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A Taylor–Galerkin finite element time marching scheme is derived to numerically simulate the flow of a compressible and nonisothermal viscoelastic liquid between eccentrically rotating cylinders. Numerical approximations to the governing flow and constitutive equations are computed over a custom refined unstructured grid of piecewise linear Galerkin finite elements. An original extension to the DEVSS formulation for compressible fluids is introduced to stabilise solutions of the discrete problem. The predictions of two models: the extended White–Metzner and FENE-P-MP are presented. Comparisons between the torque and load bearing capacity predicted by both models are made over a range of viscoelastic parameters. The results highlight the significant and interacting effects of elasticity and compressibility on journal torque and resultant load, and the stability of the journal bearing system.

Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen‐Loeve expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss‐Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.

The linear stability analysis of Rivlin–Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.

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 completely characterized in terms of the arrangement of diamonds. The approach is shown to be very flexible in terms of the meshes that can be generated. It also aids comparisons between the grids used in many mantle convection studies. A spectral element discretization of Poisson's equation is performed to demonstrate the efficacy of the grid generation technique. Rapidly convergent approximations are obtained that demonstrate that fewer degrees of freedom are required to obtain a desired level of accuracy compared with low-order finite element approximations.

This paper is concerned with the numerical solution of high-dimensional Fokker-\ud Planck equations related to multi-dimensional diffusion with polynomial coefficients\ud or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows\ud from the classification of one-dimensional Pearson diffusion. There are six important\ud classes of Pearson diffusion - three of them possess an infinite system of moments\ud (Gaussian, Gamma, Beta) while the other three possess a finite number of moments\ud (inverted Gamma, Student and Fisher-Snedecor). Numerical approximations to the\ud solution of the Fokker-Planck equation are generated using the spectral method.\ud The use of an adaptive reduced basis technique facilitates a significant reduction in\ud the number of degrees of freedom required in the approximation through the determination\ud of an optimal basis using the singular value decomposition (SVD). The\ud basis functions are constructed dynamically so that the numerical approximation\ud is optimal in the current finite-dimensional subspace of the solution space. This is\ud achieved through basis enrichment and projection stages. Numerical results with different\ud boundary conditions are presented to demonstrate the accuracy and efficiency\ud of the numerical scheme.

Proper generalised decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGD provide a much stabler algorithm than an equivalent Galerkin PGD and provide proofs of convergence of the algorithms.

This paper is concerned with a comparison of the performance and efficiency of mixed finite element methods for solving single phase fluid flow in porous media. Particular attention is given to the accurate determination of the groundwater fluxes. The linear systems generated by the mixed finite element method (MFEM) are indefinite. Symmetric positive definite linear systems are generated through the introduction of Lagrange multipliers giving rise to the mixed hybrid finite element method (MHFEM). The convergence behaviour of the numerical approximations is investigated over a range of conductivity coefficients from heterogeneous, isotropic and diagonal to discontinuous, anisotropic and full, on both triangular and quadrilateral, structured and distorted meshes. The robustness and efficiency of various preconditioned solvers is investigated in terms of optimality with respect to both mesh size and conductivity coefficient.

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 Karhunen–Loeve expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss–Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen–Loeve expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables.

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 employed in which the conservation equations are solved for velocity and pressure and the constitutive equation (Oldroyd-B and Giesekus) are solved for the polymeric component of the extra-stress tensor. The computations are stabilized using the DEVSS-G/DG formulation of the problem. Excellent agreement with the literature is achieved for the drag coefficient in the case of an Oldroyd-B fluid. Causes for the breakdown in numerical convergence with mesh refinement beyond some critical value of the Weissenberg number are explored. The high resolution property of spectral/hp approximations has enabled an instability that develops in the shear layer on the cylinder and is convected downstream to be identified. The onset of this instability is shown to occur at the critical value of the Weissenberg number predicted by the theory of Dou and Phan-Thien [9]. The influence of the Reynolds number and, for the Giesekus model, the mobility parameter on the drag coefficient is also investigated and discussed.

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 fluids are presented. The standard formulation of the boundary element method (BEM) is in terms of a boundary integral equation with a singular kernel. A reformulation of the BEM in terms of a non-singular kernel is shown to provide enhanced stability. In situations when a liquid jet forms and impacts the far side of the bubble there is a transition to a toroidal form. This topological singularity in bubble geometry is modelled by placing a vortex ring inside the bubble to account for the circulation in the fluid and the discontinuity in potential following jet impact. The bubble dynamics are dependent on the initial stand-off distance from the boundary as well as the viscous and elastic properties of the fluid. It is shown that, while the viscosity of the fluid inhibits jet formation, the dynamics are particularly dependent on the relative strength of viscous, elastic and inertial forces. In particular, if the Deborah number is large enough elastic effects effectively negate fluid viscosity and behaviour similar to the inviscid case is recovered in terms of liquid jet formation.

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 extend the model to complex fluids whose stress tensors are characterized by more complicated constitutive relations. The lattice Boltzmann methodology may be generalized by considering the full collision matrix and tuning the matrix elements to obtain the desired macroscopic properties. The generalized hydrodynamics of a generalized lattice Boltzmann equation (LBE) was studied by Lallemand and Luo [Phys. Rev. E 61, 6546 (2000)] by solving the dispersion equation of the linearized LBE. In this paper, an alternative approach to solving the dispersion equation based on a formal perturbation analysis is described. The methodology outlined is systematic, can be readily applied to other lattices, and does not require the reciprocals of the relaxation times to be small.

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 determine the normal velocity at the bubble surface. Viscoelastic effects are incorporated into the model through the normal stress balance across the bubble surface. The model predicts that bubble dynamics, and jet formation, in particular, is strongly influenced by the initial bubble sizes, the initial stand-off distances and on the fluid parameters. Viscoelastic effects are shown to be responsible for some interesting dynamics. Although inter-bubble interaction is shown to play an important role, it is not always dominant for viscoelastic fluids.

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 is given to the construction of the most general form of the compressible Oldroyd B model that is consistent with the compressible Newtonian and upper-convected Maxwell models in the appropriate limits. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscoelastic effects that are predicted include increased bubble oscillation with increasing Weissenberg number and considerable bubble deformation and cusping near the wall. Most importantly, it has been shown that viscoelasticity has the ability to prevent jet formation and therefore is likely to have a mitigating effect on cavitation damage.

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. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.

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 simulations are performed over an wide range of parameters. The influence of Weissenberg number and viscosity ratio on the swelling ratio and exit pressure loss are investigated and comparisons with previous results in the literature are presented. The range of validity of Tanner’s theory is investigated.

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 is derived. The governing equations are solved using the boundary element method in which both the bubble surface and the potential are represented by cubic splines. The numerical scheme is validated through comparisons with results in the literature for the inviscid case. The introduction of viscoelasticity introduces some interesting bubble dynamics including the occurrence of oscillations during collapse. Most importantly, it is shown that viscoelasticity can serve to suppress the formation of a liquid jet. The subsequent reduced pressures compared with the inviscid case suggest that viscoelasticity has a mitigating effect on cavitation damage.

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 characterized by a variable viscosity and viscoelastic fluids characterized by fluid elasticity. A comprehensive review is given of developments in this area and sample numerical results are presented in the case of inelastic fluids.

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 configuration probability density function. The solution of this equation often suffers from the so-called problem of "curse of dimensionality." An efficient solution procedure is described based on the use of an adaptive reduced basis function technique in conjunction with high-order spectral approximations. A substantial reduction in the number of degrees of freedom to achieve a required level of accuracy is obtained compared with standard tensor product representation of the basis. Some sample numerical results are presented to highlight properties of the model and the numerical scheme.

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 velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius

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 adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker-Planck equation, using the d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.

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 enables computational stability to be achieved for higher Weissenberg number for steady flows. Increased efficiency is also achieved so that computational times are of the same order (2 or 3 times) as equivalent differential models and of similar accuracy. The memory cost for storage of the deformation fields is approximately 300 times that of the stress tensor for each grid point to yield approximations that lie within 0.01% of predictions of equivalent differential models. The good agreement between equivalent integral and differential models found here suggests that more general integral models with no differential equivalent may be used reliably for steady flow simulations. A modification of the deformation fields method that avoids the introduction of a cut-off time and which is applicable to transient flows is also presented.

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 overcoming this problem is based on the introduction of image particles to populate the deficient regions and the use of linear extrapolation to determine the velocity of these image particles from that of fluid particles. A consistent treatment of boundary conditions, utilising the momentum equation to obtain approximations to the velocity of image particles, is described. The method ensures second order approximation of the boundary conditions. It is validated for Poiseuille and Couette flow, for which analytical series solutions exist and shows second order convergence under certain conditions.

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 element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.

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 analyzed the XPP model and found interesting multiple solutions for certain choices of the parameters which indicate possible problems with the model. The operator-integration-factor-splitting technique is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

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 capacity of the journal bearing is sensitive to all of these attributes of the model, but piezoviscosity is found to be the most significant. The equations of motion are adapted to a moving frame to explore the stability of the journal in a more dynamic setting and it is found that a free journal using this model will spiral outward exhibiting half speed whirl. The model is discretized in 2D, semi-implicitly in time and using the spectral element method in space. Numerical results are presented that highlight the contributions of the different elements in the model to journal stability.

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 the journal bearing system is shown to be dependent on the fluid model, the relaxation time and also the gap size.

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 is illustrated for Newtonian and Oldroyd B fluids in both the eccentrically rotating cylinder and flow past a cylinder benchmark problems. In the context of a generalised Newtonian fluid, the relation between the flowdependent stress measure to other field variables under certain flow conditions, is examined and is indicative of its importance in complex flow.

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 method, collectively called the Godunov marker-particle projection scheme. The results show good qualitative agreement with previous research as well as demonstrating the efficacy of the method.

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 simple steady shear and uniaxial extensional flow, multiple solutions are found. Furthermore, the extended pom-pom model can predict positive as well as negative values for the second normal stress difference. Experimentally, the second normal stress difference is negative for polymer melts.

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 that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.

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-Lagrangian treatment of the material derivatives is combined with a spectral element discretization in space. The roles of the speed of sound and the eccentricity ratio on the load-bearing capacity of the journal bearing are investigated. Compressibility is shown to enhance the load-bearing capacity and this effect is amplified as the eccentricity ratio approaches unity. It is shown that for speeds of sound in the region of those of multigrade oils, the dominant component of the force on the journal acts along the line joining the centres of the bearing and journal and in the direction away from the narrow gap.

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 described. Comparisons are made between simulations based on the Oldroyd B constitutive model and those based on Brownian configuration fields using Hookean dumbbells. Comparisons are also made between simulations based on different force laws. The performance of the method is discussed.

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 and on the entry and exit development lengths. Although most of the results refer to planar (2D) flows, some results are presented to illustrate how the 3D case convergences towards two-dimensional flow as the relevant aspect ratio of the geometries becomes large. Some results are presented for certain aspects related to the flow kinematics, in order to illustrate the link between the pressure field and the resulting flow field.

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 components of the a posteriori error estimator. The non-linear projections do not need to be explicitly computed to construct the a posteriori error estimates.

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 is chosen for the pressure space. In the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice. However, this space is not used in practical implementations of finite or spectral element methods for the Stokes problem. In this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner. The dependence of the condition number of the Uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated.

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 is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

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 (SXPP), comparing the results of two different finite volume schemes. 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. This draws upon fluctuation distribution schemes (upwinding), different combinations of ‘flux’ and ‘median-dual-cell’ spatial discretisations and time-term treatments. Here, unstructured and structured meshes may be used, based largely on triangular grids. A comparison of the two finite volume approaches will be presented, concentrating upon the new features posed by the pom-pom class of models.

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 stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.

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 advantages of adopting the ALE formulation are highlighted. A comparison with the predictions of lubrication theory is also presented and the limitations of the lubrication approximation are demonstrated when inertial effects are significant. A comprehensive set of results is presented illustrating the salient features of the spectral element mesh generation schemes described in the paper and the way in which these impinge on the efficiency of the iterative solution of the discrete equations at each time step. Copyright © 2005 John Wiley & Sons, Ltd.

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 explicit treatment of the convection and deformation terms and an implicit treatment of the linear terms. The generalized three field Stokes problem generated in the latter is solved using a nested preconditioned conjugate gradient method. The performance of a number of preconditioners is investigated. A modified formulation of the Stokes problem is presented that ensures a unique pressure approximation with zero mean is obtained at each time step. The performance of the schemes and models when applied to the benchmark problem of start-up of plane Poiseuille flow is investigated.

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 the hybrid scheme, a semi-implicit formulation is employed in the momentum equation, based on the Taylor-Galerkin approach with a pressure-correction step to enforce incompressibility. The basis of the pure finite volume approach is a backward Euler scheme with a semi-Lagrangian step to treat the convection terms in the momentum and constitutive equations. Two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids. The schemes are constructed to be second-order accurate in space. The issue of stability is also addressed with respect to the treatment of convection. Numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions.

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 molecule. The variant of the pom-pom model used in this paper is the extended pom-pom model [J. Rheol. 45 (4) (2001) 823]. The model has been rewritten to make it suitable for implementation in an existing spectral element algorithm, which solves the constitutive equation together with the momentum and continuity equations The temporal discretization of the convective terms in the momentum equation and the constitutive equation is performed using a second-order time-splitting technique. Spatial discretization is performed using spectral element techniques. The system of discretized equations is solved using a preconditioned conjugate gradient method, and details of the preconditioners are given. Results are presented for the start-up of Poiseuille flow in a planar channel, and flow past a confined cylinder. The influence of the model parameters on the solution is described.

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 discretisation in which the radial basis functions satisfy the conditions along the axis of symmetry of the tube explicitly. Compatible approximation spaces are chosen for velocity, pressure and extra-stress. The convection problem is solved in a type-sensitive manner using a high-order Runge–Kutta method. The weak formulation of the generalised Stokes problem is discretised and solved using a nested conjugate gradient technique. The effect of elasticity on flow resistance is investigated and comparisons made with other results in the literature.

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 Williams [J. Non-Newtonian Fluid Mech. 87 (1999) 214]. This method combines the advantages of fixed grids inherent in Eulerian methods with modifications to treat the convective terms using particle tracking methods. The method alleviates some of the difficulties encountered when discretizing differential constitutive equations of hyperbolic type for viscoelastic models. The semi-Lagrangian technique is applied to the convective terms in the momentum and constitutive equations. It is a natural way of treating these terms without resorting to upwinding techniques.

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 formulation. The convection terms are treated using an operator-integration-factor splitting method. The remaining terms in the system of equations constitute a generalized Stokes problem. This problem is formulated in terms of an Uzawa operator applied to the pressure. Efficient preconditioners are developed for inverting this operator that are independent of the numerical and physical parameters of the problem.

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 global system of equations. This difficulty is resolved by incorporating the factor r into the weight function for spectral elements adjacent to the axis of symmetry and using appropriate basis functions in these elements in the radial direction. Properties of the Jacobi polynomials are used to construct the elements of the modified method. Numerical results are presented demonstrating some of the features of the proposed approach.

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, pressure-thickening and temperature-thinning viscosity on the dynamics and stability of a journal bearing. It is also shown that the thermal conductivity of the bearing can have a major effect on the minimum oil film thickness attained in the bearing.

A transient thermal analysis for a dynamic journal bearing system is presented in this paper based on a non-isothermal non-Newtonian fluid model. A spectral element approach is used to solve the full set of coupled equations (kinematics and constitutive) governing the flow of the lubricant, and an operator-splitting spectral element approximation is used to solve the dynamic energy equation. The transient thermal effects in the lubricant viscosity, non-Newtonian lubricant film thickness, variation of the journal rotating speed and time dependent applied load are investigated in this paper. The numerical results indicate that the thermal dynamic analysis plays an important role in the design of the journal bearing systems, especially, under large applied loads and high rotating speeds.

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 problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. Proper matching conditions between spectral elements are addressed leading to a discontinuous extra-stress approximation.

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 conjugate gradient method. The performance of the algorithm is examined. The dependence of the drag on the cylinder as a function of the Reynolds number is investigated.