H.R. Tamaddon-Jahromi

Numerical prediction of extensional flows in contraction geometries: hybrid finite volume/element method

Abstract We examine the flow of viscoelastic fluids with various shear and elongational properties in axisymmetric and planar 4:1 contractions, under creeping flow conditions. Particular attention is paid to the influence of elongational viscosity upon vortex enhancement/inhibition. Simulations are performed with a novel hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. Both abrupt and rounded-corner configurations are considered. The Oldroyd-B fluid exhibits vortex enhancement in axisymmetric flows, and vortex reduction in planar flows, qualitatively reproducing experimental observation for some Boger fluids. For shear-thinning fluids (Phan-Thien/Tanner models, PTT), both vortex enhancement and inhibition is observed. This follows trends in extensional viscosity. Lip-vortex activity has been observed in planar and sharp-corner instances, but not in axisymmetric or rounded-corner flows. Finally, cross-flow extensional-stress contours in the salient-corner neighbourhood reflect the size and curvature of the associated vortex structure.

The numerical prediction of planar viscoelastic contraction flows using the pom-pom model and higher-order finite volume schemes

This study investigates the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We extend our earlier work for Poiseuille flow in a planar channel and the single equation form of the extended pom-pom (SXPP) model [M. Aboubacar, J.P. Aguayo, P.M. Phillips, T.N. Phillips, H.R. Tamaddon-Jahromi, B.A. Snigerev, M.F. Webster, Modelling pom-pom type models with high-order finite volume schemes, J. Non-Newtonian Fluid Mech. 126 (2005) 207-220], to determine steady-state solutions for planar 4:1 sharp contraction flows. The numerical techniques employed are time-stepping algorithms: one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum-continuity equations by a fractional-staged Taylor-Galerkin pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. A comparison of the two finite volume approaches is presented, concentrating upon the new features posed by the pom-pom class of models in this context of non-smooth flows. Here, the dominant feature of larger shear and extension in the entry zone influences both stress and stretch, so that larger stretch develops around the re-entrant corner zone as Weissenberg number increases, whilst correspondingly stress levels decline.

Numerical modelling of thixotropic and viscoelastoplastic materials in complex flows

This study is concerned with the finite element/finite volume (fe/fv) simulation of thixotropic and viscoelastoplastic material systems through two model approaches: (i) a new micellar thixotropic constitutive model for wormlike micellar systems (that introduces viscoelasticity into the network structure construction/destruction kinetic equation) and (ii) adopting a Bingham–Papanastasiou model. The computational approach is based on a hybrid parent/subcell scheme, which is cast about a semi-implicit incremental pressure correction (ipc) scheme. The appearance of plastic behaviour arises through the micellar polymeric viscosity, by increasing the zero-shear viscosity (low solvent fractions), whilst the Bingham–Papanastasiou introduces plastic features through the solvent viscosity. The characteristics of thixotropic wormlike micellar systems are represented through the class of Bautista–Manero models. Correction is incorporated, based on physical arguments for fluidity, in which absolute values of the dissipation function are adopted in complex flow, thereby accessing low-solvent fractions and high-elasticity levels. Considering elastic and plastic influences separately, solutions are compared and contrasted for contraction–expansion flow, identifying such flow field features as vortex dynamics, stress field structure, yield front patterns and enhanced pressure drop. Particular attention is paid to the influence of enhanced strain hardening that is introduced through stronger thixotropic structural features. Vortex activity decreases as either We is increased at a fixed τ0 or τ0 is increased at a fixed We. Exaggerated strain-hardening properties are observed to have a major impact on vortex activity. Patterns and trends in normal stress difference fields reflect those in re-entrant corner vortex patterns. Yield front patterns are significantly influenced with yield stress τ0 variation and more so than elevation in elasticity. Findings on excess pressure drop (epd) versus increased yield stress (τ0) follow a linear trend. Consistently, it is evident that any variation that leads to a more solid-like behaviour produces epd enhancement. In addition, relatively more structured fluids display distinctly larger epd values throughout the τ0 range covered.

Numerical vs experimental pressure drops for Boger fluids in sharp-corner contraction flow

This paper addresses the problem of matching experimental findings with numerical prediction for the extreme experimental levels of pressure-drops observed in the 4:1 sharp-corner contraction flows, as reported by Nigen and Walters [“Viscoelastic contraction flows: Comparison of axisymmetric and planar configurations,” J. Non- Newtonian Fluid Mech. 102, 343–359 (2002)]. In this connection, we report on significant success in achieving quantitative agreement between predictions and experiments. This has been made possible by using a new swanINNFM model, employing an additional dissipative function. Notably, one can observe that extremely large pressure-drops may be attained with a suitable selection of the extensional viscous time scale. In addition, and on vortex structure, the early and immediate vortex enhancement for Boger fluids in axisymmetric contractions has also been reproduced, which is shown to be absent in planar counterparts.

Oldroyd-B numerical solutions about a rotating sphere at low Reynolds number

This study investigates the numerical solution of a viscoelastic flow for an Oldroyd-B model, due to the rotation of a sphere about its diameter. Analysis of the elastic-viscous problem had been reported by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964), Walters and Savins (J Rheol 9:407–416, 1965) and Giesekus (Rheol Acta 9:30–38, 1970). In this respect, three different flow patterns (types 1–3) predicted by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964) have been successfully reproduced when using an Oldroyd-B fluid to represent a Boger fluid. Initially, solutions for the Oldroyd-B model were calibrated in the second-order regime against the analytical solution. Then, the work is extended to cover three different flows regimes (second-order regime, transitional and general flow) and two settings of polymeric solvent-fraction. Analysis based on the bounding sphere-radius, associated with type 2 flow, and through different flow regimes revealed that the distinctive symmetrical shape formed in the second-order regime was not preserved, but an elliptical shape was acquired. Moreover, for general and transitional flow regimes, a new and third vortex was identified in the polar region of the sphere. The adjustment of this feature between two different fluid compositions, with solutions for high-solvent and high-polymeric versions (low-high polymeric contributions), was contrasted. The second normal stress difference (N2) on the field was increased as the m parameter developed across the different flow regimes. Different torque values for several m values were compared against the theory, demonstrating the expected linear behaviour. The numerical algorithm involved a hybrid sub-cell finite-element/finite volume discretisation (fe/fv), which solved the system of momentum-continuity-stress equations. It is based on a semi-implicit time-stepping Taylor-Galerkin/pressure-correction parent-cell finite element method for momentum continuity, whilst invoking a sub-cell cell-vertex fluctuation distribution finite volume scheme for the stress. The hyperbolic aspects of the constitutive equation were addressed discretely through finite volume upwind Fluctuation Distribution techniques and inhomogeneity calls upon Median Dual Cell approximation.

Numerical simulation of tube-tooling cable-coating with polymer melts

This study investigates the numerical solution of viscous and viscoelastic flows for tube-tooling die-extrusion coating using a hybrid nite element/nite volume discretisation (fe/fv). Such a complex polymer melt extrusion-draw-coating flow displays a dynamic contact line, slip, die-swell and two separate free-surfaces, presenting an inner and outer conduit surface to the melt-coating. The practical interest lies in determining efficient windows for process control over variation in material properties, stressing levels generated and vacuum pressure levels imposed. The impact of shear-thinning is also considered. Extensive reference is made throughout to viscous inelastic counterpart solutions. Attention is paid to the influence and variation in relevant parameters of Weissenberg number (We), solvent-fraction (β) and second normal difference (N2) (ξ parameter for EPTT). The impact of model choice and parameters upon field response is described in situ through, pressure-drops, rates of deformation and stress. Various numerical alternative strategies, their stability and convergence issues are also addressed. The numerical scheme solves the momentum-continuity-surface equations by a semi-implicit time-stepping Taylor-Galerkin/pressure-correction (TGPC) finite element (parent-cell) method, whilst invoking a sub-cell cell-vertex fluctuation distribution finite volume scheme for the constitutive stress equation. The hyperbolic aspects of the constitutive equation are addressed discretely through upwind Fluctuation Distribution techniques, whilst temporal and source terms are consistently accommodated through medium-dual-cell schemes. The dynamic solution of the moving boundary problem may be resolved by either separating the solution process for each free-surface section (decoupling), or coupling both sections and solving simultaneously. Each involves a surface height location method, with dependency on surface nodal velocities and surface element sections; two such schemes are investigated. Dedicated and localised shock-capturing techniques are introduced to handle solution singularities as disclosed by die-swell, slip and moving contact lines.

On the use of continuous spectrum and discrete-mode differential models to predict contraction-flow pressure drops for Boger fluids

Over recent years, there has been slow but steady progress towards the qualitative numerical prediction of observed behaviour when highly elastic Boger fluids flow in contraction geometries. This has led to an obvious desire to seek quantitative agreement between prediction and experiment, a subject which is addressed in the current paper. We conclude that constitutive models of non-trivial complexity are required to make headway in this regard. However, we suggest that the desire to move from qualitative to quantitative agreement between theory and experiment is making real progress. In the present case with differential models, this has involved the introduction of a generalized continuous spectrum model. This is based on direct data input from material functions and rheometrical measurements. The class of such models assumes functional separability across shear and extensional deformation, through two master functions, governing independently material-time and viscous-response. The consequences of such...

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 (ε), 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 (ε > 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 (ε < 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.

Time-dependent algorithms for viscoelastic flow: Bridge between finite-volume and finite-element methodology

Publisher Summary nThis chapter considers the accuracy of discrete schemes when tracking transient solutions to viscoelastic flow problems, and for this purpose, it selects a model problem, equipped with an analytical solution. The chapter indicates the transient, planar poiseuille flows for viscoelastic fluids. It proposes a novel time-dependent hybrid finite volume (fv)/finite element (fe) algorithm. This approach combines a Taylor-Galerkin fe-treatment for mass and momentum conservation equations and a cell-vertex fv-discretisation of the hyperbolic stress constitutive equation. A consistent formulation for the constitutive equation is the key. This incorporates fe and fv-treatment of the various terms. In this manner, an accurate transient algorithm emerges that reproduces analytical solution structure, both in core-flow and across shear-boundary zones. The chapter is also are concerned with isothermal, incompressible, viscoelastic flows, described via the combination of mass and momentum conservation equations, alongside a stress constitutive equation. In addition, recourse to consistent mass-matrix iteration for time-terms, in place of a finite volume “mass-lumping approach,” has introduced fresh avenues to explore in the search for transient solutions within the viscoelastic domain.

A computational extensional rheology study of two biofluid systems

The main focus of the present computational modelling work is to determine the extensional rheological response of some model biofluids, with a view to ultimately aiding experimentally based analyses and clinical practice. This is accomplished in the present study through model extensional flows and rheological investigation, addressing filament stretching and contraction flows, and upon which significant advances are presented. As such, two biofluid flow systems within the human body are of current interest: (i) respiratory disorders and sputum in the lung airways (associated with filament stretching), where stretchiness of mucus sputum in situ is vital, with clinical focus on chronic obstructive pulmonary disease (COPD/sputum); and (ii) bile flow in the biliary system (contraction flow), with clinical focus on disorders of primary sclerosis cholangitis and common bile duct narrowing. Both sputum and bile biofluid systems are represented through kinetic theory rheological fluid modelling, with capability to represent material structure entanglement, branching and anisotropy. This is practically achieved by appealing to the class of pom-pom differential constitutive models, extracted from polymer melt physics and deployed here through a single extended pom-pom (SXPP) approximation. This class of models is sufficiently rich to enable description of both network structure and rheological properties, exhibiting viscoelastic response (memory), with strain-hardening/softening and shear-thinning properties.