A finite difference technique for solving a time strain separable K-BKZ constitutive equation for two-dimensional moving free surface flows

This work is concerned with the numerical solution of the K-BKZ integral constitutive equation for two-dimensional time-dependent free surface flows. The numerical method proposed herein is a finite difference technique for simulating flows possessing moving surfaces that can interact with solid walls. The main characteristics of the methodology employed are: the momentum and mass conservation equations are solved by an implicit method; the pressure boundary condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by a second-order Runge–Kutta method. This new technique is verified by solving shear and uniaxial elongational flows. Furthermore, an analytic solution for fully developed channel flow is obtained that is employed in the verification and assessment of convergence with mesh refinement of the numerical solution. For free surface flows, the assessment of convergence with mesh refinement relies on a jet impinging on a rigid surface and a comparison of the simulation of a extrudate swell problem studied by Mitsoulis (2010) [44] was performed. Finally, the new code is used to investigate in detail the jet buckling phenomenon of K-BKZ fluids.Indisponível

The extrudate swell singularity of Phan-Thien-Tanner and Giesekus fluids

The stress singularity for Phan-Thien-Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like rλ0, where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like r-(1-λ0) and dominates the polymer stresses, which are like r-4(1-λ0)/(5+λ0) for PTT and r-(1-λ0)(3-λ0)/4 for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.</p

The extrudate swell singularity of Phan-Thien-Tanner and Giesekus fluids

The stress singularity for Phan-Thien-Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like rλ0, where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like r-(1-λ0) and dominates the polymer stresses, which are like r-4(1-λ0)/(5+λ0) for PTT and r-(1-λ0)(3-λ0)/4 for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.</p

Numerical studies of Newtonian and viscoelastic fluids

The direct numerical simulation (DNS) approach is used to understand the flow behavior of Newtonian and viscoelastic fluids in porous materials, four-to-one contraction and the flow of a Newtonian fluid past an airfoil. In simulations the viscoelastic fluid is modeled by the finitely extensible nonlinear elastic (FENE) dumbbell and Oldroyd-B models. The finite element method (FEM) is used to discretize the flow domain. The DNS results show that the permeability of a periodic porous medium depends on the wavelength used for arranging particles in the direction of flow. Specifically, it is shown that for a given particle size and porosity the permeability varies when the distance between particles in the flow direction is changed. The permeability is locally minimum for kD [approximately equal to] 7.7 and locally maximum for kD [approximately equal to] 5.0; where k is the wave number and D the diameter. A similar behavior holds for a viscoelastic fluid, except that the variation of permeability with kD is larger than for the Newtonian case. For flow in the four-to-one contraction, it is found that the stress near the 3π/2 comer is singular and that the singularity is stronger than for a Newtonian liquid. In the region away from the walls, the stress varies as r -0.47 and near the walls it varies as r -0.61. Since the singularity is integrable, the flow away from the comer is not effected [sic] when the flow around the comer is resolved by using a radial mesh with sufficient resolution in the tangential direction at the comer. The DNS approach is also used to demonstrate that the boundary layer separation on the upper surface of the airfoil can be suppressed by placing injection and suction regions on the upper surface. The simulations are performed for Re &leq; [less than or equal too] 500 and angle of attack up to 40°. Analysis of numerical results shows that the pressure contribution to drag decreases when the boundary layer separation is avoided. The viscous contribution to drag, however, increases and thus there is only a negligible decrease in the total drag for Re &leq; [less than or equal too] 500. Another beneficial effect of suction and injection is that the pressure contribution to the lift increases and the stall is avoided

Numerical shape optimisation in blow moulding

Blow moulding is a popular manufacturing process for the production of plastic and glass containers, e.g. bottles, jars, jerrycans. In a blow moulding process a so-called preform of molten material is brought into a mould and subsequently inflated with air as to take the mould shape. Blow moulding processes typically vary in the way the preform is produced and brought into the mould. The stretch blow moulding process is a variation of the blow moulding process in which the preform is simultaneously inflated with air and stretched with a stretch rod. A two-dimensional axial-symmetrical blow moulding simulation model is developed. The numerical simulation model is based on Finite Element Methods and uses Level Set Methods to track the moving interfaces between the melt and air. Level Set Methods mark the location of the interfaces implicitly by a so-called level set function and therefore do not require re-meshing of the finite element mesh. The e??ciency of the simulation model is illustrated by applying it to the stretch blow moulding of a plastic water bottle and the blow moulding of a glass beer bottle. The model is validated by means of volume conservation and comparison with data provided by industry. Two mathematical problems are considered in blow moulding. The forward problem is to find the final container that is blow moulded from a given preform under certain operating conditions. In practice often a container with a certain wall thickness distribution is desired. Then the corresponding initial operating conditions, such as the shape of the preform and the initial temperature distribution, are sought in order to produce a container with exactly this thickness distribution. In this case the inverse problem is considered, to find the shape of the preform, given a designed container, such that the container can be blow moulded from the preform. The solvability and sensitivity of the inverse problem are analysed. It is shown that under some circumstances the melt-air interfaces can reach a force equilibrium state during blow moulding. Consequently, constraints on the mould surface and process time are necessary so that the inverse problem is solvable and not excessively sensitive to perturbations in the shape. The sensitivity of the inverse problem with respect to perturbations in the shape can be estimated by means of an approximation of the melt flow. Numerical shape optimisation is used to find a solution of the inverse problem. The optimisation method describes the unknown preform surface by a parametric curve, e.g. spline, Bezi´er curve, and computes the optimal positions of the control points of the curve as to minimise the objective function. The objective function represents the distance between the inner surface of the computed container, which is the solution of the forward problem for the approximate preform, and the inner surface of the designed container. Gradient-based optimisation algorithms are discussed to find the optimal positions of the control points. In gradient-based optimisation information about the gradient of the objective function with respect to changes in the parameters, i.e. the positions of the control points, is used to find the optimum. However, computing the gradient is extremely computationally expensive and can form the computational overhead. Therefore, finite difference approximations of the Jacobian are combined with secant updates. An error analysis is performed to choose an optimal error tolerance for the optimisation algorithm. The optimisation methods are applied to glass blow moulding and results are compared with each other. An initial guess for the iterative optimisation algorithms is constructed by an analytical approximation of the optimum. The approximation is derived by omitting the mass flow in polar direction in spherical coordinates, so that the inverse problem can be solved analytically

Constitutive modelling of branched polymer melts in non-linear response

This thesis is concerned with modelling long chain branched polymer melts using the McLeish and Larson Pompom constitutive equations. Usually the non-linear terms in this model are fitted to uniaxial extensional data due its sensitivity to levels of branching, but in this thesis I will study a number of other non-linear flows using this model. For each flow the results are compared to experiments on a set of polyethylene melts. The first flow types I examine are simple shear flows. In a shear step-strain flow the stress relaxation of branched polymers is observed to be time-strain separable, whereby the relaxation modulus can be separated into the product of separate functions of time and strain. I show that although the Pompom model is not time-strain separable in general, there exist subsets of parameter values for which time-strain separability is valid. For these sets a branched damping function is derived which is analogous to the Doi-Edwards damping function for linear polymer melts. The other simple shear flow examined is oscillatory shear. Commonly, oscillatory shear is probed at low strain amplitudes over a range of frequencies to measure the usual dynamic moduli of linear viscoelasticity. In this work the effect of strain amplitude is explored up to absolute strains of order unity. The non-linear stress response is analysed from the higher harmonics in the Fourier series. In particular it is shown that the third Fourier components are dependent on the Pompom non-linear stretch relaxation time and a low-strain asymptote is obtained. Subsequently, this thesis focuses on the stagnation point flow generated in a cross-slot geometry. The stress calculated from the Pompom constitutive model is compared to experimental flow induced birefringence images. It is shown for linear and lightly branched materials that the Pompom model predicts both the form of the birefringence pattern and stress values obtained from the stress-optical law. However, for more highly branched polymers the Pompom model fails to predict the change to birefringence patterns. Subsequent analysis shows that there could exist a transient overshoot in extension which the Pompom model cannot capture as it stands. In the final part of my thesis I suggest an empirical alteration to the Pompom constitutive model to capture this transient extensional overshoot, which is able to resolve the differences between experimental flow induced birefringence images and theoretical simulations

Numerical study of filament-stretching and step-strain in viscoelastic fluid flows.

This thesis is concerned with the numerical prediction of two-dimensional viscoelastic filaments under stretching and step-strain within cylindrical-like domains. A hybrid finite element/finite volume (fe/fv) scheme has been implemented in this study to solve the governing equations (mass and momentum conservation and constitutive model). A time-stepping procedure is utilised in the fe/fv algorithm. A number of rheological models have been employed to stimulate the desired rheological response. Amongst these is the Oldroyd-B model. This is considered as a strong strain-hardening model being widely used due to its sound physical background and its ability to reproduce qualitative response of polymer melts in rheometrical flows. The linear version of Phan-Thien/Tanner (LPTT) and Giesekus models are also studied to compare simulation results for both dilute and concentrated polymeric systems against the Oldroyd-B model. For fluids with higher degree of strain-hardening, larger stress values cause a reduction in stretching period. In addition, Boger-like response has been represented under increasing levels of solvent within the system. Filament-stretching has been studied under two modes of stretching, exponential and linear for multi-mode and single-mode representations, that has included a numerical study on mesh refinement and algorithms developed for free-surface movement. Bead-like structure formation has been studied for a variety of surface tension coefficients in the absence/presence of body forces. ALE methods and free-surface techniques have been analysed for Volume-of-Fluid (VOF) mesh and Compressed-Mesh (CM) procedures. VOF mesh procedures are outperformed by their CM counterparts. For free-surface curvature to be determined precisely, a particle-tracking approach has been found to be preferable to a kinematic condition for surface-level. Variation of anisotropy levels and xi-parameter settings has been studied for the Gieskus and LPTT models, respectively. A further chapter is included where the recently addressed subject of step-strain is considered, to simulate sudden cessation of stretching across the three viscoelastic models. Sudden decline and sharp rise in axial stress have been observed and interpreted alongside filament radial evolution in the context of step-strain. The effect of inertia has been neglected but the effect of capillary and body forces has been brought into consideration. Larger stress values are observed for fluids with a higher degree of strain-hardening, and consequently, cause an increase in the step- strain period. Similar dynamic trends are followed for LPTT fluids with parameter settings of xi={lcub}0.0,0.13{rcub} under the context of step-strain. Here, rheological differences would emerge in shear. A paper which has been recently submitted for publication is included in the appendix. There, different aspects of gradual plate halt are discussed