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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.

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

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.

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.

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.

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.

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.

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 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.

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.

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 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 particles on a regular grid are traced backwards over a single time-step. The method is applied to the 4 : 1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions. The development of vortex behaviour with increasing values of We is analyzed.

The flow past a sphere in an infinite expanse of viscoelastic fluid is simulated numerically using a pseudospectral technique. The problem is formulated using primitive variables. A time-splitting scheme is used that ensures that the velocity field is divergence-free at the end of each time step. The flow domain is treated without truncation using rational Chebyshev polynomials as basis functions in the radial direction. The drag on the sphere is computed for a range of values of the non-dimensional elasticity number We . A quantitative comparison of the drag coefficient with that obtained by a perturbation analysis is given. As We is initially increased, a reduction in the drag from its Newtonian value is observed. For larger values of We , drag enhancement is detected. In addition, for small values of We the streamlines are shifted slightly downstream in agreement with experimental observations. The numerical algorithm is described in detail and the numerical results compared qualitatively with experiments.

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