About

164

Research items

5,856

Reads

3,002

Citations

Introduction

Research Experience

Oct 2004

Education

Oct 1979 - Sep 1982

Projects

Project (1)

This project is aimed at the development of thermodynamically consistent continuum models for non-Newtonian (viscoelastic) fluids. New models are developed using energy/entropy formulation within the Generalised Bracket framework. Benchmark problems for viscoelastic flows with high compression/extension and temperature variation are considered.

Research

Research items (164)

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg number. This study presents a new approach to preserve the symmetric positive definiteness of the conformation tensor and to b...

A novel approach to the construction of three-dimensional grids in spherical geometries is described. The grids are based on a range of underlying regular polyhedra. The faces of the polyhedra are arranged into a number of diamonds in the latitudinal and longitudinal directions to facilitate the creation of a structured mesh. Each polyhedron is com...

The discretisation of benchmark viscoelastic flow problems in axisymmetric geometries using the spectral element method is considered. The computations are stabilized using the DEVSS-G/DG formulation of the governing equations. A decoupled approach is employed in which the conservation equations are solved for velocity and pressure and the constitu...

When a cavity forms near a solid boundary a liquid jet can form directed towards the boundary, causing the generation of high pressures at the wall (potentially causing damage) and the formation of a toroidal bubble. In this paper several recent developments in the boundary element modelling of the dynamics of cavitation bubbles in viscoelastic flu...

A spectral element formulation of the immersed boundary method (IBM) is presented. The spectral element formulation (SE-IBM) is a generalisation of the finite element immersed boundary method (FE-IBM) based on high-order approximations of the fluid variables. Several schemes for tracking the movement of the immersed boundary are considered and a se...

This paper is concerned with the numerical solution of high-dimensional Fokker–Planck equations related to multi-dimensional diffusion with polynomial coefficients or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows from the classification of one-dimensional Pearson diffusion. There are six important classes of Pear...

Spectral/hp element methods and an arbitrary Lagrangian-Eulerian (ALE)
moving-boundary technique are used to investigate planar Newtonian extrudate
swell. Newtonian extrudate swell arises when viscous liquids exit long die
slits. The problem is characterised by a stress singularity at the end of the
slit which is inherently difficult to capture and...

This paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single-phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated K...

The benchmark problem of flow of a viscoelastic fluid around a confined cylinder is considered. The governing equations are discretised using spectral/hp element methods. These allow the spatial and temporal variations in the solution that are characteristic of viscoelastic flows, to be resolved accurately and efficiently. A decoupled approach is e...

This paper is concerned with the development of a high-order numerical scheme for two-phase viscoelastic flows. In the companion paper, herein referred to as Part 1, the scheme is applied to the modelling of two-phase Newtonian flows. The particular problem of the collapse of a 2D bubble in the vicinity of a rigid boundary is considered. Attention...

The dynamics of bubbles immersed in a viscoelastic fluid directly beneath an initially plane free surface is modelled using the boundary integral method. The model predicts a range of dynamics that is dependent on the Deborah number, the Reynolds number and the proximity of the bubble to the free surface. The motion of the free surface jet caused b...

This paper is concerned with the development of a high-order numerical scheme for the modelling of two-phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two-phase viscoelastic flows. The particular problem of the collapse of a two-dimensional bubble in the vicinity of a rigid boundary is considered. The...

Multi-bubble dynamics is investigated in the presence of a rigid boundary. A simplified configuration is considered in which
two bubbles, not necessarily of the same size, are positioned along an axis of symmetry. The problem is formulated in terms
of a generalized Bernoulli equation for a velocity potential. A boundary element method is adopted to...

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxin...

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences an...

An understanding of the phenomena associated with cavitation is important in many areas of science and engineering. This paper
is concerned with the influence of viscoelasticity on the dynamics of cavitation bubbles near rigid boundaries. Viscoelastic
effects are modelled using a Maxwell constitutive equation, and a generalized Bernoulli equation i...

A new method for generating a numerical grid on a spherical surface is presented. This method allows the grid to be based on several different regular polyhedrons (including octahedron, cube, icosahedron, and rhombic dodecahedron). The type of polyhedron on which the grid is based can be changed by altering only a few input parameters. Each polygon...

This chapter discusses the applications of Langevin and Fokker–Planck equations in polymer rheology. It presents the stochastic simulation techniques for solving the Langevin equation. It introduces the stochastic differential equations for dilute polymer solutions modeled by dumbbells. Micro-macro techniques for simulating flows of polymeric fluid...

The lattice Boltzmann method has been established as an efficient technique for solving a wide range of complex problems in fluid dynamics including multiphase flows. In this paper, the extension of the technique to the simulation of non-Newtonian fluids is described. Two classes of non-Newtonian fluids are considered: inelastic fluids characterize...

This paper is concerned with two dimensional numerical simulations of plane extrusion of a viscoelastic fluid. The fluid is modelled using the Oldroyd-B and UCM constitutive equations. The problem is discretized using the spectral element method and the free surface is evolved using a Arbitrary Lagrangian Eulerian (ALE) technique. Numerical simulat...

The start-up of plane Couette flow of a finitely extensible nonlinear elastic (FENE) fluid is considered. A numerical method for solving this problem based on a decoupled micro-macro approach is described. The polymeric stress is determined in the microscopic part of the calculation by computing the solution of a Fokker Planck equation for the conf...

This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, whi...

This paper is concerned with the numerical prediction of the extrudate swell behaviour of branched polymer melts in a planar
configuration. The multi-mode extended pom-pom (XPP) model is used to describe the polymer dynamics. A second-order operator-integration-factor
splitting scheme is used for the temporal discretisation of the problem, whilst a...

This paper is concerned with two dimensional numerical simulations of plane extrusion of a Newtonian fluid. The problem is discretized using the spectral element method and the free surface is evolved according to an ALE treatment. Numerical simulations are performed over a wide range of Reynolds and Weber numbers to highlight the effects of inerti...

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the veloc...

The numerical solution of the one-dimensional Fokker–Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an...

This paper presents a modification to the numerical treatment of integral constitutive equations initiated by Peters et al. [E.A.J.F. Peters, M.A. Hulsen, B.H.A.A. van den Brule, Instationary Eulerian viscoelastic flow simulations using time separable Rivlin–Sawyers constitutive equations. J. Non-Newtonian Fluid Mech., 89 (2000) 209–228] that enabl...

The particle deficiency problem in the presence of a rigid wall for smoothed particle hydrodynamics (SPH) is considered. The problem arises from insufficient information being available to perform accurate interpolation of data at particles located nearer to the boundary than the support of the interpolation kernel. The standard method for overcomi...

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a
cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer
dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral e...

The formulation of constitutive equation (model) is essential i n o rder to simulate or predict t he behavior of viscoelastic material i n many c omplicated industrial fl ow processes. The generation o f such model i s tedious and time-consuming op eration. The techniques of artificial intelligence (AI) have proven to b e amenable to solving rheolo...

We present stable and accurate spectral element methods for predicting the steady-state flow of branched polymer melts past a confined cylinder. The fluid is modelled using a modification of the pom-pom model known as the single eXtended Pom-Pom (XPP) model, where we have included a multi-mode model of a commercial low-density polyethylene. We have...

Compressibility plays a significant role in the load-bearing capacity of a journal bearing. This paper offers more realistic modelling of the lubricant than presented in an earlier paper (Int. J. Numer. Meth. Fluids 2007; 55(11):1091–1120) by including variable sound speed, piezoviscosity and both temperature and shear thinning. The load-bearing ca...

Traditionally, the components of the stress with respect to a relevant coordinate system are used for the purpose of stress
visualisation and interpretation. A case for using a flow dependent measure to interpret and visualise stress is made for
two dimensional flow, together with a suggestion for extending the idea to three dimensions. The method...

The quasilinear system of partial differential equations governing the flow of a UCM fluid is known to be of mixed elliptic-hyperbolic type. The compatibility equations associated with the hyperbolic part of the system are derived in this paper. There are two characteristic variables that are transported along the characteristics. These are both as...

The influence of viscoelasticity on the performance of statically and dynamically loaded journal bearings is considered. The lubricant in the system is modelled using either the Oldroyd-B or linear PTT models. Significant viscoelastic effects are presented for both moderate and narrow gap journal bearing configurations. The dynamical behaviour of t...

The problem of droplet deformation and break-up is considered. A hybrid Eulerian–Lagrangian method is used in which the velocity and pressure are discretized on a fixed mesh and Lagrangian particles are used to implicitly track the interface between the two phases. The Navier–Stokes equations are solved using an approximate Godunov projection metho...

The simplest and most efficient lattice Boltzmann model that is able to recover the Navier-Stokes equations is based on a single-parameter scattering matrix, where the parameter is the first nonzero eigenvalue of the collision matrix. This simple model, based on a single relaxation time, has many shortcomings. Among these is the lack of freedom to...

A recently derived axisymmetric lattice Boltzmann model is evaluated numerically. The model incorporates a spatially and temporally varying source term into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog analysis so that th...

A compressible viscous isothermal model is presented for studying journal-bearing lubrication. The viscosity in the model thickens with increasing density. The governing equations are written in terms of velocity, the natural logarithm of the density and the kinematic extra-stress tensor. A semi-Làgrangian treatment of the material derivatives is c...

DOI:https://doi.org/10.1103/PhysRevE.76.059902

The extended pom-pom (XPP) model was first introduced to eradicate perceived deficiencies of the original pom-pom model such as the lack of a second normal stress difference and a discontinuity in the derivative of the extensional viscosity. However, in this paper it is shown that the XPP model itself possesses some disconcerting attributes. In sim...

A modified lattice Boltzmann model based on the two-dimensional, nine-velocity lattice-Bhatnagar-Gross-Krook fluid is presented for axisymmetric flows. A spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the sour...

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so...

The lattice Boltzmann equation is often promoted as a numerical simulation tool that is particularly suitable for predicting the flow of complex fluids. This paper develops a two-dimensional 9-velocity (D2Q9) lattice Boltzmann model for immiscible binary fluids with variable viscosities and density ratio using a single relaxation time for each flui...

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. Sniger...

In the last decades several new and advanced numerical strategies have been proposed for solving the flow models of complex fluids. Most of them were based in the classical discretization techniques (finite elements, finite volumes, finite differences, spectral methods, meshless approachesE) applied on the macroscopic descriptions of such flows (di...

Spectral element methods are developed for simulating viscoelastic flows based on a micro–macro modelling approach in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The application of the technique to the start-up of Couette flow and flow between eccentrically rotating cylinders are describ...

This paper is concerned primarily with the numerical prediction of the pressure field associated with the flow of an Oldroyd-B fluid through 4:1 contractions, 1:4 expansions and combined 4:1:4 contraction/expansions. Particular interest lies in the effect of the ratio of solvent to total viscosity parameter on the profile of the pressure gradient a...

In this article we propose and analyze an a posteriori error estimator for a three-field model of a generalized Stokes problem. The components of the a posteriori error estimator are defined via a non-linear projection of the residues of the variational equations. Both upper and lower bounds for the approximation error are derived in terms of the c...

This paper considers the spectral element approximation of the Stokes problem and the conditioning of the resulting discrete problem. The well-posedness of the variational formulation of the Stokes problem and, in particular, the uniqueness of the pressure has been demonstrated when the subspace of square-integrable functions having vanishing mean...

This paper examines the numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods. The flow domain is divided into a number of conforming rectangular subregions. Within each of the subregions the solution is approximated by a tru...

In this paper we examine the question of whether the viscoelastic properties of multigrade oils can have a measurable effect
on the performance of dynamically loaded journal bearings. Our investigation takes the form of a computational study in which
the full set of coupled equations (kinematic and constitutive) governing the flow of the lubricant...

Based on over 15 years’ experience in the design and delivery of successful first-year courses, this book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management and business studies. The book starts with a summary of basic skills and takes its readers as far as constrained optimisation helpi...

Spectral element methods are developed for solving two-phase flow problems of relevance to two-phase power generation systems (e.g. IC and gas turbines engines), environmental protection (e.g. halon-free explosion suppression) and a range of non-Newtonian fluids. In particular, a direct numerical simulation is presented for the benchmark problem of...

Stable and accurate spectral element methods for predicting the flow of branched polymer melts past a confined cylinder are presented. The fluid is modelled using a modification of the pom–pom model known as the extended pom–pom (XPP) model. Steady and transient flows are considered in this paper. The operator integration factor splitting technique...

Spectral techniques for solving problems in non-Newtonian fluid mechanics are introduced. Following the work of Coleman (J. Non-Newtonian Fluid Mech.; 15, 227–238 [1984]), the governing equations for the creeping flow of a co-rotational Maxwell fluid are written in terms of the Airy stress function and a stream function. This ensures that the conti...

A spectral algorithm based on the influence matrix technique is desclibed for solving numerically the flow of incompressible viscous fluids. The algorithmic development is for both Newtonian and non-Newtonian flows. To investigate the performance of the method several test problems are solved. Accurate results are obtained with relatively few degre...

This paper concerns the modelling of dynamically loaded journal bearing systems using a moving spectral element method. The moving grid method employed in this paper is the arbitrary Lagrangian–Eulerian (ALE) method. The ALE methodology is compared with a quasi-Eulerian approach in the context of dynamically loaded journal bearings and the advantag...

We are concerned with the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We take our earlier work for transient start-up flow in a channel and extend this beyond Oldroyd-B modelling to consider a different fluid model of the pom-pom class. This includes Single Extended form of the pom-pom model (SXP...

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT
model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both
polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stabilit...

Spectral element methods are developed for solving transient flows of viscoelastic fluids. The fluids are modeled using the upper-convected Maxwell and Oldroyd B constitutive relationships. Several temporal schemes for dealing with the time dependent nature of the problems are considered. The computation associated with each time step comprises an...

In this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented. Two contrasting finite volume schemes are described: a hybrid cell-vertex scheme and a pure cell-centred counterpart. Both schemes employ a time-splitting algorithm to evolve the solution through time towards steady state. In the case of th...

The flow of a viscoelastic fluid through an undulating tube is considered. The fluid is modelled using the UCM constitutive equation. The governing set of equations is solved using a time-splitting technique. This is based on separate treatments of the convection and generalised Stokes operators. The spatial discretisation is based on a spectral di...

The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom model [J. Rheol. 42 (1998) 81]. The pom-pom model is a coarse-grained molecular model that was developed for describing branched polymers. A description of the configuration distribution is given in terms of the orientation and stretch of the pom-pom m...

A rheological equation of state is required to simulate the behaviour of a viscoelastic material in many complicated industrial ow processes.

The relaxation spectrum is an important tool for studying the behaviour of viscoelastic materials. The most popular procedure is to use data from a small-amplitude oscillatory shear experiment to determine the parameters in a multi-mode Maxwell model.

This paper is devoted to the development of efficient iterative methods for solving the systems of algebraic equations arising from a spectral element discretization of the equations governing the flow of an Oldroyd B fluid. The governing equations are written in terms of velocity, pressure and extra-stress, giving rise to the so-called three-field...

In this paper, the differences in the development of vortex structure with increasing elasticity between creeping and inertial flows in both planar and axisymmetric contraction configurations are studied for an Oldroyd B fluid. The numerical calculations are performed using a semi-Lagrangian finite volume method described in a paper by Phillips and...

In this paper, we develop an SUPG spectral element scheme suitable for computations of viscoelastic flows at high Deborah numbers. The novelty of the scheme lies in the derivation of the upwinding factors used in the perturbed test tensors of the weak form of the equations. These factors are related to the Deborah number and to the local mesh spaci...

Human placentae and two of the cell types in placentae (cytotrophoblasts and macrophages) were examined by RT-PCR for transcripts of the eight TNF superfamily ligands known to induce death of activated immune cells, tumour cells, and virus-infected cells (TNFalpha, LT alpha, LT beta, FasL, TRAIL, TWEAK, LIGHT, 4-1BBL). Transcripts for all ligands w...

Semi-Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area-weighting. Numerical results are presented illustrating som...

A new finite volume method for solving the incompressible Navier-Stokes equations is presented. The main features of this method are the location of the velocity components and pressure on different staggered grids and a semi-Lagrangian method for the treatment of convection. An interpolation procedure based on area-weighting is used for the convec...

The system of partial differential equations governing the flow of an upper converted Maxwell fluid is known to be of mixed elliptic-hyperbolic type. The hyperbolic nature of the constitutive equation requires that, where appropriate, inflow conditions are prescribed in order to obtain a well-posed problem. Although there are three convective deriv...

The approximation of the Stokes problem in axisymmetric geometries using the spectral element method is considered. The presence of the volume element r dr dz in the weak formulation of the problem is shown to be a potential source of difficulty. The discrete equations associated with nodes on the axis of symmetry can lead to a degeneracy in the gl...

In dynamically loaded journal bearings it is important to predict and assess the performance of lubricants within the bearing with respect to friction, wear and load bearing capacity over a wide range of operating conditions. In this paper, a mathematical model is described which allows for a systematic study of the effects of shear-thinning, press...

A transient thermal analysis for a dynamic journal bearing system is presented based on a non-isothermal non-Newtonian fluid model. A spectral element approach is used to solve the full set of coupled equations (kinematic and constitutive) governing the flow of the lubricant, and an operator-splitting spectral element approximation is used to solve...

The relaxation spectrum is an important tool for studying the behaviour of viscoelastic materials. The most popular procedure is to use data from a small-amplitude oscillatory shear experiment to determine the parameters in a multi-mode Maxwell model. However, the discrete relaxation times appear nonlinearly in the mathematical model for the relaxa...

The numerical approximation of the mixed velocity–pressure–stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed...

The two-dimensional flow of a Newtonian fluid past a cylinder is investigated. The governing equations are discretized using a second-order temporal discretization based on the method of characteristics and a spectral element spatial discretization. The resulting discrete systems are symmetric and (semi-) positive definite and are solved using the...

A semi-Lagrangian finite volume scheme for solving viscoelastic flow problems is presented. A staggered grid arrangement is used in which the dependent variables are located at different mesh points in the computational domain. The convection terms in the momentum and constitutive equations are treated using a semi-Lagrangian approach in which part...

Mechanisms accounting for protection of the fetal semiallograft from maternal immune cells remain incompletely understood. In other contexts, interactions between TRAIL (TNF-related apoptosis-inducing ligand/Apo-2L) and its receptors kill activated lymphocytes. The purpose of this study was therefore to investigate the potential of the TRAIL/TRAIL-...

Although the development of rheological software has kept pace with the advances in modern rheometrical instruments the process of data analysis and interpretation remains fallible and can be prone to some significant errors. The reason for this is that the ability and knowledge of human beings to utilize the instruments and data at their disposal...

In mice and humans, expression of the tumour necrosis factor receptor-1 (TNF-R1) gene in placental trophoblast cells is constitutive whereas expression of the TNF-R2 gene is developmentally programmed. In order to study the individual functions of TNF-R1 and -R2 in this lineage, cell lines were generated from placental explants of homozygous mating...

The effects of non-Newtonian lubricants on the dynamics of a 3D journal bearing are investigated using a moving spectral element method. Comparisons are made with the findings reported for the 2D case. The variation of L/D, the ratio of the length of the bearing to its diameter, is shown to have a significant effect on the stability properties of t...

The numerical approximation of the mixed velocity-pressure-stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces, a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-pose...

The discretization of the mixed velocity–pressure–stress formulation of the Stokes problem using the spectral element method is considered. The compatibility conditions between the discrete velocity and extra stress spaces are examined. A sufficient condition for compatibility, namely that the discrete extra stress space contains the gradient of th...

In this paper we examine the question of whether the viscoelastic properties of multigrade oils can have a measurable effect on the performance of dynamically loaded journal bearings. Our investigation takes the form of a computational study in which the full set of coupled equations (kinematic and constitutive) governing the flow of the lubricant...

We present a semi-Lagrangian finite volume scheme for solving viscoelastic flow problems. A staggered grid arrangement is used in which dependent variables are located at different mesh points in the computational domain. The convection terms in momentum and constitutive equations are treated using a semi-Lagrangian approach in which particles on a...

The development of accurate and stable timedependent approximations to viscoelastic flow problems remains a priority and challenge for researchers in this field. The mathematical statement of viscoelastic flow problems is invariably in terms of velocity, pressure and extra-stress, the so-called mixed formulation. This class of flow problems is char...

A muli-domain spectral collocation method is developed for the approximation of the Stokes problem in two dimensions. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well-posed. The collocation scheme possesses the property that the continuity equation is satisfied at all the collocation points...

The performance of journal bearings is dependent on many factors such as design, materials, load cycle and lubricant. The behaviour of lubricants in automotive bearings is of critical importance as the size of the bearing is restricted due to engine compartment space constraints and the loadings are increased with rising power output. The tradition...