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].     

Numerical simulations of non-isothermal three-dimensional flows in an extruder by a finite-volume method

This study considers numerical applications of a finite-volume method to steady non-isothermal flows in geometries close to a single-screw extruder. Two geometrical configurations of the channel, with gap and zero gap, are investigated. The simulations concern incompressible fluids obeying different constitutive equations: Newtonian, generalized Newtonian with shear-thinning properties (Carreau–Yasuda law), and two viscoelastic differential models, the upper convected maxwell (UCM) and the Phan–Thien/Tanner (PTT). The temperature dependence is described by a Williams–Landel–Ferry (WLF) equation. For discretizing the equations and unknowns, we use a staggered grid with a QUICK scheme for the convective-type terms and solve the set of governing equations by a decoupled algorithm, stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) technique, in the viscoelastic case for the UCM model. The numerical results enable us to state the influence of temperature and rheological properties on the flow characteristics in the geometries investigated and underline the complex behaviour of the materials in such configurations.     

Numerical simulations of dispersive mixing of viscoelastic suspensions in a four-roll mill

The dispersive mixing of particles suspended in Newtonian and viscoelastic fluids in a four-roll mill is studied by direct numerical simulations. A fictitious domain method is used to handle the particle motion. To quantify the mixing, a proper mixing distribution function is defined. The combined effect of fluid rheology and particle-particle/particle-wall hydrodynamic interactions is addressed. At variance with the Newtonian case where the particle distribution remains uniform, the viscoelasticity-induced migration leads to a significant segregation process. The effect of the Deborah number (the product of the fluid relaxation time and the roll angular velocity), shear-thinning, particle concentration, and size on the microstructure evolution is thoroughly investigated.     

Numerical study of the rheology of rigid fillers suspended in long-chain branched polymer under planar extensional flow

We report a detailed numerical study of the rheology of two-dimensional rigid fillers suspended in branched polymer melt under planar extensional flow. The polymer melt is modelled using the pom-pom constitutive equation. The numerical method uses a finite element solution of the flow in a unit cell within the self-replicating lattice for planar extensional flow identified by Kraynik and Reinelt [Int. J. Multiphase flow 18 (1992) 1045]. This is implemented using a quotient space representation that maps all points in space onto points within the unit cell. We show that the Kraynik and Reinelt cell allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The influence of both the pom-pom parameters and the particle area fraction on the rheology of the suspension are investigated. We find a reduction in the degree of extension-rate thickening with the increase of both particles concentration and Weissenberg numbers in agreement with published experimental and numerical findings on other viscoelastic models. This reduction is found to be due to flow disturbance created by the particles which disrupts the alignment of backbone tube segments with the extensional axis.     

On the skin friction coefficient in viscoelastic wall-bounded flows

Analysis of the skin friction coefficient for wall bounded viscoelastic flows is performed by utilizing available direct numerical simulation (DNS) results for viscoelastic turbulent channel flow. The Oldroyd-B, FENE-P and Giesekus constitutive models are used. First, we analyze the friction coefficient in viscous, viscoelastic and inertial stress contributions, as these arise from suitable momentum balances, for the flow in channels and pipes. Following Fukagata et al. (Phys. Fluids, 14, p. L73, 2002) and Yu et al. (Int. J. Heat. Fluid Flow, 25, p. 961, 2004) these three contributions are evaluated averaging available numerical results, and presented for selected values of flow and rheological parameters. Second, based on DNS results, we develop a universal function for the relative drag reduction as a function of the friction Weissenberg number. This leads to a closed-form approximate expression for the inverse of the square root of the skin friction coefficient for viscoelastic turbulent pipe flow as a function of the friction Reynolds number involving two primary material parameters, and a secondary one which also depends on the flow. The primary parameters are the zero shear-rate elasticity number, El₀, and the limiting value for the drag reduction at high Weissenberg number, LDR, while the secondary one is the relative wall viscosity, μ_w. The predictions reproduce both types A and B of drag reduction, as first introduced by Virk (Nature, 253, p. 109, 1975), corresponding to partially and fully extended polymer molecules, respectively. Comparison of the results for the skin friction coefficient against experimental data shows good agreement for low and moderate drag reduction which is the region covered by the simulations.     

Development of a Low-Reynolds-number k-ω Model for FENE-P Fluids

A low-Reynolds-number k-ω model for Newtonian fluids has been developed to predict drag reduction of viscoelastic fluids described by the FENE-P model. The model is an extension to viscoelastic fluids of the model for Newtonian fluids developed by Bredberg et al. (Int J Heat Fluid Flow 23:731–743, 2002). The performance of the model was assessed using results from direct numerical simulations for fully developed turbulent channel flow of FENE-P fluids. It should only be used for drag reductions of up to 50 % (low and intermediate drag reductions), because of the limiting assumption of turbulence isotropy leading to an under-prediction of k, but compares favourably with results from k-ε models in the literature based on turbulence isotropy.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimization of the vane geometry

The use of nonstandard geometries like the vane is essential to measure the rheological characteristics of complex fluids such as non-Newtonian fluids or particle dispersions. For this geometry which is of Couette type, there is no analytical simple model defining the relation between the shear stress and the torque or relating the angular velocity to the shear rate. This study consists on calibrating a nonstandard vane geometry using a finite volume method with the Ansys Fluent software. The influence of geometrical parameters and rheological characteristics of the complex fluids are considered. First, the Newtonian fluid flow in a rotative vane geometry was simulated and a parametric model is derived therefrom. The results show an excellent agreement between the calculated torque and the measured one. They provide the possibility to define equivalent dimensions by reference to a standard geometry with concentric cylinders where the relationships between shear stress (resp. shear rate) and the torque (resp. the angular rotation) are classical. Non-Newtonian fluid flows obeying a power law rheology with different indices were then simulated. The results of these numerical simulations are in very good agreement with the preceding Newtonian-based model in some ranges of indices. The absolute difference still under 5 % provided the index is below 0.45. Finally, this study provides a calibration protocol in order to use nonstandard vane geometries with various heights, gaps, and distance to the cup bottom for measuring the rheology of complex fluids like shear thinning fluids and concentrated suspensions.     

黏弹流体流动的数值模拟研究进展

综述了黏弹流体流动数值模拟的研究进展,突出介绍近十年来有限元法在黏弹流体流动数值模拟研究中取得的成果,通过动量方程的适当变形和本构方程离散权函数的合理选择,可以显著增强数值计算的稳定性。得到较高Weissenberg数下的解,同时文中对黏弹流体流动数值模拟中本构方程的应用、非等温情况和三维空间下的研究进行了介绍。     

A rheological model with constant approximate volume for immiscible blends of ellipsoidal droplets

We develop a set of evolution equations describing the effects of a general deformation field on the shape, size, and orientation of constant-volume droplets suspended in a Newtonian fluid. The rheological characteristic functions of this incompressible and immiscible polymer blend model are also derived in tandem with the abovementioned set. The constant-volume constraint is implemented using a recent methodology (Edwards BJ, Dressler M, Grmela M, Ait-Kadi A (2002) Rheological models with microstructural constraints. Rheo Acta (in press)) and discussed relative to the similarly volume-constrained model of Almusallam et al. (Almusallam AS, Larson RG, Solomon MJ (2000) A constitutive model for the prediction of ellipsoidal droplet shapes and stresses in immiscible blends. J Rheol 44:1055–1083). Sample results are reported for step-strain and shear relaxation profiles.     

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.     

Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability

Traditional torsional resonators, often obtaining the viscoelastic moduli of complex fluids only at one or several given discrete frequencies, lack the continuously varying frequency capability. This is an obvious disadvantage of the traditional torsional resonator technique. This paper presents an improved strategy, based on our previous discrete-frequency-measuring method (Wang et al., J Rheol 52:999–1011, 2008), to overcome such restriction and thus accomplish the continuously varying frequency capability of the traditional torsional resonator for measuring the viscoelastic properties of complex fluids. The feasibility of this strategy is demonstrated with the Newtonian fluids (several water–glycerol solutions) of viscosities varying from 10 to 1,400 cp by using our homemade torsion resonator apparatus in the 10 ~ 2,500 rad/s frequency range (continuous frequencies). Some results for typical viscoelastic polymers (two polyethylene oxide (PEO) aqueous solutions) are also given. Additionally, a comparison of the PEO results is made with the common rheometer technique. It is demonstrated that this improved strategy could enable the traditional torsional resonators, with one oscillating resonance mode, to work as the microrheological technique and the common rheometer technique in the continuous frequency range.     

Drop oscillations under simple shear in a highly viscoelastic matrix

Recent two-dimensional numerical simulations and experiments have shown that, when a drop undergoes shear in a viscoelastic matrix liquid, the deformation can undergo an overshoot. I implement a volume-of-fluid algorithm with a paraboloid reconstruction of the interface for the calculation of the surface tension force for three-dimensional direct numerical simulations for a Newtonian drop in an Oldroyd-B liquid near criticalities. Weissenberg numbers up to 1 at viscosity ratio 1 and retardation parameter 0.5 are examined. Critical capillary numbers rise with the Weissenberg number. Just below criticality, drop deformation begins to undergo an overshoot when the Weissenberg number is sufficiently high. The overshoot becomes more pronounced, and at higher matrix Weissenberg numbers, such as 0.8, drop deformation undergoes novel oscillations before settling to a stationary shape. Breakup simulations are also described.     

Selective withdrawal of polymer solutions: Computations

This paper reports numerical simulations of selective withdrawal of Newtonian and polymeric liquids, and complements the experimental study reported in the accompanying paper (Zhou and Feng [2]). We use finite elements to solve the Navier–Stokes and constitutive equations in the liquid on an adaptively refined unstructured grid, with an arbitrary Lagrangian–Eulerian scheme to track its free surface. The rheology of the viscoelastic liquids are modeled by the Oldroyd-B and Giesekus equations, and the physical and geometric parameters are matched with those in the experiments. The computed interfacial deformation is in general agreement with the experimental observations. In particular, the critical condition for interfacial rupture is predicted to quantitative accuracy. Furthermore, we combine the numerical and experimental data to explore the potential of selective withdrawal as an extensional rheometer. For Newtonian fluids, the measured steady elongational viscosity is within 47% of the actual value, apparently with better accuracy than other methods applicable to low-viscosity liquids. For polymer solutions, an estimated maximum error of 300% compares favorably with prior measurements.     

An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.     

A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

A new low-Reynolds-number k–ε turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–ε turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLT_ij). The NLT_ij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.     

A two-phase solver for complex fluids: Studies of the Weissenberg effect

In this work a new two-phase solver is presented and described, with a particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed code is based on a combination of classical Volume-of-Fluid and Continuum Surface Force methods, along with a generic kernel-conformation tensor transformation to represent the rheological characteristics of the (multi)-fluid phases. Benchmark test problems are solved in order to assess the numerical accuracy of distinct levels of physical complexities, such as the interface representation, the influence of advection schemes, the influence of surface tension and the role of fluid rheology. In order to demonstrate the new features and capabilities of the solver in simulating of complex fluids in transient regime, we have performed a set of simulations for the problem of a rotating rod inserted into a container with a viscoelastic fluid, known as the Weissenberg or Rod-Climbing effect. Firstly, our results are compared with numerical and experimental data from the literature for low angular velocities. Secondly, we have presented results obtained for high angular velocities (high elasticity) using the Oldroyd-B model which displayed very elevated climbing heights. Furthermore, above a critical value for the angular velocity, it was observed the onset of elastic instabilities driven by the combination of elastic stresses, interfacial curvature and secondary flows, that to the authors best knowledge, were not yet reported in the literature.     

Viscoelastic effects in double‐pipe single‐pass counterflow heat exchangers

The conducto‐convective heat loss from a viscoelastic liquid, in the core of a double‐pipe heat exchanger arrangement, to a cooler Newtonian fluid flowing in the outer annulus is investigated with direct numerical simulations. A numerical algorithm based on the finite difference method is implemented in time and space with the Giesekus constitutive model for the viscoelastic liquids. The flow of both the annulus and core‐fluids is considered to be Poiseuille flow, driven by respective pressure gradients. In general, the results show that a viscoelastic core‐fluid leads to slightly lower (albeit comparable) attainable temperatures in the core‐fluid stream as compared with a corresponding Newtonian fluid. Copyright © 2008 John Wiley & Sons, Ltd.     

Non-Newtonian deterministic lateral displacement separator: theory and simulations

Deterministic lateral displacement devices have been proved to be an efficient way to perform continuous particle separation in microfluidic applications (Huang et al. Science 304:987–990, 2004). On the basis of their size, particles traveling through an array of obstacles follow different paths and can be separated in outflow. One limitation of such a technique is that each device works for a specific critical size to achieve particle separation, and a new device with different geometrical properties needs to be fabricated, as the dimensions of the particles to be separated change. In this work, we demonstrate the possibility to tune the critical particle size in a deterministic lateral displacement device by using non-Newtonian fluids as suspending liquid. The analysis is carried out by extending the theory developed for a Newtonian constitutive law (Inglis et al. Lab Chip 6:655–658, 2006) to account for fluid shear-thinning. 3-D finite element simulations are performed to compute the dynamics of a spherical particle flowing through the deterministic ratchet. The results show that fluid shear-thinning, by altering the flow field between the obstacles, contributes to decrease the critical particle diameter as compared to the Newtonian case. Numerical simulations demonstrate that tunability of the critical separation size can be achieved by using the flow rate as control parameter. A design formula, relating the separation diameter to the fluid rheology and the relevant geometrical parameters of the device, is derived. Such a formula, originally developed for a power-law model, is proved to work for non-Newtonian liquids with a general viscosity trend.     

Viscoelastic medium flows in convergent-divergent channels

The problem of viscoelastic fluid flow in a convergent-divergent channel is considered. A mathematical model and the results of numerical investigations are presented. A model of the differential type is used to describe the rheological properties of viscoelastic fluids. A comparative analysis of the flow parameters of generalized Newtonian and viscoelastic fluids in the channels considered is carried out on the basis of the results of numerical calculations.