Roger I. Tanner

A new constitutive equation derived from network theory

Abstract A constitutive equation is derived from a Lodge—Yamamoto type of network theory for polymeric fluids. The network junctions are not assumed to move strictly as points of the continuum but allowed a certain “effective slip”. The rates of creation and destruction of junctions are assumed to depend on the instantaneous elastic energy of the network, or equivalently, the average extension of the network strand, in a simple manner. Agreement between model predictions and the I.U.P.A.C. data on L.D.P.E. is good.

The Solution of Viscous Incompressible Jet and Free Surface Flows Using Finite Element Methods.

Abstract : The authors discuss the creation of a finite element program suitable for solving incompressible, viscous free surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. The authors conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related stick-slip problem and the tube inlet problem, both of which also contain stress singularities, are also given. (Modified author abstract)

Numerical study of the Bingham squeeze film problem

Abstract In studies of the flow of a Bingham fluid in a parallel-plate plastometer there has been disagreement about whether or not a yield surface exists, and if it does exist what shape the yield surface has. The present authors have re-exemined the problem using a finite element program and have concluded that a small plug of unyielded fluid exists adjacent to the centre of the plates. This result has been verified by replacing the unyielded plug with a solid body.

Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method

Abstract In this paper, a general class of viscoelastic model is used to investigate numerically the pattern and strength of the secondary flows in rectangular pipes as well as the influence of material parameters on them. To solve the coupled governing equation system, an implicit finite volume method based on the SIMPLEST algorithm, which is applicable for both time-dependent and steady-state flow computations, has been developed and extended for viscoelastic flow computations by applying the decoupled techniques. The main feature of the method is to split the solution process into a series of steps in which the continuity of the flow field is enforced by solving a Poissons equation for the pressure, and at the end of the steps, both the pressure and velocity fields are made to satisfy one and the same momentum equation. For viscoelastic flow computations, artificial diffusion terms are introduced on both sides of the discretized constitutive equations to improve numerical stability. It is found that there are in total two vortices in each quadrant of the pipe at different aspect ratios (from 1 to 16), and at each ratio the pattern of secondary flows takes the same form for different material parameters, but their strength is very sensitive to the viscoelastic material parameters. Numerical results indicate that the presence of secondary flow strongly depends on the primary flow rate and the elasticity of the fluid, namely, the first and the second normal stress differences as well as their functional departure from the constant multiple viscosity.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

Abstract We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when Re>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/H in the experiment is at least 10.

Galerkin/least-square finite-element methods for steady viscoelastic flows

The elastic viscous split stress formulation (EVSS) and the discrete EVSS formulation (DEVSS) are effective in stabilizing numerical simulations of viscoelastic flows and have been widely used. Following the concept of Galerkin least-square perturbations proposed by Hughes et al. [Comput. Meth. Appl. Mech. Eng. 73 (1989) 173–189] and Franca et al. [SIAM J. Numer. Anal. 28(6) (1991) 1680–1697; Comput . Meth. Appl. Mech. Eng. 99 (1992) 209–233; Ibid. 104 (1993) 31–48] we are able to give the DEVSS formulation a new explanation as a perturbation to the Galerkin method based on the strain-rate residual, and furthermore, introduce another stabilized formulation, here named as MIX1, based on the incompressibility residual of the finite element discretizations. The three formulations (EVSS, DEVSS, MIX1), combined with a h–p type finite element algorithm that employs the SUPG technique to solve the viscoelastic constitutive equations are then tested on three benchmark problems: the flow of the upper-convected Maxwell fluid between eccentric cylinders, the flow of the Maxwell fluid around a sphere in a tube and the flow of the Maxwell and Oldroyd-B fluids around a cylinder in a channel. The results are checked with previous published works; good agreement is observed. Our numerical experiments convincingly demonstrate that the MIX1 is an accurate algorithm and convergent in terms of the p-extension, it has the same level of stability and robustness as the DEVSS method and is superior to the EVSS method in some respects. More important is that with MIX1 method one needs not solve for the strain-rate tensor as in EVSS and DEVSS methods, therefore, the CPU time consumption in the MIX1 method especially when using a coupled iteration scheme can be radically reduced. The success of the MIX1 method presents a challenge to the widely accepted concept of making the momentum equation explicitly elliptic.

A streamline element scheme for solving viscoelastic flowproblems part II: Integral constitutive models

Abstract A finite element method for using integral constitutive models in viscoelastic flow simulation is presented. This method is based on a streamline element scheme (S.E.S) which was reported in Part I of this paper. The technique of particle tracking and strain history calculation is discussed in detail. In calculating the infinite memory integral, either Gaussian or Laguerre numerical quadrature formulae are used in our scheme. Some simple and complex flow problems involving the upper-convected Maxwell integral model are solved as test problems and afterwards the effort is concentrated on the K.B.K.Z. model. Much numerical work is devoted to simulating the axial extrusion swell experiments with LDPE sample A of the IUPAC Working Party on Structure and Properties of Commercial Polymers, using a specific version of the K.B.K.Z. model with multiple relaxation times in the memory function designed by Papanastasiou, Scriven and Macosko. It is shown by the numerical results that if the shear viscosity function of the model is kept unchanged, the calculated swelling ratio is very sensitive to the elongational behaviour of the model; it increases (or decreases) monotonically with the increase (or decrease) of the elongational viscosity in the corresponding stretching rate region. When both the shear and elongational response of this model agree well with experiments, the numerical predictions of the swelling ratio also agree well with experimental data at low and high apparent shear rates, while in the medium region the numerical calculation underestimates the swelling ratio. It is also seen that, using this model in our method, the extrusion calculation is surprisingly stable, even at very high Weissenberg numbers or very high extrusion swelling ratios, thus showing the very promising potential of integral models in the field of viscoelastic flow computation.

Plane Creeping Flows of Incompressible Second‐Order Fluids

It is shown that any plane creeping Newtonian velocity field is also a solution for second‐order fluids under identical velocity boundary conditions.

Numerical analysis of three-dimensional Bingham plastic flow

Abstract The present paper considers the problem of predicting motionless regions and true plug (constant velocity areas where the stress is below the yield stress) regimes for fully three-dimensional Bingham plastic flows. Numerical solutions are obtained using a finite-element biviscosity formulation. Comparisons are drawn to alternative numerical approximations and a review of common finite elements is made with a view to finding accurate, stable and economical schemes. A number of elements are compared and we conclude that some of the Fortin elements are most useful on the grounds of computational overhead and solution accuracy. These are used to investigate axial flows of a Bingham body in an annulus. Finally, we show that the numerical solutions of the axial flow of a Bingham plastic in a narrow eccentric annulus show good agreement with observations and theoretical predictions.

An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG

Abstract We report an adaptive viscoelastic stress splitting (AVSS) scheme, which was found to be extremely robust in the numerical simulation of viscoelastic flow involving steep stress boundary layers. The scheme is different from the elastic viscous split stress (EVSS) in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. Two decoupled versions of the scheme were implemented for the Upper Convected Maxwell (UCM) model: the streamline integration (AVSS/SI), and the streamline upwind Petrov-Galerkin (AVSS/SUPG) methods of integrating the stress. The implementations were benchmarked against the known analytic Poiseuille solution, and no upper limiting Weissenberg number was found (the computation was stopped at Weissenberg number of O (10 4 )). The flow past a sphere in a tube was solved next, giving convergent results up to a Weissenberg number of 3.2 with the AVSS/SI and 1.55 with the AVSS/SUPG (both were decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled streamline integration method was about 0.3). These results show that the decoupled AVSS is more stable than the corresponding EVSS, and the SI is more robust than SUPG in solving the constitutive equation of hyperbolic type. In addition, we found a very long stress wake behind the sphere, and a weak vortex in the rear stagnation region at a Weissenberg number above W i of about 1.6, which does not seem to increase in size or strength with increasing W i .