fv(sc) scheme includes parent and subcell triangular reference, solution-dependent source terms, and consistent weighting of individual stress equation terms. This implementation has been established as second-order through analytic solutions, overcoming traditional low-order fv-expectation, whilst proving competitive on stability capture for complex flows (smooth/non-smooth). Such advance has been achieved through parent-subcell representation and integral evaluation (non-conservative area integrals for stress flux with piecewise constants for stress gradients), satisfaction of the linearity preservation property, and consistent stress flux-source treatment (so, beyond pure advection considerations). Stress source is strongly influenced by the quality of representation of velocity gradients, which may be extracted through either global (weighted-residual) or local (pointwise) procedures. The latter choice has proved particularly favorable on consistency properties between function spaces and has generated a superconvergent-localised-recovery procedure, in continuous piecewise-quadratic form (velocity representation also quadratic order). Re subcell construction, four subcell linear (stress) fv-triangles are formed within each parent quadratic (velocity) fe-triangle connecting its mid-side nodes. Hence, stress variables are located at fv-cell vertices within the fv-discretisation, yet also fall on fe-nodal values, so avoiding projection error within the fe-discretisation. Cell-vertex fv-upwinding is referred to as fluctuation distribution, a technique that distributes control volume contributions to cell-nodal equations according to prescribed theory. Fluctuation distribution schemes, originally designed for pure advection problems, may satisfy properties of conservation, linearity preservation and/or positivity, and appear in various forms, for example – minimum diffusion (low-diffusion) B-scheme (LDB); positive streamwise invariant (PSI) scheme. Then, the stress nodal update over a time-step has contributions per node from its fluctuation distribution component over the fv-cell, and a uniform distribution component over its subtended median dual cell (unique per node, non-overlapping nearest neighbor region). This procedure demands appropriate area-weighting to maintain consistency, that for temporal accuracy has been extended to time-terms likewise. With the candidate stress equation considered as split into time derivative, flux ( . )= ∇R u ττττ and source (Q, all other terms), and integrated over associated control volumes, the concise generalized fv-nodal update equation (for node l) may be expressed per stress component as,

( ) ( )1

1 1l l l l

nT T T T MDCl

T l T T l T l T lT MDC T MDC

ˆ b bt

+

∀ ∀ ∀ ∀

∆τΩ + − Ω = + − ∆ ∑ ∑ ∑ ∑δ α δ δ α δ (20)

where ( )T T Tb R Q= − + , ( )lMDCl MDC MDCb R Q= − + . Here, TΩ is the area of the fv-triangle T , whilst TlΩ̂ is that of its median-dual-cell (MDC). Parameter Tδ directs the balance taken between the contributions from the median-dual-cell and the fv-triangle T , with0 1T≤ ≤δ . This expression recognizes segregated fluctuation distribution and median dual cell contributions, area weighting and upwinding factors (Tlα -scheme dependent).

The combined fe/fv(sc) scheme forms a time-stepping process, with a three fractional-staged pc-formulation per time-step. On each time-step cycle, the first stage solves a set of equations for stress-

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velocity update, subject to the current pressure state (and past state for incremental-pc) in the momentum equation. Second, the forward time-step pressure is updated by imposing the continuity constraint through a Poisson equation (suitably adjusted with time derivative of density for compressible flow). At a third stage, the first fractional-stage velocity field is corrected to be compliant with the updated pressure field.

As discussed further below, generalization of the subcell formulation under the hybrid fe/fv(sc) is also complemented by extension to counterpart stress fe-subcell discretisation and constructs. This calls upon fe-upwinding for stress at the sub-cell level, following the Petrov-Galerkin streamwise style (SUPG). In addition, further consistent variational weighting is included to account for cross-stream influences, introducing the cross-stream/discontinuity capturing variant (DC). Comparison in the application of these various sub-cell formulations is illustrated below under case studies when considering compressible and incompressible implementations.

4. The contraction flow benchmark

For present illustrative purposes and application of computational rheology techniques, it is suitable to focus attention upon a particular benchmark problem, the classical contraction flow. Under isothermal viscoelastic flows, this problem has emerged as a stringent test-bed for stability and accuracy of numerical schemes. The relevance of this entry flow is two-fold. From a practical point of view, it occurs in several polymer processing operations as in extrusion and injection molding. In rheometry, such flows are involved in the accurate measurement of material properties. From a numerical standpoint, the simplicity of the geometry, that generates complex deformation (both shear and extension), provides a well-suited framework to investigate complex viscoelastic flows. A challenging feature of the abrupt contraction flow problem is the presence of a stress singularity at the re-entrant corner, the discrete treatment of which impacts upon numerical stability properties. In this respect, many fluid models suffer a limiting Weissenberg number (Wi), beyond which numerical solution procedures fail (lack of convergence ensues). This issue is frequently encountered and has been linked to equation-type and solver compatibility. The phenomenon has become known as ‘the high Wi problem-HWNP’, drawing considerable attention over the last three decades or so, whilst today remaining an open and challenging research issue. This subject matter is covered in the comprehensive literature reviews of Crochet et al. in 1984, White et al. in 1987, Baaijens in the late 90’s, and more recently, Walters & Webster (2003), Denn (2004) and Rodd et al. (2005). Solutions to the HWNP have gradually evolved since the IUTAM meeting in 1978 and the underlying reasons for convergence breakdown are now reasonably well understood (see, the historical expositions by Tanner & Walters and Crochet & Walters). Certainly, numerical convergence with mesh refinement and scheme stability are viewed today as an absolute necessity in this connection. For brevity, the present coverage of the expansive literature on contraction flows is primarily restricted to creeping flow consideration and 4:1 contractions by default.

4.1 Advance through experimental investigation

Historically, the experimental literature on the contraction flow problem is vast, from which a number of key features emerge. The salient-corner vortex, which is present in creeping Newtonian flows, can grow in size and strength with increasing flow rate for elastic liquids. This leads to the phenomenon of ‘vortex enhancement’. Similarly, under certain flow conditions and levels of elasticity, a lip-vortex may be formed near the abrupt contraction. When this appears, it is usually a dominant feature, so that upon

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further increase in flow rate, the lip-vortex may encapsulate the salient-corner vortex. Interestingly, for constant-viscosity Boger fluids, vortex enhancement is clearly evident in circular contractions, but is absent in the case of planar counterparts. On the other hand, shear-thinning polymer solutions show vortex enhancement in both geometrical contraction variants. Early experimental studies were conducted by groups of Walters, Boger and McKinley. Commencing with experimental results in the late 80’s, from the group of Walters, both lip-vortex and salient-corner vortex behavior were studied in planar contractions. These experiments emphasized the complex nature of the vortex enhancement mechanism and identified dependency upon several factors: material properties, geometric contraction ratio, type of contraction (abrupt or rounded), and fluid inertia. So, for example, no evidence of lip-vortex activity was found in a 4:1 abrupt planar contraction for Boger fluids, or 0.5% and 0.3% shear-thinning aqueous solutions of polyacrylamide. However, a lip-vortex was generated for a 0.2% polyacrylamide solution. Upon rounding the corner, a completely different flow structure emerged, manifesting only a small salient-corner vortex and dispensing with the lip-vortex. Note that for a Boger fluid, it was necessary to increase the contraction ratio to 80:1 to observe lip-vortex response.

Provocatively, from the work of Boger and co-workers in the early 90’s, two fluids of constant-viscosity (PAA/corn-syrup and PIB/PB solutions) were compared with essentially the same characteristic times, through abrupt circular contractions. The observations revealed two distinct flow patterns with increasing shear-rate. The first fluid showed continual salient-corner vortex growth, with the separation line changing in shape from concave to convex. In contrast, the second test-fluid exhibited a combination of lip- and salient-corner vortices: with shear-rate increase, the lip-vortex completely engulfed the decreasing salient-corner vortex. These studies concluded that knowledge of steady and dynamic shear properties alone was insufficient to characterize elastic liquid behavior in circular contractions. Moreover, both fluids showed different behavior in vortex development upon rounding the re-entrant corner, which could only be explained through their extensional properties.

Other experimental research at this time (1991) identified a sequence of flow transitions that lead to time-periodic, quasi-periodic and aperiodic dynamics near the contraction zone and to the formation of elastic lip-vortices. Such dynamics were noted to be steady for contraction ratios above five, below which time-dependant 3D flow was observed near the contraction zone. More recent experiments from the Mackley group (2005 and 2007) have emphasized the effect of excess pressure loss (Couette correction), flow instabilities and the dynamics of vortex enhancement. In these studies, the significant differences in vortex growth pathways have been explained through transient extensional rheology measurements. Moreover, a wide range of Weissenberg and Reynolds numbers were introduced defining their ratio through the elasticity numberE Wi Re= , based on micro-fabricated contraction geometries.

Recent experiments from 2005 (cited in review of Rodd et al.) have also utilized micro-fabricated geometries succeeding in generating large shear-rates and high elasticity states, that far exceed those achievable in traditional macro-scale entry-flow experiments. These conditions suggest that elastic instabilities, close to the contraction zone, can be manifested by the development of additional streamline curvature and fluctuation in the local velocity at critical elasticity levels. This allows coherent and stable lip-vortices to emerge and subsequently develop into asymmetric viscoelastic corner-vortices, that continue to grow upstream with increasing elasticity. The influence and role of inertia was also investigated in these micro-fabricated geometries. For lower elasticity levels, inertio-elastic instabilities upstream of the contraction plane, replace the lip-vortices observed at higher elasticity numbers. Over a moderate elasticity range, elastic vortices grow steadily upstream. These vortices are essentially symmetric but as inertial effects become increasingly important they become temporally unsteady and spatially bi-stable structures. Upon approaching critical elasticity states,

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diverging streamlines eventually develop, just upstream of these elastic vortex structures. This body of evidence provides many challenges to be addressed under predictive methodology through the techniques of computational rheology – as discussed next.

4.2 Progress and application through numerical methods

Considering the historical development of numerical techniques in computational rheology, early successful implementations for viscoelastic problems were performed in the mid-late 70’s for steady flows based on finite difference schemes combined with stress splitting methodology. About the same period (1977), finite-element techniques came into vogue, based on differential constitutive equations and likewise, solutions for integral models began to appear in the early 80’s (see Crochet et al., 1984). At this time and continuing with steady solutions, some abrupt entry flow calculations were conducted and pressure-drops reported based on Oldroyd-B and FENE models in both planar and axisymmetric geometries. Such representative work exemplified finite difference discretisation, through a fourth-order differential stream function formulation and a decoupled method of solution. There, full viscous stress-splitting was employed in the momentum equation and a time-stepping technique was utilized as an iterator for the constitutive equation. Elsewhere, Fortin and co-workers proposed a stable and cost-effective mixed fe-method, a variant of the stress-splitting EVSS methodology, from which early numerical solutions for the network PTT model appeared.

Into the mid-80’s, Crochet and coworkers published two valuable contributions championing a coupled finite element method, a steady formulation, and a coupled-system Newton–Raphson iterative solver. First, Hermitian elements were introduced, augmented with a continuous representation for velocity gradients, and found to enhance the limit of numerical convergence in Wi over a more conventional Lagrangian element choice. A lip-vortex was detected with the Hermitian scheme at moderate Wi-levels, and over the same range of Weissenberg number, salient-corner vortex growth was observed. Subsequently, a variationally-inconsistent streamline upwinding (SU) technique with 4×4 linear stress sub-elements was introduced by this group, which was hailed for its enhanced numerical stability, achieving steady Oldroyd solutions for the first time at large Wi-levels without loss of convergence. Under the SU method, a weak lip-vortex appeared at moderate Wi-levels; this then disappeared, as elasticity increased further, and was dismissed as a numerical artifact. The size and intensity of the salient-corner vortex remained weak and insensitive to increasing elasticity. Based on this technique, further numerical studies explored in detail the flow of Oldroyd-B, Phan Thien/Tanner and Giesekus-Leonov fluids through circular abrupt contractions. These developments led onto related upwinding studies in the late 80’s by the group of Tanner, via a decoupled finite-element approach, on Galerkin, SUPG and SU methods, concentrating on accuracy and convergence behavior with a full viscous stress-split in the momentum equation. This research exposed the inconsistent nature of the SU method, which was observed to smooth solutions through numerical diffusion (lowering in order of accuracy to first-order) when attaining elevated Wi-levels. In such comparatives, based on a single mesh and with elasticity increase, there was no evidence of lip-vortex activity reported, and the salient-corner vortex remained weak in size and strength.

With regards to compatibility issues upon discretisation, in particular to space inclusion for stress and velocity gradients, further comparative studies were performed in the 90’s, based on two variants of stress splitting-EVSS technology (discontinuous Galerkin DG and continuous SUPG variants). The DG/EVSS scheme was observed to reflect significant improvement over the SUPG/EVSS variant at higher Wi-levels (with smoothness in solutions and enhanced robustness, see on to fv-implementations). Moreover, space–time/Galerkin least squares techniques with discontinuous stress were also

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introduced, with EVSS-type construction. Also in the early 90’s, some significant advance was made through an alternative decoupled methodology centred on a Lagrange–Galerkin technique for velocities and a Lesaint–Raviart discontinuous method for stress, employing linear interpolation. This was considered as a modified version of the Fortin group approach that employed quadratic stress approximation. Such a decoupled approach highlighted the fact that for the creeping viscoelastic problem there is, in addition to the standard Babuska–Brezzi (LBB) condition, a further compatibility relationship imposed upon trial spaces in a velocity–pressure–stress formulation, requiring the gradient of velocity space to lie within that of stress. Note that by design, localized recovery schemes for velocity gradients satisfy this requirement at the continuous level, whilst discontinuous stress approximations (such as DG), strive to do so at the discontinuous level. With regard to order of stress interpolation, linear interpolation with SU smoothing has been compared against a characteristic-based algorithm, with quadratic interpolation. The quadratic form hinted at a minuscule lip-vortex at near critical Wi-levels, whilst the linear alternative was able to triple critical elasticity solution level attainment of the quadratic variant without generating lip-vortices. Along these lines and more recently (2007), Arbitrary Lagrangian-Eulerian (ALE) schemes have begun to appear and show some considerable promise.

In the late 80’s/early 90’s, generalized Taylor-Galerkin and Taylor-Petrov-Galerkin pressure-correction (fe)-schemes were introduced. This generated Oldroyd-B and Phan-Thien/Tanner solutions for sharp-corner contraction configurations. For Oldroyd models, implementation indicated the onset of instability as elasticity was increased to a limiting Weissenberg number, with no lip-vortices apparent for Wi

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The literature on finite volume (fv) implementations for viscoelastic flow splits into treatment of the full system through such techniques and also counterpart hybrid variants. A series of papers have appeared from the 90’s onwards on fv-schemes, providing a comprehensive overview for this aspect of the subject. In the early 90’s, a full fv-system was introduced through the SIMPLER algorithm, based on a staggered grid. With increase in elasticity, the lip-vortex once present increased in size and intensity. The salient-corner vortex was hardly influenced by increasing Wi. The effect of adding inertia was also noted, whereupon such vortices decreased in size and strength, though lip-vortices still persisted. Salient-corner vortices were compressed further into the corner recess with increase in flow rate. Under low elasticity considerations, the lip-vortex was segregated from the salient-corner vortex, whilst, at higher elasticity, lip and salient-corner vortices co-existed within a single recirculation cell. Similarly, some SIMPLER implementations in the late 90’s utilized a first-order implementation that applied a semi-Lagrangian treatment for convection terms. With mesh refinement and for an Oldroyd-B fluid, this method demonstrated that the size of the salient-corner vortex remained fairly constant with Wi increase, both for creeping and inertial flows. On coarser meshes, an absence of lip-vortex activity was observed, whilst lip-vortex growth was reported on finer meshes.

Finite volume implementations continued to be applied (in 1998) based upon an algebraic extra-stress formulation, derivable from the Oldroyd-B constitutive equation. This approach has been extended further, via a second-order fv-implementation, employing a staggered-grid technique, reproducing precisely the extra-stress tensor components under pure shear and elongational flow. Notably, this new technique has also been successfully applied to three-dimensional (3D) unsteady flows using Cartesian coordinates on a non-uniform staggered grid. Likewise, in the late 90’s, a new constitutive equation for incompressible materials was introduced, treating stress as an isotropic function of tensors, rate-of-strain and relative-rate-of-rotation. There, vortex-size was observed to increase with rise in level of extensional viscosity, whilst it only mildly decreased when the parameter relating to normal stress coefficient was increased.

In a series of successive and consistent studies, Oliveira and co-workers (2000-2007) have compared first- and second-order pure fv-schemes (collocated variable). Their first-order implementation was able to converge at fairly large Wi-levels on a refined mesh, but proved to be inaccurate, reflecting large spurious vortex structures, reminiscent of Amstrong-Brown results in the mid 80’s for second-order fluids. Yet, a second-order alternative scheme failed to converge at Wi above unity on the same refined mesh. Oliveira and co-workers noted that this premature numerical breakdown corresponded to oscillations in the stress field, associated with high-order upwinding. These oscillations were overcome by appealing to a limiting (min-mod) procedure applied to stress convection. Recently (2003), this research group has produced a refined-mesh solution reference, charting in detail the development of both vortex-size and intensity for Oldroyd-B and PTT fluids. This has highlighted the necessity of suitable mesh refinement (as in 3D-fv noted by others) within the re-entrant corner zone to sharply capture the singularity, though this often reduces the critical level of Wi attained, when compared to that reached on cruder quality meshes. Additional work has involved variation in contraction ratio for abrupt planar and axisymmetric contraction flows, providing yet further evidence for lip-vortex growth. In a related manner, Webster and co-workers (2006) have analyzed some generally applicable stabilization strategies through hybrid finite element/volume (fe/fv) implementations and Oldroyd fluids. Results have indicated that most improvement is encountered with a strain-rate stabilization scheme (as under DEVSS), more than doubling critical Wi-levels attained, whilst constraining stress-peaks. Overall with increasing Wi, salient-corner vortex reduction was observed (with all schemes) and lip-vortex growth likewise. Notably, lip-vortex features were found to be significantly affected by the particular treatment of the re-entrant corner.

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In addition, some transient schemes for this benchmark problem have appeared throughout the mid-90’s. Representative solutions based on hybrid schemes were reported which employed a time-explicit fe-method for momentum and time-implicit fv for pressure and stress of cell-centred type. There, higher-order upwinding was achieved through the application of a TVD flux-corrected (inconsistent) transport scheme to the advection terms of the stress equation. With this approach, a lip-vortex was observed as a pseudo-transient phenomenon, being induced by an instantaneous increase (doubling) of elasticity level. Curiously, the lip-vortex subsequently disappeared through the time-stepping process as steady-state conditions were approached. A similar trend was also observed through the transient appearance/disappearance of a lip-vortex for a Giesekus fluid with a rounded-corner, based on a method-of-lines time-integration technique and a finite-difference discretisation. Considering further advance in methodology and more recently in 2005, a modified form of the time-dependent DEVSS-G/SUPG scheme (see also under DEVSS-G/DG) was adopted, associated with a penalty function method and based on a CONNFFESSIT approach. There, inertial effects were studied, revealing the importance of convection and the significant influence this has in changing flow patterns. Further findings (2002) on transient solutions with fe/fv schemes are discussed below under case-studies.

There has also been some treatment of compressibility within the viscoelastic context, as exemplified through advanced developments based on the fe/fv pc-formulation (2004 onwards). Close examination of flow patterns and vortex trends indicate the broad differences anticipated between incompressible and weakly-compressible viscoelastic solutions, with vortex reduction on rising Wi a common feature. Compressible solutions have provided larger vortices (salient and lip) than their incompressible counterparts, with larger stress peaks in the re-entrant corner neighborhood. Furthermore, compressible implementations have been shown to yield solutions for a range of compressible settings, tending to the incompressible limit at vanishing Mach number without loss of stability or accuracy (see on to application section). General observations from this work reveal that subcell linear approximation for stress within the constitutive equation (either fe or fv) yields a more stable scheme, than that for its quadratic counterpart, whilst still maintaining second-third order accuracy. In addition, the more compatible form of stress interpolation within the momentum equation has been shown to be linear upon subcells (note earlier comments upon interpolation order choice and Lagrange-Galerkin findings).

4.3 Some analytical and semi-analytical developments

Alongside numerical solution for viscoelastic flow, there has also been some theoretical advance made on aspects such as boundary-initial conditions, loss of evolutionary form, singular solution representation and problem transformation. For example, analytical derivations have appeared for solutions to flow about re-entrant corners (mid 90’s), with direct relevance to the numerical solution of contraction flows. This has proven instrumental in proof-checking discrete solutions for sufficiency in mesh refinement (mesh independence). In particular, analysis has focused on asymptotic description at high Weissenberg number limits, corner singularities, and instability. Alternative constitutive models have been considered, such as UCM, Oldroyd-B and PTT, with similarity solutions appearing, for example, with Oldroyd fluids and flow about a 270° re-entrant corner. Here, the velocity was found to vanish as (r5/9) and the stress to be singular like (r−2/3), as a function of radial distance to the corner. This knowledge has been employed effectively to validate mesh convergence for a host of numerical solutions that have appeared.

Inspired by singularity treatment applied to crack failure problems, a particular singular function approach combining theory with discretisation was explored in the 90’s. In particular, this followed a boundary integral method to solve a Laplacian problem over an L-shaped domain. The approach was

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based on a fe-formulation in which the solution was approximated by standard polynomial basis functions, supplemented by the leading terms of the local asymptotic solution expansion. This promising method eliminated the need for high-order integration, improved overall accuracy, and yielded highly accurate estimates for singular coefficients. These ideas yet await vigorous implementation across a broader spectrum of applications.

Recently under problem transformation (2004 and 2005), a possible source for numerical instabilities and scheme failure at high Weissenberg numbers has been identified as being related to exponential growth in stress. To overcome this, a log-conformation representation for stress has been proposed, under which the extensional components of the deformation field act additively, rather than multiplicatively. Theoretically, this provides access to solutions beyond conventional limiting Weissenberg numbers. Such modifications have been successfully implemented through various discretisation alternatives (second-order finite-differences, finite-elements via DEVSS/DG, and finite volume schemes), flow problems and models. The solutions that are beginning to appear under such transformation scheme are already showing promise, through greater understanding is still sought for sharp-corner solutions and severely strain-hardening models (as with Oldroyd-B).

Summarizing over the various developments outlined in this section (experimental, numerical, and theoretical) much attention and progress has been made in the field, with considerable interest in the contraction flow benchmark. In the following sections, some novel viscoelastic applications are discussed through the vehicle of a local subcell discretisation. These case studies show-case some recent results and concentrate upon topical themes of transient flow solutions, modeling with pom-pom representations, pressure-drop estimation, and the inclusion of compressibility effects upon viscoelastic flows.

5. Specific applications – topical case studies

5.1 Modeling of some transient flows

The numerical simulation of transient flows of viscoelastic materials is an important area of study in many industrial and natural processes. This is due to the fact that the nature of common flows of complex materials is essentially time-dependent, revealing many important viscoelastic phenomena. To conduct transient simulations for viscoelastic flows, it is important to develop accurate and robust numerical procedures that can handle transient boundary conditions and appropriate constitutive equations. More recently, significant progress has been achieved in the development of numerical methods and rheological description of complex materials, but less attention has been paid to the computation of transient flows. A principal difficulty encountered in the computation of unsteady incompressible/compressible viscous flow arises through the mixed-type of differential equations, which appears in the coupling of the momentum equation with the incompressibility equation, and consequently the treatment of pressure terms. Moreover, in the viscoelastic context, introducing a supplementary hyperbolic extra-stress equation subset dramatically increases numerical tractability. In addition, for some transient computations and models, specification of a consistent set of boundary conditions still remains the subject of open debate, particularly for constitutive models with complex mathematical structure, such as the pom-pom model. Inconsistency and sharp change between initial conditions (IC) and boundary conditions (BC) can produce heavy oscillations in the ensuing computed solutions. This can stimulate large numerical perturbations (particularly in pressure) that may subsequently degrade solution accuracy.

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For viscoelastic flows, two main approaches have been deployed in the literature to incorporate inflow and outflow type BCs. Typically, these are considered as simple shear-flow conditions (often steady), reducing to one-dimensional ODEs (ordinary differential equations) for both velocity and stress. In the first approach and for some constitutive models of straightforward mathematical structure, transient analytical solutions may be extracted. This is true for example with constant shear viscosity Maxwell/Oldroyd-B models and Poiseuille-type channel flow. Such dynamics, or their limiting steady-state forms, may then be imposed at domain inlet for both velocity and stress, and at outlet in velocity alone. This determines the controlled flow-rate scenario commonly encountered. Here, inconsistent application of steady-state BCs (as standard) may result in large oscillations in pressure-transients and induce their associated evolving solution dynamics. The second approach for transient simulation is a controlled force-driven configuration, see work of the Crochet group (80s/90s). Force-control may be instigated through the application of a controlled pressure-drop across a flow (say, impulsively imposed), built into the problem via force-type BCs (boundary integrals in a weak-form solution). Such conditions may be configured to smoothly propagate and mimic more closely the actual physics of the flow. One may note that many flows arise under driving force conditions, such as under pressure-difference or gravity. As such, a key aspect is to impose a pressure-difference across the flow as a boundary condition, and compute velocity and stress upon open boundaries (inlet-outlet), consistent with the flow dynamics.

Transient solutions, with temporal sweeping of large vortices may be discussed through some recent Oldroyd results with the fe/fv scheme. Commencing from quiescent initial fields, two principal driving boundary condition settings have been analyzed: steady-state flow-rate control (Q, anchored velocity BC – impulsive step-change); and transient flow-rate control (Qt analytic, Waters & King). In practical terms, this has demonstrated how the type of boundary conditions imposed completely dictates the resulting solutions, exposing some novel differences between the transients generated. By way of example of characteristic transient solutions, Figure 1 illustrates the evolution in streamline patterns for the 4:1 rounded-corner (rc) planar problem (Oldroyd-B, Wi=1.0, planar). Figure 2 describes salient-corner vortex intensity measure, from an early temporal phase towards steady-state conditions for the steady-state Q-control (ss-Q) and transient (Qt)-protocols. Vortex intensity (ψint) and size-measures together identify growth to extrema at different rates: more rapid early rate goes with (ss-Q); closely followed by the (Qt)-protocol. The (Qt)-version reveals rich structure, comprising of multiple sub-phases within the mid-term period displaying: reversed flow, followed by vortex detachment, prior to reattachment (Figure 2). These patterns are found to correlate with the peaks-and-troughs in the evolving velocity and pressure profiles, and are also observed under inertial flow settings. Important temporal features to observe are the shape and oscillation in the evolving vortices, and the corresponding rates of vortex-size adjustment involved across settings (Figure 1). The separation streamline shape adjusts from: concave to linear to convex varying with protocol. Nevertheless, identical temporal limiting vortex structures are ultimately derived under ( )ss Q− and (Qt)-settings (same steady-states reached). Some recent work has extended these ideas to consider the implementation of the more general force-controlled conditions ( )F p− ∆ in contrast to those under flow-rate control. This has revealed some fresh transient features for both Oldroyd-B and pom-pom solutions. In particular, only attached vortex structure are now detected for Oldroyd-B fluids, that provocatively show exaggerated growth-decay modes over flow-rate driven solutions (tripling maximum vortex intensities). This is a strong pointer to the evolutionary solution differences to be expected when monitoring driving boundary conditions in this manner.

In contrast to the above, it is also informative re dynamics of the corner solution, to comment upon similar solutions for the abrupt-corner counterpart problem. There, with (Qt)-protocol and at Wi=2.0,

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lip-vortex dynamics are observed to be stimulated from an extension of the early-time secondary-flow structures generated, up to and beyond the re-entrant corner. In the (ss-Q) case, any lip-vortex presence is much less conspicuous at steady-state. This emphasizes the temporal nature of lip-vortex dynamics, as also noted experimentally, and drawn out with sharp-corner solutions. It is also shown that the steady-state structure, around and after the re-entrant corner is established somewhat earlier in normal-stress than shear-stress.

5.2 Pom-pom (kinetic-based) modeling

Recently, a new molecular class of tractable differential constitutive models has emerged, capable of describing the behavior of polymer melts. These fluid models are based on the developments of Doi and Edwards for tube-kinetic-theory models, relevant for entangled melts. The double-convection-reptation (DCR) model of Ianniruberto and Marrucci, the pom-pom model introduced by McLeish and Larson, and the subsequent extended versions proposed by Verbeeten et al., are all examples of such kinetic models. These models incorporate two new important aspects absent in phenomenological equations. The first aspect is that via proposed molecular relaxation mechanisms, the models include dependency on polymer molecular structure. The second aspect is the introduction of two relaxation times, one for backbone-orientation and another for backbone-stretch. Chain stretch, reptation and Rouse friction are all concepts included within the DCR model for linear systems (a multi-mode convection-constraint release ‘CCR’-version). Moreover, the introduction of the CCR-mechanism has been a major step forward, since it resolves the problem of excessive shear-thinning predicted by the Doi-Edwards model at high shear-rates. Furthermore, a two-mode DCR model exhibits a narrow window of shear-thickening. The pom-pom model itself has emerged to represent an idealised branched polymer molecule, composed of a backbone with an equal numbers of dangling arms (q ) at both ends.

The ratio of the two relaxation times introduces the parameter (ε), which may be used to relate to the degree of system entanglement. High and low values of ε correspond to less-entangled (dilute polymer) and highly-entangled (concentrated polymer) systems, respectively. In the original formulation, three dynamic variables appear in the differential approximation: backbone-orientation tensor (S), backbone-stretch (λ ), and arms-withdrawal (cS ). Arms-withdrawal takes place when the molecule reaches its maximum stretched state, provoking an abrupt finite extensibility constraint which results in a discontinuity in steady-state extensional viscosity. The decoupled nature of orientation and stretch relaxation times is a key feature in this class of models. It is significant that such a simple molecular topology, with only one mode, has been able to qualitatively predict transient viscosities (shear and elongational) of the IUPAC A LDPE (low density polyethylene) melt.

Nevertheless, the original pom-pom model suffers from three major drawbacks: discontinuity in steady-state extensional viscosity, the equation for orientation is unbounded, and the model predicts a zero second-normal stress-difference in shear flow. Despite a number of modifications - extension to multi-mode, drag-strain coupling, modification of the backbone-orientation relaxation-time to allow for retraction as well as stretch - these disadvantages still remain. The following adjustments have been proposed for an extended version of the model: first, the backbone-orientation is not required to obey Doi & Edwards theory for linear polymers; second, although the drag-strain coupling is retained, the finite extensibility constraint is discarded. In addition, two versions of the pom-pom model have been introduced, the Single and Double eXtended pom-pom models, (SXPP and DXPP, respectively). Their main difference lies in the evolution equation for stretch ( )λ : for the DXPP model, λ is computed through a partial differential equation, whilst for the SXPP model, λ is derived algebraically. Importantly, these new types of model recognize the dependency of the rheology on the internal

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architecture of the polymeric system. As such, they are able to quantitatively predict the rheology of some commercially available melts (LDPE). Furthermore, modifications to the thermodynamic admissibility of the pom-pom model have been validated theoretically. A significant issue in this regard lies in the influence of the Giesekus-like anisotropy parameter, 0α ≠ . It has been shown that setting low values of the anisotropy parameter,α , (with some second-normal stress-difference (2N )), produces a stabilizing effect on Couette and Poiseuille flows. Yet, for sufficiently large α and numbers of branched arms (q), analytical singularities have been shown to appear. Positively, a multi-mode pom-pom model, with a physically reasonable distribution of branching, is able to reproduce the material functions of a LDPE melt, covering four decades in deformation-rates. With such pom-pom modeling, an excellent fit for steady-state shear and extensional viscosity has been obtained for three polymer melts: LDPE, metallocene-catalyzed LLDPE (mLLDPE) and Polyvinyl butyral (PVB).

A number of computational studies have contributed to this general area utilizing a variety of different approaches and versions of pom-pom models. Using the original differential form of the pom-pom model, it has been shown that incrementing the branching (increasing dangling arms and strain-hardening properties) produces larger vortices at any given Weissenberg number for transient flow through a 4:1 planar contraction. This agrees with experimental observations for vortex growth of branched and linear polyethylene melts, where only smaller vortices were found for the linear melt. To improve stability of the XPP model, the stretch dynamics have been modified through variation of the extra function, ( )f τ , introducing a modified eXtended pom-pom (mXPP) model. Moreover, steady-state multi-mode XPP response in a 3D contraction geometry, has been compared against experimental information for a LDPE melt, reporting errors within 15% in vortex cell-size measurements. Predictions with an alternative pom-pom version, termed the Double-Convected pom-pom (DCPP) model, have been compared against birefringence measurements for a LDPE melt, obtaining good quantitative agreement. Elsewhere, excellent comparison with experimental observations has also been reported for a LDPE melt, covering flow through a cross-slot device, confined flow around a cylinder and for a 3.29:1 planar contraction flow. A drawback is that unrealistic values of stretch,λ , were detected at stagnation point stations in flow around a cylinder and close to the sharp re-entrant corner in the contraction flow. In addition, the flow through a planar 4:1:4 rounded-corner contraction/expansion using integral and differential versions of the pom-pom model, has also revealed non-physical stretch values ( 1λ < ) near the constriction. Further related work on this topic may be found in the extensive and growing literature currently appearing. For example, a new proposal from the Tanner group is to unite the molecular derivation of the pom-pom model with the network construction of the PTT-model. This has generated the new PTT-like modification to the XPP model, the PTT-XPP model, which captures the steady elongational behavior of the extended pom-pom model but is less rapid in shear-thinning response at larger shear-rates.

5.2.1 Some steady-state SXPP solutions

To motivate discussion on pom-pom solutions, one may illustrate some sample steady-state results (4:1 rc-planar) with the hybrid fe/fv scheme for the single-equation version (SXPP). This serves to demonstrate extensional deformation in contraction flows, the adjustment of structure-related parameters, and the generation of large vortices. At the same time, the opportunity arises to compare and contrast flow response for the kinetic-based theoretic models (pom-pom) against their network-based counterparts (PTT). For example, one may assess the influence on flow characteristics of varying the number of dangling side-branch arms (2 15q≤ ≤ ) at each end of the pom-pom molecule, which affects the level of system entanglement and degree of hardening. With XPP (unlike PTT) models, the

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degree of hardening can be effectively adjusted without influencing the shear response, over a working range of solvent viscosity fractions and deformation-rates. This may be achieved by first fixing the ratio of relaxation times (backbone-stretch to orientation), and then, allowing the number of side-branch arms to vary. Proceeding in this manner and for fluids devoid of strain-softening, vortex cell-sizes are observed to be almost identical for pom-pom fluids with five or more side-arms. With two side-arms, two distinct and separate entry-flow zones are detected where strain-softening may occur; yet even these are practically removed once the number of branched-arms exceeds five.

With increase in the number of arms, molecular backbone-stretch (λ) itself, is observed to continually increase and double over the entry region. This response is heightened particularly from configurations with (q=2) to (q=5), remaining fairly static in (q)-change thereafter (Figure 3, right column). Regions of relatively low molecular stretch can be identified in the shear inflow zone (upstream channel before entry), along/near the centerline far upstream/downstream from the constriction, and in the recirculation region. A ‘banded entry-flow’ zone of stretched material can also be identified, as recognized by others, corresponding to dominate extension over shear deformation (noted in extensional viscosity distribution). Moreover, incrementation in the number of arms results in a general increase in 1N over

the contraction zone and along the downstream wall (Figure 3, left column).

Turning to controlled vortex generation for these complex flows, one may distinguish between the significant and separate influence of extensional viscosity E(η ) and Trouton ratio (Tr), generated by

pom-pom (molecular, kinetic) versus PTT (phenomenological, network) models. Similar trends in solution response may be extracted, in terms of vortex enhancement and inhibition, depending upon the precise extensional rheology forged (four pom-pom fluid options posed, Fluids- I-IV with * notation, matching EPTT( 0.02ε = and 0.25) peaks in Eη and Tr, Figure 4). The parameters for these four different model fluids are: Fluid-I(q=8, ε=0.99), Fluid-II(q=5, ε=0.50), Fluid-III(q=2, ε=0.33), and Fluid-IV(q=2, ε=0.075). Strong strain-hardening and matching on peak-extensional viscosity (Fluid-I in Figure 4a) equates closest to the dynamic response of EPTT(ε=0.02), a setting which stimulates the largest vortex response, but also the lowest level of Wicrit (critical Wi for converged solution). In this same regime, but matching instead on peak-Trouton ratio (Fluid-II in Figure 4c), lessens vortex activity yet doubles Wicrit. Under the moderate strain-hardening regime, typified through EPTT(ε=0.25) (supplying Fluids-III, IV), the correspondingly more modest extensional properties have been observed to ultimately lead to vortex inhibition as Trouton ratio declines (Fluid-IV, Tr-match, as in Figure 4d). Here, vortices significantly diminish from their hardening counterparts, even in the intermediate instance with match on peak-extensional viscosity (Fluid-III in Figure 4b). Concerning large vortex generation, the strongly strain-hardening model variants are to be preferred. In combination with the knowledge on molecular stretch, this may provide a way forward in practical applications, where one seeks to take advantage of uncoiling and recovering structures in a complex flow, alongside the presence of extravagant vortices.

5.3 Pressure drop estimation in contraction flows

One of the principal features of significance in contraction flows is the study of pressure-drops, covering various geometric configurations and fluid rheology. Experimental observations provide the problem motivation and the background data. For example, experimental observations on ‘excess pressure-drop’ (epd) for severe strain-hardening Boger fluids of practically constant shear-viscosity, reveal significant enhancement above Newtonian fluids in axisymmetric geometries. Correspondingly, under planar configurations, no such enhancement is apparent. This discrepancy has presented an

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outstanding challenge to the field that has effectively eluded predictive capability to date in contraction flows, for all relevant viscoelastic models and schemes. Turning to detailed experimental findings, Nigen & Walters in 2002 compared pressure-drops under increasing flow-rates for two sets of constant shear viscosity fluids, Boger fluids (B1 and B2, polyacrylamide/water-glucose) and for Newtonian liquids (NS1 and NS2, glucose-water). Long and short exit die-lengths under circular and planar contraction configurations were considered for contraction ratios between two and thirty-two, establishing a linear relationship in pressure-drop versus flow-rate data. At low flow-rates, Boger and Newtonian liquid results were inseparable. Significantly, for axisymmetric but not planar flows and flow-rates one order larger, differences in pressure-drop between these two fluids began to emerge, with more exaggerated differences for short over long-exit die-lengths. Likewise, Rothstein and McKinley in 1999 and 2001 investigated experimentally the creeping flow of a dilute monodisperse polystyrene/polystyrene Boger fluid. These authors covered a large range of Deborah numbers ( )here, De Wi≡ for axisymmetric contraction/expansion geometries, varying contraction ratios between two and eight and degrees of re-entrant corner curvature. Once more, large epd was observed above that for a Newtonian fluid (axisymmetric configurations), independent of contraction ratio and re-entrant corner curvature. Furthermore, Rodd et al. in 2005 considered microfabricated contraction/expansion geometries (16:1:16, planar, sharp), investigating vortex generation and pressure-drops and the complex relationships between inertial and elastic influences. For a large range of Weissenberg number, the length-scale of the geometry was found important to generate strong viscoelastic effects, which were non-reproducible with macro-scale geometries and the same fluid. In this manner, significant upstream vortex growth was generated alongside increase in excess pressure-drop of some 200 percent.

Some early pioneering predictive work (in 1977), described the effects of increasing elasticity in viscoelastic flows for L -shaped and expansion/contraction/expansion geometries through the strength and size of resulting vortices. For the L -shaped geometry, a decrease in pressure-drop as elasticity increased was reported for a 4-constant, shear-thinning Oldroyd model. In contrast, pressure-drop was found to rise with increase of inertia by almost five orders more than under the regime of increasing elasticity. Elsewhere, the source of decline in pressure-drop for both Oldroyd-B and FENE-CR without inertia, has been identified with the presence of the long exit-tube and establishment of Poiseuille flow. For the Oldroyd-B model, the extra pressure-drop decreased linearly with Wi ; and likewise for FENE-CR, until Wi was sufficiently high for finite extensibility to take affect, whereupon there was slight rise before levelling-off. Moreover, in the work of Webster and co-workers, the ratio of the total non-Newtonian pressure-drop has been compared against its Newtonian counterpart for five different fluids (Oldroyd-B and four PTT model variants, 1 9=s

β), in axisymmetric and planar configurations, with

4:1 abrupt and rounded-corner contraction geometries. In all instances, as noted by others likewise, total elastic pressure-drop was observed to diminish monotonically with increasing Wi from its Newtonian counterpart, with practically identical trend for sharp and rounded-corner geometries. Hence, this is the general consensus predictive position for the orifice-tube and 4:1 contraction problem – with lack of correspondence to axisymmetric experiments.

Generalization to the 4:1:4 contraction/expansion flow and adjustment of material parameters has assisted in providing useful insight to resolve this experiment-simulation discrepancy, in comparative form to the 4:1 geometric counterpart problem. That is, in providing quantitative pressure-drop predictions for axisymmetric contraction flows, showing how enhancement in epd may be extracted for these strongly strain-hardening Boger fluids. Reference may be made to some recent results for constant shear-viscosity/strain-hardening fluids in contraction and contraction/expansion flow geometries. Some of this work has utilized Oldroyd model compositions of various solvent:polymeric viscosity fractions,

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yet there is also appeal to alternative strain-hardening model representations of FENE, Phan-Thien/Tanner and pom-pom type. To motivate the discussion and for illustration, some results are taken from the hybrid fe/fv algorithm of incremental pressure-correction time-stepping structure, reflecting some novel features with respect to the discrete treatment of pressure. Szabo et al. in 1977 made an incisive contribution to the topic in making this switch from the 4:1 conventional contraction flow to the rounded-corner 4:1:4 axisymmetric contraction/expansion problem. There, a split Lagrangian-Eulerian scheme was considered with the FENE-CR model, reflecting constant-viscosity strain-hardening properties. A key advantage of the 4:1:4 geometry over the 4:1 benchmark was recognized in that shorter downstream distances are demanded to relax the stress beyond the constriction. In addition, a second property of the 4:1:4 configuration relies on equal entry-exit channel flow conditions in stress, so that the total dissipation-rate may be equated to the flow-rate times the pressure-drop. Importantly, the effect on epd has been demonstrated of increasing the FENE-CR extensibility parameter (L) from 3.26, to 5, to ∞ (lowering L reduces polymeric stress, as does increasing solvent fraction sβ ). For each choice of L, epd initially decreased monotonically away from the Newtonian reference-line, to a local minimum, prior to upturn. The maximum departure from the Newtonian reference increased with rise in L, with no minimum detected for the limiting Oldroyd-like instance. Provocatively, at the intermediate value of L=5, a small epd-enhancement was ultimately obtained at ( )or 10Wi De ≈ , after intersecting with the Newtonian reference-line (see Figure 5). This observation has stimulated further research in seeking substantial epd-enhancement under viscoelastic modeling. For shear-thinning pom-pom model fluids, further evidence has been provided elsewhere for pressure-drop predictions in a planar 4:1:4 geometry. There, total pressure-drop has been found to decrease monotonically with increasing elasticity, as observed likewise for Phan-Thien/Tanner (PTT) models. Moreover, the POLYFLOW finite element package with Oldroyd models and an EVSS formulation (Binding et al., 2006), has provided some epd-results for 4:1:4 planar and axisymmetric contraction/expansion geometries with solvent viscosity ratios of 0.9s =β and 1 9. With increasing De and the larger

0.9s =β ratio, an increase in epd, with upturn and enhancement, was reported under planar circumstances. Counterpart axisymmetric solutions with 0.9s =β revealed much reduced epd, in agreement with Aguayo (2006) who also considered larger ratios of 0 95 0 99 0 999( . , . , . )=s

β. With the

smaller value of 1 9=sβ

, Szabo et al., Binding et al. and Aguayo all concur, noting that epd decreases

monotonically with rise in elasticity level for both planar and circular geometric configurations (see Figure 5 for comparison in epd ratio of some axisymmetric results).

The outstanding question and computational challenge on contraction flow pressure-drops remains: when and under what conditions could a constant shear-viscosity/strain-hardening viscoelastic fluid, such as Oldroyd-B, faithfully reflect experimental enhanced epd, as experienced under axisymmetric flow settings for Boger fluids? The implication from general findings with the fe/fv-scheme in pressure drop studies (Aguayo, 2006) is that if enhanced epd is sought, levels of stress must be raised across the constriction, for which the fluid rheology plays a key role, and in particular strong strain-hardening properties are crucial. Other major findings for Oldroyd fluids may be summarized as follows. Significant differences in excess pressure-drop (epd)-data may be identified between axisymmetric and planar configurations for Oldroyd-B fluids in 4:1:4 contraction/expansion flows. For 4:1:4 axisymmetric settings only, epd enhancement above Newtonian and local minima/upturn may be identified, say for high solvent fraction fluids. Downstream vortex enhancement is stimulated upon surpassing local conditions of minima in epd, whilst enhanced-epd provides preferentially larger vortices downstream than upstream. Identification of boundary stress influences along the wall (ξ -coordinate) of Figure 6, supplies an explanation for the local epd-minima in axisymmetric 4:1:4

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configurations, as seen in extravagant extensional stress fields located within the corner neighborhood over the constriction-outlet. In order to stimulate epd-enhancement in the 4:1 contraction, when taken comparatively against the 4:1:4 problem and stress peak data (see below), one can predict that it would be necessary to elevate deformation-rates and elasticity levels by some two