Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers (We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.     

Numerical simulation of viscoelastic flows with Oldroyd-B constitutive equations and novel algebraic stress models

The purpose of the present study is to compare numerical simulations of viscoelastic flows using the differential Oldroyd-B constitutive equations and two newly devised simplified algebraic explicit stress models (AES-models). The flows of a viscoelastic fluid in a 180° bent planar channel and in a 4:1 planar contraction are considered to illustrate and support the underlying theory. The flow in the bent channel is used to illustrate the frame-invariant property of the new models in a pure shear flow exhibiting strong streamline curvature. The flow in the 4:1 contraction serves as a benchmark test in a situation where strong elongation occurs. For both geometries, it is found that the predictions of the new AES-models are in good agreement with Oldroyd-B up to Deborah numbers of order 0.5, with a significant reduction in computational effort.     

A new mixed finite element for calculating viscoelastic flow

A new mixed finite element has allowed us to calculate flows of Maxwell-B and Oldroyd-B fluids at very high values of the Deborah number, De. The element is divided into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretizing the constitutive equation. The method is applied to the stick-slip problem, the flow through a tapered contraction and the flow through four-to-one abrupt plane and circular contractions. Important corner vortices develop at high values of De in the circular contraction. We have not encountered upper limits for the Deborah number in our calculations with Oldroyd-B fluids.     

Application of Boundary-Fitted Coordinate Transformations to groundwater flow modeling

The Boundary-Fitted Coordinate (BFC) Transformation method is a very powerful, efficient and accurate method of modeling heat or fluid flow in two- or three-dimensional domains with complex boundary shapes and abrupt changes in internal properties. Since the late 1970's it has become the modeling method of choice among many aerodynamicists and heat-flow modelers. It is being presented here for the first time as a new approach to modeling groundwater flow, based on successful research results in two dimensions. The BFC transformation method was employed to simulate two hypothetical well-flow scenarios in isotropic and anisotropic domains, and actual groundwater flows in the area of West Lafayette, Indiana. The numerical solutions in those cases were at least as accurate as and/or consistent with those obtained by purely finite difference and finite element methods, but with the added advantage of more accurate representation and implementation of the boundary condition in the region of great sensitivity. The BFC method successfully applied to two-dimensional simulations should be easily extended to simulations of three-dimensional flow and transport and thus, this research is continuing in that direction.     

Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems

This study extends our previous analysis on pressure-drops for strain-hardening Boger-type fluids in contraction flow settings, into those fluids that manifest mild shear-thinning properties. Numerical simulations are compared and contrasted for a variety of constitutive equations, categorised through their differences in viscometric functional response, considering application on 4:1:4 contraction-expansion flow and 2:1 flow past a sphere. Here, prior results on pressure-drop enhancement for constant shear-viscosity fluids have revealed the counter-influences of first normal stress differences and extensional viscosity. The present comparative work advances this study by selectively including the effects of shear-thinning. Suitable models to accomplish this are chosen from the class of Phan-Thien/Tanner (PTT) models, with cross-reference to FENE-models and Oldroyd-B. Furthermore, the work explores the falling sphere problem with comparison of the drag coefficient factor for various implementations. The numerical computations are performed by appealing to a well-founded hybrid finite element/finite volume algorithm, using structured triangular meshing, semi-implicit time-stepping and subcell technology. The cell-vertex finite volume scheme is particularly suited to the solution of the stress subsystem, and invokes fluctuation-distribution for upwinding and median-dual-cells for source-term representation.     

Numerical simulation of transient planar flow of a viscoelastic material with two moving free surfaces

In this study, we examine the numerical simulation of transient viscoelastic flows with two moving free surfaces. A modified Galerkin finite element method is implemented to the two-dimensional non-steady motion of the fluid of the Oldroyd-B type. The fluid is initially placed between two parallel plates and bounded by two straight free boundaries. In this Lagrangian finite element method, the spatial mesh deforms in time along with the moving free boundaries. The unknown shape of the free surfaces is determined with the flow field u, v, τ, p by the deformable finite element method, combined with a predictor-corrector scheme in an uncoupled fashion. The moving free surfaces and fluid motion of both Newtonian and non-Newtonian flows are investigated. The results include the influence of surface tension, fluid inertia and elasticity.     

A simple method for simulating viscoelastic fluid flows in slit channel

A low-cost semi-analysis finite element technique, named the finite piece method (FPM) is presented in this article. It aims to solve three-dimensional (3D) viscoelastic slit flows. The viscoelastic stress of the fluid is modelled using an K-BKZ integral constitutive equation of the Wagner type. Picard iteration is used to solve non-linear equations. The FPM is tested on flow problems in both planar and contraction channels. The accuracy of the method is assessed by comparing flow distributions and pressure with results obtained by 3D finite element method (FEM). It shows that the solution accuracy is excellent and a substantial amount of computing time and memory requirement can be saved.     

Laminar flow in three-dimensional square–square expansions

In this work we investigate the three-dimensional laminar flow of Newtonian and viscoelastic fluids through square–square expansions. The experimental results obtained in this simple geometry provide useful data for benchmarking purposes in complex three-dimensional flows. Visualizations of the flow patterns were performed using streak photography, the velocity field of the flow was measured in detail using particle image velocimetry and additionally, pressure drop measurements were carried out. The Newtonian fluid flow was investigated for the expansion ratios of 1:2.4, 1:4 and 1:8 and the experimental results were compared with numerical predictions. For all expansion ratios studied, a corner vortex is observed downstream of the expansion and an increase of the flow inertia leads to an enhancement of the vortex size. Good agreement is found between experimental and numerical results. The flow of the two non-Newtonian fluids was investigated experimentally for expansion ratios of 1:2.4, 1:4, 1:8 and 1:12, and compared with numerical simulations using the Oldroyd-B, FENE-MCR and sPTT constitutive equations. For both the Boger and shear-thinning viscoelastic fluids, a corner vortex appears downstream of the expansion, which decreases in size and strength when the elasticity of the flow is increased. For all fluids and expansion ratios studied, the recirculations that are formed downstream of the square–square expansion exhibit a three-dimensional structure evidenced by a helical flow, which is also predicted in the numerical simulations.     

Analysis of fluid flow in constricted tubes and ducts using body-fitted non-staggered grids

A finite volume method for the calculation of laminar and turbulent fluid flows inside constricted tubes and ducts is described. The selected finite volume method is based on curvilinear non-orthogonal co-ordinates (body-fitted co-ordinates) and a non-staggered grid arrangement. The grids are either generated by transfinite interpolation or an elliptic grid generator. The method is employed for calculation of laminar flows through a tube, a converging-diverging duct and different constricted tubes by both a two- and a three-dimensional computer program. In addition, turbulent flow through an axisymmetric constricted tube is calculated. Both the power law scheme and the second-order upwind scheme are used. The calculated results are compared with the experimental data and with other numerical solutions.     

Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method

Mould filling processes, in which a material flow front advances through a mould, are typical examples of moving boundary problems. The moving boundary is accompanied by a moving contact line at the mould walls causing, from a macroscopic modelling viewpoint, a stress singularity. In order to be able to simulate such processes, the moving boundary and moving contact line problem must be overcome. A numerical model for both two- and three-dimensional mould filling simulations has been developed. It employs a pseudo-concentration method in order to avoid elaborate three-dimensional remeshing, and has been implemented in a finite element program. The moving contact line problem has been overcome by employing a Robin boundary condition at the mould walls, which can be turned into a Dirichlet (no-slip) or a Neumann (free-slip) boundary condition depending on the local pseudo-concentration. Simulation results for two-dimensional test cases demonstrate the model's ability to deal with flow phenomena such as fountain flow and flow in bifurcations. The method is by no means limited to two-dimensional flows, as is shown by a pilot simulation for a simple three-dimensional mould. The reverse problem of mould filling is the displacement of a viscous fluid in a tube by a less viscous fluid, which has had considerable attention since the 1960's. Simulation results for this problem are in good agreement with results from the literature. © 1998 John Wiley & Sons, Ltd.     

Transient finite element analysis of generalized Newtonian coextrusion flows in complex geometries

A two-dimensional transient finite element model capable of simulating problems related to two-layer polymer flows has been developed. This technique represents an effective tool which can be used to study the possibility of the onset of interfacial instability in coextrusion flows, considering melt rheology as well as the fluid–geometry interaction. A code has been developed to solve the transient problem of the flow of bi-component systems of Newtonian and generalized Newtonian fluids through parallel plates and complex geometries, such as: 2:1 abrupt expansion, 2:1 (30°) expansion, 4:1 abrupt contraction and 4:1 tapered (30°) contraction. Solutions are compared with experimental data from the literature and results provided by linear stability analysis (LSA) for the case of parallel plate flows. Numerical results are in agreement with LSA results for the parallel plate geometry cases studied. The expansion geometries tend to stabilize flows in the parallel plate section downstream of the expansion. Contractions may give rise to break-up of the interface depending on the flow conditions. © 1998 John Wiley & Sons, Ltd.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

Numerical investigation of transient contraction flows for worm-like micellar systems using Bautista–Manero models

This study is concerned with the numerical modelling of the Modified Bautista–Manero (MBM) model, for both steady-state and transient solutions in planar 4:1 contraction flow. This model was proposed to represent the structured composition and behaviour of worm-like micellar systems which have importance in industrial oil-reservoir recovery applications. A parameter sensitivity analysis for the rheology of this model is presented in both transient and steady response, covering pure shear and uniaxial extension. In addition, some features in evolutionary flow-structure are demonstrated in contraction flows due to the influence and imposition of start-up transient boundary conditions. The different effects of various model parameter choices are described through transient field response, stress and viscosity fields in the contraction flow setting. Distinction may be drawn between fluid response in the strong/moderate extension hardening regimes by matching both steady-state and transient shear and extensional viscosity peaks, contrasting between micellar (MBM) models against network-based counterparts Phan-Thien/Tanner (PTT). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm.     

A coupled fluid-elasticity model for the wave forcing of an ice-shelf

A mathematical model for predicting the vibrations of ice-shelves based on linear elasticity for the ice-shelf motion and potential flow for the fluid motion is developed. No simplifying assumptions such as the thinness of the ice-shelf or the shallowness of the fluid are made. The ice-shelf is modelled as a two-dimensional elastic body of an arbitrary geometry under plane-strain conditions. The model is solved using a coupled finite element method incorporating an integral equation boundary condition to represent the radiation of energy in the infinite fluid. The solution is validated by comparison with thin-beam theory and by checking energy conservation. Using the analyticity of the resulting linear system, we show that the finite element solution can be extended to the complex plane using interpolation of the linear system. This analytic extension shows that the system response is governed by a series of singularities in the complex plane. The method is illustrated through time-domain simulations as well as results in the frequency domain.     

Viscoelastic flow in a two-dimensional collapsible channel

We compute the flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens blood model) in a two-dimensional channel partly bounded by a tensioned membrane, a benchmark geometry for fluid–structure interactions. The predicted flow patterns are compared to those of a Newtonian liquid. We find that computations fail beyond a limiting Weissenberg number. Flow fields and membrane shape differ significantly because of the different degree of shear thinning and molecular extensibility underlying the three different microstructural models.     

Magnetohydrodynamic transient flows of a non-Newtonian fluid

Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.     

Comparison of three solvers for viscoelastic fluid flow problems

The computational efficiency of three numerical schemes has been examined for the solution of a linearized system of equations resulting from the finite element discretization of a viscoelastic fluid flow problem. The first scheme is a modified frontal solver, which solves the linear system of equations directly. The other two, one based on a biconjugate gradient stabilized (BiCGStab) method and another based on a generalized minimal residual (GMRES) method, are iterative schemes. The stick-slip problem and the four-to-one contraction problem were analyzed and the viscoelastic fluid was assumed to obey the Oldroyd-B model. The two iterative schemes are superior to the direct scheme in terms of CPU time consumed and the BiCGStab scheme is even faster than the GMRES scheme. The range of convergence for both iterative schemes is compatible with that of the direct scheme.     

Highly parallel time integration of viscoelastic flows

A highly parallel time integration method is presented for calculating viscoelastic flows with the DEVSS-G/DG finite element discretization. The method is a synthesis of an operator splitting time integration method that decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. Both steps are performed in parallel. The discontinuous finite element discretization of the hyperbolic constitutive equation leads to highly-parallel element-by-element calculation of the stress at each time step. The Stokes-like problem is solved by using the BiCGStab Krylov iterative method implemented with the block complement and additive levels method (BCALM) preconditioner. The solution method is demonstrated for the calculation of two-dimensional (2D) flow of an Oldroyd-B fluid around an isolated cylinder confined between two parallel plates. These calculations use extremely fine finite elements and expose new features of the solution structure.