Brownian Configuration Fields Research Articles

Multiscale SPH simulations of viscoelastic injection molding processes based on bead-spring chain model

In this paper we present an extension of a multiscale smoothed particle hydrodynamics (SPH) method to transient viscoelastic injection molding processes. Specifically, we use the method presented by Xu and Yu, J. Non-Newtonian Fluid Mech. 229 (2016) pp. 27–42, in which the bead-spring chain model is employed to describe the viscoelastic behavior of the fluid without resort to a closed-form constitutive equation. Moreover, to resolve the heavy computation and the time-consuming problem, an efficient parallel algorithm, including parallelizations of both SPH particles and Brownian configuration fields, is developed. To validate the algorithm, we first simulate the viscoelastic Couette flow based on Hookean dumbbell, FENE dumbbell, and FENE chain models. The SPH results are compared with the analytical solutions or those obtained by other methods. The parallel performances of the algorithm are analyzed. Then, we extend the method to the challenging injection molding problem. A number of numerical examples including the injection molding of an F-shaped cavity in two dimensions and a door handle in two and three dimensions are investigated. Some molecular information such as the molecular stretch, the orientation angle, and the mean configuration thickness are displayed. It is found that the injection molding processes could be successfully investigated by SPH from a micro perspective. The proposed parallel algorithm is essential for the efficient simulation of injection molding processes. The maximum speedup can reach more than 130 for the case of the door handle in three dimensions.

Large-amplitude oscillatory shear flow simulation for a FENE fluid

In this work, the FENE dumbbell model under small- and large-amplitude oscillatory shear flows using a micro-macro approach is presented. This approach involves the evolution of an ensemble of Brownian Configuration Fields which describes the polymer dynamics of the microscopic scale and the momentum equation describes the macroscopic scale. The Lissajous curves for the shear stress and the first normal stress difference versus the instantaneous strain or strain rate for the elastic or viscous projection are shown. The influences of the solvent/polymer viscosity ratio, the maximum extension length, and the relation between strain rate and frequency are analyzed. An important finding is the self-intersection of the Lissajous curves, which forms secondary loops for short extension lengths and high Weissenberg/Deborah dimensionless numbers ratio.

Exploring the Molecular Distributions in Dilute Polymer Solutions Using a Multi-Scale Numerical Solver.

Simulating the rheological behaviors of polymer solutions is intrinsically a multi-scale problem. To study the macroscopic and microscopic characteristics in the fluid flow of dilute polymer solutions, we designed a multi-scale solver, which couples the Brownian Configuration Fields with the macroscopic hydrodynamic governing equations. Numerical simulation results using the multi-scale solver exhibited good accordance with the macroscopic only approach. Through a scalar field D we also quantitatively studied the flow behaviours in 2D planar channels, and analyzed the correlation between the molecular distribution and the macroscopic fluid flow in polymer solutions. Our results verified the correctness of the solver, which could provide valuable guidance for multi-scale simulations of complex fluids based on OpenFOAM.

The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors

The aim of this paper is to present a numerical study of oscillatory motion of Oldroyd-B fluid in a uniform magnetic field through a small circular pipe. First, we derive the orientation stress tensor by considering the Brownian force. Then, the orientation stress tensor is incorporated by taking Hookean dumbbells on Brownian configuration fields in the Oldroyd-B model. The Oldroyd-B model is then reformulated coupled with the momentum equation and the total stress tensor. Finally, we analyze the orientation stress tensor in the pipe by the numerical simulations of the model and showed that the effect of orientation stress tensor is considerable although the Brownian force is sufficiently small.

SPH simulations of 2D transient viscoelastic flows using Brownian configuration fields

Based on a micro–macro approach, the smoothed particle hydrodynamics (SPH) method is developed to simulate 2D transient viscoelastic fluid flows, in which the polymer dynamics is described by the evolution of a number of continuous Brownian configuration fields. This approach combines a macroscopic description of the kinematics with a coarse-grained microscopic description of the evolution of the extra-stress, and accordingly a closed-form constitutive equation is not necessary any longer. In order to validate our approach, both Poiseuille flow and Couette flow using Hookean dumbbell model are first studied. Results for the velocity, the shear stress and the first normal stress difference are in good agreement with the reference data from the literature. The simulations are also extended to the finitely extendable non-linear elastic (FENE) dumbbell model. In particular, the influence of dimensionless finite extensibility parameter b on the evolutions of the velocity and the polymer stress tensor is analyzed in detail. Results for the lid driven cavity flow of FENE dumbbell model are also shown. All numerical results demonstrate that the SPH method is a powerful and flexible tool for meshfree simulations of 2D transient viscoelastic fluid flows using Brownian configuration fields.

Blood flow of an Oldroyd-B fluid in a blood vessel incorporating a Brownian stress

The mechanical behavior of non-Newtonian fluids can be modeled by several constitutive differential equations. The Oldroyd model is viewed as one of the successful models for describing the response of a subclass of polymeric liquids, in particular the non-Newtonian behavior exhibited by these fluids. In this paper, we are concerned with the study of the unsteady flows of an incompressible viscoelastic fluid of an Oldroyd-B type in a blood vessel acting on a Brownian force. First we derive the orientation stress tensor considering Hookean dumbbells on Brownian configuration fields. Then we reformulate the three-dimensional Oldroyd-B model with the total stress tensor which consists of the isotropic pressure stress tensor, the shear stress tensor, and the orientation stress tensor. Finally we present the numerical simulations of the model and analyze the effect of the orientation stress tensor in the vessel.

ORIENTATIONAL STRESS TENSOR OF POLYMER SOLUTION WITH APPLICATIONS TO BLOOD FLOW

Blood circulating efficiently inside the veins and arteries, provides essential nutrients and oxygen to tissues and organs in the entire body. To highlight the fundamental properties of blood and gain insight into the regularizing effect of various formulations, we need to develop mathematical models. In order to do this, first we present the polymer dynamics in terms of an ensemble of Hookean dumbbells with Brownian configuration fields to derive the orientation stress tensor. Then, we describe the continuity and the momentum equations for time-dependent incompressible flow and the Oldroyd-B model. Finally, we present our numerical analysis of the model and give the effect of the orientation stress tensor in a blood vessel. The development of mathematical models and numerical procedures for the approximation of the blood specific flow equations have enabled us to understand the complex behaviors of blood flows inside the vessel.

Continuum and multi-scale simulation of mixed kinematics polymeric flows with stagnation points: Closure approximation and the high Weissenberg number problem

Abstract It is a well known fact that the upper We limit encountered in continuum level viscoelastic flow simulations with typical constitutive equations for dilute polymeric solutions that predict bounded extensional viscosities in geometries with internal stagnation points on solid surfaces is a strong function of the strain hardening nature of the fluid. Specifically, the upper We limit decreases as the level of strain hardening of the fluid is increased. To provide further insight into the high We limitations in continuum level computations of this class of flows, we have performed extensive continuum and multiscale flow simulations in two benchmark flow problems, namely sedimentation of sphere in a tube and flow past a cylinder in a channel, utilizing the FENE-P (continuum and Brownian Configuration Fields (BCF)) and Giesekus (continuum) constitutive equations as well as the FENE (BCF) dumbbell micromechanical model. Extremely large stress gradients in the axial normal stress along the plane of symmetry in the wake of the cylinder and sphere are observed in FENE-P and Giesekus predictions using both multiscale and continuum numerical techniques at We of O(1) where the numerical simulations begin to breakdown for significantly strain hardening fluids, i.e., b > 300, α < 0.005. The existence of very large localized polymeric stresses and stress gradients is shown to be the consequence of significant over prediction of macromolecular extension by the closed form constitutive equations in the extensionally dominated region of the flow. The inability of the aforementioned constitutive equations to accurately describe the flow microstructure coupling downstream of stagnation points in complex kinematics flows should motivate use of regularization techniques for the polymeric stress or utilization of more sophisticated constitutive equations in flows with strong straining components.

Micro–macro simulations of a shear-thinning viscoelastic kinetic model: applications to blood flow

This article is concerned with the numerical simulation of the flow of blood in small channels. Under certain flow conditions, blood is expected to behave like a viscoelastic fluid. Therefore a non-Newtonian, viscoelastic and shear-thinning finitely extensible non-linear elastic (FENE) dumbbell suspension model is used. The FENE dumbbell model is a coarse-grained molecular model that was developed for polymer solutions. The FENE model in this paper uses the variance reduction techniques of Brownian configuration fields and equilibrium control variates. The model has been implemented into a spectral element algorithm, which solves the stochastic differential equation together with the momentum and continuity equations. The temporal discretization of the convective term in the momentum equation is performed using a first-order time-splitting technique. An Euler–Maruyama predictor–corrector scheme has been used for the temporal discretization of the stochastic differential equation for the evolution of the configuration fields. Spatial discretization is performed using spectral element techniques, and the system of discretized equations is solved using a preconditioned conjugate gradient method. Results are presented for the start-up of Poiseuille flow in a planar channel, and for simulations of blood flowing through a small channel with a symmetric and a non-symmetric stenosis. The influence of the model parameters on the solution is described.

Viscoelastic flow past confined objects using a micro–macro approach

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.

Numerical stability of the method of Brownian configuration fields

We investigate numerical aspects of the Brownian configuration fields method, and in particular its numerical stability as the Weissenberg number increases. Our results show the method to be immune to the type of instability leading to numerical blowup in the simulation of macroscopic models. We discuss this finding in the light of the stability criterion proposed in Fattal et al. [R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg using the log-conformation representation, J. Non Newtonian Fluid Mech. 126 (2005) 23–37].

Spatially correlated noise and variance minimization in stochastic simulations

Brownian simulation methods have become a popular approach in computational rheology with the introduction of the CONNFFESSIT algorithm and the method of Brownian configuration fields in the 1990s. Jourdain et al. [B. Jourdain, C.L. Bris, T. Lelievre, On a variance reduction technique for micro–macro simulations of polymeric fluids, J. Non-Newton. Fluid Mech. 122 (2004) 91–106] pointed out that both methods can be viewed as variants that differ in the spatial correlation of the noise, which can be viewed as a computational parameter for statistical error minimization. We formulate an optimization problem of variance minimization with respect to the choice of noise correlation. Our analysis takes place in an infinite-dimensional function space. We solve the optimization problem analytically for the shear flow of a Hookean dumbbell model at steady state. Interestingly, we find that spatially uncorrelated noise, i.e., CONNFFESSIT minimizes the statistical error, although the precise meaning of this statement can only be interpreted as a limit of finite-dimensional approximations.

On the reduction of stochastic kinetic theory models of complex fluids

Kinetic theory models involving the Fokker–Planck equation are usually solved in the framework of stochastic approaches, which allows us to circumvent the difficulties related to the multidimensional character of that equation. In fact, the Fokker–Planck equation governs the evolution of the distribution function that defines the molecular configuration at each point of the physical space and at each time. As the molecular conformation is usually defined by several coordinates, the resulting distribution function will depend on the physical and configuration coordinates and the time. Although different numerical strategies have recently been proposed for solving that equation with efficiency and accuracy (Ammar et al 2006 J. Non-Newtonian Fluid Mech. 134 136–47, Ammar et al 2006 J. Non-Newtonian Fluid Mech. 139 153–76) the stochastic approach is today the most common for solving general kinetic theory models. This paper presents some preliminary results that provide evidence for the potential applicability of model reduction techniques based on the Karhunen–Loève decomposition or on separated representations for reducing the computational efforts related to the solution of such models in the Brownian configuration fields framework.

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, W i = λ γ ˙ (where λ is the characteristic relaxation time of polymer, and γ ˙ is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.

A spectral element approach to the simulation of viscoelastic flows using Brownian configuration fields

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.

Generalized Brownian configuration fields for Fokker–Planck equations including center-of-mass diffusion

An alternative argument is presented to show a connection between Fokker–Planck equations in non-homogeneous flows and the Brownian configuration fields (BCFs) pioneered by Hulsen et al. [M.A. Hulsen, B.H.A.A. van den Brule, A.P.G. van Heel, Simulation of viscoelastic flows using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1–2) (1997) 79–101]. The method is used to show how these BCFs can be generalized to include center-of-mass diffusion of the polymer chains.

Convergence Analysis of BCF Method for Hookean Dumbbell Model with Finite Difference Scheme

A convergence analysis of the Brownian configuration fields (BCF) method [M. A. Hulsen, A. P. G. van Heel, and B. H. A. A. van den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79-101] for the Hookean dumbbell model with finite difference scheme in dimension 2 or 3 is given in this paper under the assumption that the continuous solution is smooth enough. An explicit solution of the Hookean dumbbell model is obtained via deformation tensor. A large deviation-type estimate for the error of polymeric stress $\mathbb{E}(\bfmath{QQ})$ is given, which is a key step in the proof. It is shown that if the number of configuration fields N, the space stepsize h, and the time stepsize $\delta t$ are chosen appropriately, the convergence of second order in space and first order in time may be proved after excluding a set of small probability. Simultaneous discretization of Monte Carlo and space and the inverse inequality trick are essential for the proof.

Size reduction methods for the implicit time-dependent simulation of micro–macro viscoelastic flow problems

Traditionally, micro–macro simulations have been performed using simple explicit time-marching algorithms, which lack the desirable stability of implicit methods. In this study, a fully implicit time integration scheme is presented and implemented for the first time for the solution of time-dependent complex flows using the Brownian Configuration Fields approach. Special techniques need to be applied to deal with the very large size of the resulting linear systems. A novel size-reduction scheme is used, allowing an independent treatment for each molecular field and suited to parallel hardware architecture. To illustrate the method, a selected number of applications using linear spring chains are presented and the results are compared with their corresponding closed form constitutive equation. The excellent agreement between the results demonstrates the feasibility of the proposed approach.

Well-Posedness for the Dumbbell Model of Polymeric Fluids

The dumbbell model is a coupled hydrodynamic-kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations. We prove well-posedness of this model by deriving directly a priori estimates on the stochastic model. Our results can be used to analyze stochastic simulation methods such as the ones that are based on Brownian configuration fields.

A method for multiscale simulation of flowing complex fluids

The Brownian configuration fields (BCF) method provides an attractive approach to simultaneous solution of flow and microstructure problems in complex fluids. At the continuum level, it is implemented in an Eulerian reference frame (i.e. on a stationary computational mesh), which makes it amenable to standard computational approaches for convection problems, and at the microstructural level it simply involves adding a convective term to the evolution of individual Brownian dynamics trajectories. We present here a variation on BCF that may hold advantages over the original formulation. It uses a hybrid representation of the configurational distribution function (CDF), splitting a time step into two parts: (1) a standard Brownian dynamics step (here the CDF is implicitly represented as a superposition of delta-functions) and (2) a convective step, where the distribution is projected onto an orthogonal polynomial basis and convection problems are solved for the generalized Fourier coefficients, then “lifted” back to the delta-function basis. This approach, which we call operator-splitting coefficient-matching (OSCM) allows the solution of fewer convection problems per time step than BCF and guarantees spatial smoothness of the convected properties—in some sense it is a filtered version of BCF. We present comparisons of accuracy and computational efficiency of BCF and OSCM for a simple model problem involving bead-spring dumbbells and chains. The efficiency of the two methods is comparable, and in situations where the finite extensibility becomes important, BCF can lose accuracy, due to a loss of correlation between trajectories at different point in the flow domain, where OSCM does not.