Sedimentation of particles in shear flows of viscoelastic fluids

An experimental investigation of sedimentation of single particles and concentrated suspensions in Couette flows of viscous and viscoelastic fluids is presented. With the passage from viscous to viscoelastic fluids, a slowing down of the sedimentation was observed. The sedimentation of suspensions in viscoelastic fluids accelerated with increase in the suspension concentration. The development of instabilities during the sedimentation was detected.     

Lattice Boltzmann method for the simulation of viscoelastic fluid flows

The simulation of viscoelastic fluids is a challenging task from the theoretical and numerical points of view. This class of fluids has been extensively studied with the help of classical numerical methods. In this paper we propose a new approach based on the lattice Boltzmann method in order to simulate linear and non-linear viscoelastic fluids and in particular those described by the Oldroyd-B and FENE-P constitutive equations. We study the accuracy and stability of our model on three different benchmarks: the 3D Taylor–Green vortex decay, the simplified 2D four-rolls mill, and the 2D Poiseuille flow. To our knowledge, the methodology described in this work is a first attempt for the simulation of non-trivial flows of viscoelastic fluids using the lattice Boltzmann method to discretize the constitutive and conservation equations.     

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.     

Three‐dimensional simulation of planar contraction viscoelastic flow by penalty finite element method

The planar contraction flow is a benchmark problem for the numerical investigation of viscoelastic flow. The mathematical model of three‐dimensional viscoelastic fluids flow is established and the numerical simulation of its planar contraction flow is conducted by using the penalty finite element method with a differential Phan‐Thien–Tanner constitutive model. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The distributions of velocity and stress obtained by simulation are compared with that of Quinzani's experimental results detected by laser–doppler velocimetry and flow‐induced birefringence technologies. It shows that the numerical results agree well with the experimental results. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids. Copyright © 2009 John Wiley & Sons, Ltd.     

非牛顿流体流动的格子Boltzmann方法研究进展

格子玻尔兹曼方法（lattice Boltzmann method,LBM）能够直接计算局部剪切速率并可以达到二次精度,因此在非牛顿流动数值模拟中展现出一定优势。尽管已证实LBM 对于非牛顿流动的适用性,但是LBM 需要通过即时调节BGK（Bhatnagar-Gross-Krook）碰撞项中的松弛时间来实时反映黏度改变,当松弛时间接近1/2 时,迭代会出现数值不稳定现象。该文对LBM 在非牛顿流体研究中的进展进行了总结,介绍了增加数值稳定性的方法并对结果的精度进行了比较,在此基础上对LBM 在非牛顿研究中的进一步发展进行了展望。     

A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Péclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Péclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows. Received:  4 February 1997 Accepted: 23 October 1997     

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.     

Instabilities in viscoelastic flow through an axisymmetric sudden contraction

Kinematics and dynamics of the viscoelastic flow in an axisymmetric 4 : 1 sudden contraction geometry are studied for a highly elastic polyisobutylene (PIB) based polymer solution (referred to as PIB-Boger fluid). The critical conditions for the onset of the elastic instabilities and the dynamics of the resulting secondary flows are measured for various flow rates. The spatio-temporal characteristics of the flow are determined by instantaneous pressure measurements and streakline photography. The nonlinear dynamics of the global flow field both upstream and downstream of the contraction plane are systematically examined. New dynamic flow behavior and elastic instabilities downstream of the contraction plane are reported. It is shown that the instantaneous pressure measurements along with flow visualization can be used as an effective tool to characterize viscoelastic flows in complex geometries.     

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.     

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.     

A double wall-ring geometry for interfacial shear rheometry

The rheological properties of complex fluid interfaces are of prime importance in a number of technological and biological applications. Whereas several methods have been proposed to measure the surface rheological properties, it remains an intrinsically challenging problem due to the small forces and torques involved and due to the intricate coupling between interfacial and bulk flows. In the present work, a double wall-ring geometry to measure the viscoelastic properties of interfaces in shear flows is presented. The geometry can be used in combination with a modern rotational rheometer. A numerical analysis of the flow field as a function of the surface viscoelastic properties is presented to evaluate the non-linearities in the surface velocity profile at a low Boussinesq number. The sensitivity of the geometry, as well as its applicability, are demonstrated using some reference Newtonian and viscoelastic fluids. Oscillatory and steady shear measurements on these reference complex fluid interfaces demonstrate the intrinsic sensitivity, the accuracy, and the dynamic range of the geometry when used in combination with a sensitive rheometer.     

Review of the entry flow problem: Experimental and numerical

This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.     

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.     

Stability of planar stagnation flow of a highly viscoelastic fluid

Linear stability analysis is used to predict the onset of instabilities in inertialess viscoelastic planar stagnation flow. Beyond a critical value of the dimensionless flow rate, or Deborah number, the creeping base flow of similarity type, which is valid in the limit of vanishingly small Reynolds numbers, becomes unstable to localized three-dimensional disturbances. Stability calculations of the local similarity type viscoelastic flow in a small region near the stagnation plane are reported for the quasi-linear Oldroyd-B constitutive equation. The stability results for a range of Deborah numbers and viscosity ratio are presented to explore systematically the effects of elasticity and other rheological properties. The onset of instability and the temporal and spatial characteristics of the secondary flow predicted here resemble other purely elastic instabilities measured and predicted for viscoelastic flows in other simple and complex geometries with curved streamlines.     

Efficient microfluidic rectifiers for viscoelastic fluid flow

In this work we propose a new type of microfluidic rectifier, which is able to operate efficiently under creeping flow conditions. The flow of Newtonian and non-Newtonian fluids was investigated experimentally in different microchannels with triangular (nozzle/diffuser) and hyperbolic shapes in order to achieve high anisotropic flow resistance between the two flow directions. The Newtonian fluid used was de-ionized water and the viscoelastic fluids were aqueous solutions of polyacrylamide and polyethylene oxide with different molecular weights. Pressure drop measurements were performed in addition to visualizations of the flow patterns by streak line photography for a wide range of flow rates. For the Newtonian flows, inertia leads to the appearance of recirculations for both flow directions, but no significant rectification effects appear. For the viscoelastic fluids, two distinct behaviors are identified: at low flow rates, the pressure drops are similar in both flow directions; above a critical flow rate (or Deborah number), the flow patterns become quite different, leading to different flow rates in the forward and backward flow directions for the same pressure drop, i.e., rectification effects emerge. In particular, the viscoelastic fluid flow becomes unsteady in the forward direction, due to the presence of elastic instabilities, which leads to a significant increase in the flow resistance. Flow resistance ratios greater than three were achieved for the hyperbolic rectifier, clearly in excess of the value for the triangular-shaped rectifier and for other geometries proposed in the literature for operation in creeping flow conditions. This high diodicity is associated with the distinct nature of the extensional flows in the forward and backward directions of the hyperbolic-type microgeometry.     

Karhunen–Loève Galerkin method with decimated sampling technique for the simulation of complex fluids defined in the phase space

In complex fluids, solute molecules with structural length scales much larger than atomic are dispersed in solvents of simple fluids such as water. The rheological properties of complex fluids are determined by dynamics of solute molecules which can be modeled by the Fokker–Planck equation defined in a six-dimensional phase space. In the present investigation, we devise a method of efficient simulation of complex fluid flows employing the Karhunen–Loève Galerkin (KLG) method. Adopting the decimated sampling of solvent flow fields, a reduced-order model for the Fokker–Planck equation is obtained, which can be employed for the the simulation of complex fluids with a decent computer time. As a specific example, we consider a flow of dilute polymeric liquids over a cylinder, whose constitutive equation is the FENE (finitely extensible nonlinear elastic) model. It is found that the KLG method with the decimated sampling technique yields accurate results at a computational cost less than a hundredth of that for the numerical simulation using the Fokker–Planck equation. The KLG method supplemented by the decimated sampling technique is an efficient method of coarse-graining for equations of complex fluids defined in the phase space.     

Simulation of viscoelastic fluids in a 2D abrupt contraction by spectral element method

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE‐P) fluid 1 . The re‐entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39 (9):1425–1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.