Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme

This is the second part of our error analysis of the stabilized Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements, which leads to an efficient computation with a small number of degrees of freedom especially in three space dimensions. In this paper, Part II, we apply a semi-implicit time discretization which yields the linear scheme. We concentrate on the diffusive viscoelastic model, i.e. in the constitutive equation for time evolution of the conformation tensor a diffusive effect is included. Under mild stability conditions we obtain error estimates with the optimal convergence order for the velocity, pressure and conformation tensor in two and three space dimensions. The theoretical convergence orders are confirmed by numerical experiments.Comment: See arXiv:1603.01339 for Part I: a nonlinear schem

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De≈1.3, for an extensibility parameter of the model of L2 = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L2 = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De≈4.0–4.5

Modelling the nonlinear dynamics of polymer solutions in complex flows

The flow of polymer solutions in the high Elasticity number, El, regime in complex geometries may lead to strong viscoelastic behaviour and eventually become unstable as the Weissenberg number, Wi, is increased beyond a critical level. So far, the success of numerical simulations in predicting the highly non-linear behaviour of polymer solutions in complex flows has been limited. In this thesis, selected constitutive models are evaluated under the high El flow regime in the cross-slot and contraction benchmark flows using a numerical technique based on the finite volume method. The numerical technique is implemented within the OpenFOAM framework and thoroughly validated in the benchmark flow. A modification to the FENE dumbbell model based on the non-affine deformation of polymer solutions is proposed, which enabled the prediction of some non-linear material functions and also enhanced numerical stability, allowing a higher Wi to be attained. Asymmetric flow instability in the cross-slot flow has been studied. Time-dependent stability diagrams were constructed based on Wi and the strain, ε, both of which govern the stretching of a polymer chain. In the contraction flow, elastic instability is simulated for the first time in this geometry. Substantial time-dependent asymmetric flow patterns were predicted as seen in experiments. The effect of the contraction ratio is investigated through a stability diagram. Three-dimensional finite element simulations were also carried out to study the effect of the aspect ratio in the contraction flow of a Phan-Thien-Tanner fluid. The simulations suggest that a lip vortex mechanism is a signature for the onset of strong viscoelastic behaviour.EThOS - Electronic Theses Online ServiceOverseas Research Scholarship AwardUniversity of ManchesterGBUnited Kingdo

Computational and rheological studies for coating flows.

Coating flows can be defined as a laminar free surface flows, whereby a liquid layer is applied onto a solid substrate. A typical industrial application consists of co-rotating cylindrical rollers, which are used to apply a liquid coating (paint) onto a moving substrate, and depending on the direction of the rollers, can be configured in either forward or reverse mode. These types of coating solution flows are industrially important applications, and convey viscoelastic aspects due to their polymeric content and unsteady polymeric behaviour. The process often possesses localized regions of high shear and extension rates (narrow nip and wetting-line zones), which may cause instabilities on the coated substrate (ribbing, leveling, striping). These non-Newtonian and viscoelastic studies for industrial reverse roll coating focus on the use of computational techniques to model these types of coating flows, alongside the analysis of the fluid flow behaviour and under varied rheological properties. Two flow problem configurations have been considered, a model benchmark problem of mixed combined-separating flow, and the industrial application of reverse roll coating flow. Predictions and corresponding solutions are reported for viscous, inelastic and complex viscoelastic fluid properties. The numerical formulation adopts a Taylor-Galerkin pressure-correction (TGPC) scheme, using a finite element method for viscous, inelastic flows and a hybrid finite element/finite volume method for their viscoelastic counterparts. The research plan is centered around computational fluid dynamics and rheological studies, with the main target focused on industrial roll-coating operations. From simple theory, Newtonian and non-Newtonian coating flows possess specific, yet disparate characteristics. This may lead to distinct and significant differences in their detailed flow behaviour, and in the stressing levels generated, dependent upon the nature of the flow configuration. The study is segmented into several stages: initially, solution was sought for a benchmark flow problem, where a semi- implicit time stepping finite element procedure was employed to simulate a mixed combined- separating flow. Here, both viscous and viscoplastic material approximations have been introduced. Secondly, the industrial application of reverse roll coating flow was addressed for viscous inelastic coating fluids. This incorporated scenarios of inclusion and not of a dynamic wetting line and consideration of the effects of a rubber elastomer-cover upon the applicator roll. Thirdly, viscoelastic paint coatings were addressed for the industrial reverse roll coating flow. Here, a hybrid finite element/finite volume sub-cell method was utilized, and with inclusion of a dynamic wetting line. Of the various viscoelastic material models available, use has been made of the Phan-Thien Taimer (PTT) network class of models, in both linear and exponential variety, and of the FENE class of models, with FENE-CR and FENE-P versions. This has offered a richness in capacity over variation of rheological properties. The choice of computational methods has been justified and the TGPC algorithm was deemed suitable for problem solution. The methodology tested on combined-separating flow provided high-quality numerical results, which compare favorably against experiments, literature and theory. When applied to the reverse roll coating problem, the TGPC algorithm has been coupled to a time-dependent free-surface update procedure, to determine the dynamic movement of the meniscus and the wetting line. Around the nip-region, the flow problem manifests strong flow features, which have been investigated for a range of rheological properties of varying shear and extensional response. The direct impact these have on localized peak nip-pressures and distributional lift levels has been observed, where several relief mechanisms have been successfully identified (important to optimize process control). The influence of solvent fraction, extensional viscosity and increasing elasticity, up to critical stress states have been analysed in considerable detail. In summary, the success of this work indicates optimal flow process settings and preferential Theological coating properties to employ, with respect to this industrial coating process. As such, it lays the foundation and guide towards achieving a stable and consistent coating application - specifically, as high-speed high-gain production is of current demanded

Simulation of Time-Dependent Viscoelastic Fluid Flows by Spectral Elements

The research work reported in this dissertation is aimed to develop efficient and stable numerical schemes in order to obtain accurate numerical solution for viscoelastic fluid flows within the spectral element context. The present research consists in the transformation of a large class of differential constitutive models into an equation where the main variable is the logarithm of the conformation tensor or a quantity related to it in a simple way. Particular cases cover the Oldroyd-B fluid and the FENE-P model. Applying matrix logarithm formulation in the framework of the spectral element method is a new type of approach that according to our knowledge no one has implemented before. The reformulation of the classical constitutive equation using a new variable namely the logarithmic formulation, enforces the eigenvalues of the conformation tensor to remain positive for all steps of the simulation. However, satisfying the symmetric positive definiteness of the conformation tensor during the simulation is the necessary condition for stability; but definitely, it is not the sufficient condition to reach meaningful results. The main effort of this research is devoted to introduce a new algorithm in order to overcome the drawback of direct reformulating the classical constitutive equation to the logarithmic one. To evaluate the capability of the extended matrix logarithm formulation, comprehensive studies have been done based on the linear stability analysis to show the influence of this method on the resulting eigenvalue spectra and explain its success to tackle high Weissenberg numbers. With this new method one can treat high Weissenberg number flows at values of practical interest. One of the worst obstacles for numerical simulation of viscoelastic fluids is the presence of spurious modes during the simulation. At high Weissenberg number, many schemes suffer from instabilities and numerical convergence may not be attainable. This is often attributed to the presence of solution singularities due to the geometry, the dominant non-linear terms in the constitutive equations, or the change of type of the underlying mixed-form differential system. Refining the mesh proved to be not very helpful. In this study, to understand more deeply the mechanism of instability generation a comprehensive study about the growth of spurious modes with time evolution, mesh refinement, boundary conditions and Weissenberg number or any other affected parameters has been performed. Then to get rid of these spurious modes the filter based stabilization of spectral element methods proposed by Boyd was applied with success

Brownian dynamics simulations of fine-scale molecular models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005.Includes bibliographical references (leaves 105-111).One of the biggest challenges in non-Newtonian fluid mechanics is calculating the polymer contribution to the stress tensor, which is needed to calculate velocity and pressure fields as well as other quantities of interest. In the case of a Newtonian fluid, the stress tensor is linearly proportional to the velocity gradient and is given by the Newton's law of viscosity, but no such unique constitutive equation exists for non-Newtonian fluids. In order to predict accurately a polymer's rheological properties, it is important to have a good understanding of the molecular configurations in various flow situations. To obtain this information about molecular configurations and orientations, a micromechanical representation of a polymer molecule must be proposed. A micromechanical model may be fine scale, such as the Kramers chain model, which accurately predicts a real polymer's heological properties, but at the same time possesses too many degrees of freedom to be used in complex flow simulations, or it may be a coarse-grained model, such as the Hookean or the FENE dumbbell models, which can be used in complex flow analysis, but have too few degrees of freedom to adequately describe the rheology. The Adaptive Length Scale (ALS) model proposed by Ghosh et al. is only marginally more complicated than the FENE dumbbell model, yet it is able to capture the rapid stress growth in the start-up of uniaxial elongational flow, which is not predicted correctly by the simple dumbbell models. The ALS model is optimized in order to have its simulation time as close as possible to that of the FENE dumbbell.(cont.) Subsequently, the ALS model is simulated in the start-up of the uniaxial elongational and shear flows as well as in steady extensional and shear flows, and the results are compared to those obtained with other competing rheological models such as the Kramers chain, FENE chain, and FENE dumbbell. While a 5-spring FENE chain predicts results that are in very good agreement with the Kramers chain, the required simulation time clearly makes it impossible to use this model in complex flow simulations. The ALS model agrees better with the Kramers chain than does the FENE dumbbell in the start-up of shear and elongational flows. However, the ALS model takes too long to achieve steady state, which is something that needs to be explored further before the model is used in complex flow calculations. Understanding of this phenomena may explain why the stress-birefringence hysteresis loop predicted by the ALS model is unexpectedly small. In general, if polymer stress is to be calculated using Brownian dynamics simulations, a large number of stochastic trajectories must be simulated in order to predict accurately the macroscopic quantities of interest, which makes the problem computationally expensive. However, recent technological advances as well as a new simulation algorithm called Brownian configuration fields make such problems much more tractable. The operation count in order to assess the feasibility of using the ALS model in complex flow situations yields very promising results if parallel computing is used to calculate polymer contribution to stress. In an attempt to capture polydispersity of real polymer solutions, the use of multi-mode models is explored.(cont.) The model is fit to the linear viscoelastic spectrum to obtain relaxation times and individual modes' contributions to polymer viscosity. Then, data-fitting to the dimensionless extensional viscosity in the startup of the uniaxial elongational flow is performed for the ALS and the FENE dumbbell models to obtain the molecule's contour length, bmax. It is found that the results from the single-mode and the four-mode ALS models agree much better with the experimental data than do the corresponding single-mode and four-mode FENE dumbbell models. However, all four models resulted in a poor fit to the steady shear data, which may be explained by the fact that the zero-shear-rate viscosity obtained via a fit to the dynamic data by Rothstein and McKinley and used in present simulations, tends to be somewhat lower than the steady-state shear viscosity at very low shear rates, which may have caused a mismatch between the value of ... used in the simulation and the true ... of the polymer solution. As a motivation for using the ALS model in complex flow calculations, the results by Phillips, who simulated the closed-form version of the model in the benchmark 4:1:4 contraction- expansion problem are presented and compared to the experimental results by Rothstein and McKinley [49]. While the experimental observations show that there exists a large extra pres- sure drop, which increases monotonically with increasing De above the value observed for a Newtonian fluid subjected to the same flow conditions, the simulation results with a closed-form version of the FENE dumbbell model, called FENE-CR, exhibit the opposite trend.(cont.) The ALS-C model, on the other hand, is able to predict the trend correctly. The use of the ALS-C model in another benchmark problem, namely the flow around an array of cylinders confined between two parallel plates, also shows very promising results, which are in much better agreement with experimental data by Liu as compared to the Oldroyd-B model. The simulation results for the ALS-C and the Oldroyd-B models are due to Joo, et al. [28] and Smith et al. [50], respectively. Overall, it is concluded that the ALS model is superior to the commonly used FENE dumb- bell model, although more work is needed to understand why it takes significantly longer than the FENE dumbbell to achieve steady state in uniaxial elongational flows, and why the stress birefringence hysteresis loop predicted by the ALS model is much smaller than that of the other rheological models.by Irina Burmenko.S.M

Viscoelasticity at microscopic and macroscopic scales: characterization and prediction

In this dissertation, we build mathematical tools for applications to the transport properties of human lung mucus. The first subject is the microscopic diffusive transport of micron-scale particles in viscoelastic fluid. Inspired by the technique of passive microrheology, we model the motion of Brownian beads in general viscoelastic fluids by the generalized Langevin equation (GLE) with a memory kernel (the diffusive transport modulus). The GLE is a stochastic differential equation, which admits a discrete formulation as an autoregressive (AR) process. We further use exponential series for the memory kernel in the GLE, in which case the GLE has an explicit formulation as a vector Ornstein-Uhlenbeck process. In this framework, we can develop fast and accurate direct algorithm for pathogen transport in viscoelastic fluids, and the Kalman filter and maximum likelihood method give a new method for inversion of the memory kernel from experimental position time series. The framework is illustrated with multimode Rouse and Zimm chain models. In the second topic, we revisit the classical oscillatory shear wave model of Ferry et al., and extend the theory for active microrheology of small volume samples of viscoelastic fluids. In Ferry's original setup, oscillatory motion of the bottom plate generates uni-directional shear waves propagating in the viscoelastic fluid. Our colleague David Hill built a device to handle small volume viscoelastic samples. We extend the Ferry analysis to include finite depth and wave reflection off the top plate. We further consider nonlinear viscoelastic constitutive laws. The last problem considered is the numerical simulation of viscoelastic fluid flow, which will eventually be used to predict bulk transport of mucus layers. We start with the analysis of the system of model equations and demonstrate the difficulty of a robust numerical scheme. We develop an extension of projection method, which involves a new treatment of stress evolution based on stress splitting in the numerical scheme and show the advantage over previous work

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

Further developments on theoretical and computational rheology

Tese financiada pela FCT - Fundação para a Ciência e a Tecnologia, Ciência.Inovação2010, POPH, União Europeia FEDERTese de doutoramento. Engenharia Química e Biológica. Faculdade de Engenharia. Universidade do Porto. 201