Cone-and-plate flow of the Oldroyd-B fluid is unstable

It is shown that the creeping cone-and-plate flow of an Oldroyd-B fluid is unstable with respect to an infinitesimal disturbance. The critical Weissenberg number for the case of the Maxwell fluid is about 2.     

A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes

A robust finite volume method for viscoelastic flow analysis on general unstructured meshes is developed. It is built upon a general‐purpose stabilization framework for high Weissenberg number flows. The numerical framework provides full combinatorial flexibility between different kinds of rheological models on the one hand, and effective stabilization methods on the other hand. A special emphasis is put on the velocity‐stress‐coupling on colocated computational grids. Using special face interpolation techniques, a semi‐implicit stress interpolation correction is proposed to correct the cell‐face interpolation of the stress in the divergence operator of the momentum balance. Investigating the entry‐flow problem of the 4:1 contraction benchmark, we demonstrate that the numerical methods are robust over a wide range of Weissenberg numbers and significantly alleviate the high Weissenberg number problem. The accuracy of the results is evaluated in a detailed mesh convergence study.     

The visco-elastic Ekman layer

Solutions are obtained for the flow in an Ekman layer in a visco-elastic fluid using various constitutive relations. For some models (those which exhibit no shear thinning in simple shear flow) it is found that the solution is identical with the Newtonian solution for all Weissenberg numbers. For other models it is shown that the solution differs from its Newtonian counterpart and may cease to exist when a critical Weissenberg number is exceeded. In these cases, when the solution exists it is found that the visco-elastic Ekman layer is thinner than the Newtonian Ekman layer.     

Extrudate swell through an orifice die

The extrudate swell of a viscoelastic fluid through an orifice die is investigated by using a mixed finite element and a streamline integration method (FESIM), using a version of the K-BKZ model. The free surface calculation is based on a local mass conservation scheme and an approximate numerical treatment for the contact point movement of the free surface. The numerical results show a vortex growth and an increasing swelling ratio with the Weissenberg number. Convergence with mesh refinement is demonstrated, even at a high Weissenberg number of O(587), where the swelling ratio reaches a value of about 360%. In addition, it is found that the effective flow channel at the entrance region next to the orifice die is reduced due to the enhanced vortex growth, which may be a source of flow instability.     

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。     

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.     

Squeezing a viscoelastic liquid from a wedge: an exact solution

The paper reports an exact solution for the squeezing flow from a wedge of a general viscoelastic liquid. To obtain numerical values for the field variables, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. It is found that both these features are important in this transient flow; stress overshoot is responsible for a stiffer response of the fluid (compared to the inelastic case) at moderate time —at large time, shear-thinning dominates and the fluid behaves like an inelastic fluid. On the other hand, the Oldroyd-B fluid always predicts a softer response than the Newtonian one. Furthermore, there is a limiting Weissenberg number above which one component of the stresses of the Oldroyd-B fluid increases unboundedly with time. This limiting Weissenberg number is approximately $\text{sol2}\text{3}$.     

Squeezing a viscoelastic fluid from a cone

The paper reports an exact kinematics for the squeezing flow from a cone of a general viscoelastic fluid. To obtain numerical values for the stresses, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. Both these features are important in this flow. For the special case of an Oldroyd-B fluid it is shown that there is a limiting Weissenberg number above which at least one component of the stresses increases unboundedly with time.     

An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG

We report an adaptive viscoelastic stress splitting (AVSS) scheme, which was found to be extremely robust in the numerical simulation of viscoelastic flow involving steep stress boundary layers. The scheme is different from the elastic viscous split stress (EVSS) in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. Two decoupled versions of the scheme were implemented for the Upper Convected Maxwell (UCM) model: the streamline integration (AVSS/SI), and the streamline upwind Petrov-Galerkin (AVSS/SUPG) methods of integrating the stress. The implementations were benchmarked against the known analytic Poiseuille solution, and no upper limiting Weissenberg number was found (the computation was stopped at Weissenberg number of O(10⁴)). The flow past a sphere in a tube was solved next, giving convergent results up to a Weissenberg number of 3.2 with the AVSS/SI and 1.55 with the AVSS/SUPG (both were decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled streamline integration method was about 0.3). These results show that the decoupled AVSS is more stable than the corresponding EVSS, and the SI is more robust than SUPG in solving the constitutive equation of hyperbolic type. In addition, we found a very long stress wake behind the sphere, and a weak vortex in the rear stagnation region at a Weissenberg number above W_i of about 1.6, which does not seem to increase in size or strength with increasing W_i.     

Electrohydrodynamic peristaltic flow of a dielectric Oldroydian viscoelastic fluid in a flexible channel with heat transfer

This paper discusses the effects of a vertical a.c. electric field and heat transfer on a peristaltic flow of an incompressible dielectric viscoelastic fluid in a symmetric flexible channel. The mathematical modeling includes interactions among the electric field, flow field, and temperature. The perturbation solution of the modeled problem is derived by considering a small wave number. The influence of pertinent parameters is demonstrated and discussed. The numerical results show that the possibility of flow reversal increases near the lower bound of the channel and decreases near the upper bound of the channel as the electrical Rayleigh number, the Reynolds number, and the Weissenberg number increase, whereas the opposite effect is observed as the temperature parameter and the Weissenberg number increase. It is observed that the size of the trapped bolus decreases at the upper bound of the channel and increases at the lower bound of the channel with increasing electrical Rayleigh number, whereas the opposite effect is observed as the temperature parameter increases. The results also show that the trapped bolus in the case of an Oldroydian fluid is smaller than that for a Newtonian fluid.     

Squeezing of an Oldroyd-B fluid from a tube: Limiting Weissenberg number

It is shown that the squeezing flow of an Oldroyd-B fluid from a tube with a prescribed time-dependent radius has an exact separable solution. In the special case where the tube radius varies exponentially with time a similarity solution exists. However, in this case there is a critical Weissenberg number above which a component of the stress tensor increases without bound in time.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow

The behaviour of a number of mono and polydisperse polystyrenes are probed experimentally in complex extensional flow within a Cross-Slot geometry using flow-induced birefringence. Polystyrenes with similar molecular weight (M _w) and increasing polydispersity (PD) illustrated the effect of PD on the principal stress difference (PSD) pattern in extensional flow. Monodisperse materials exhibited only slight asymmetry at moderate flowrates, although increased asymmetry and cusping was observed at high flowrates. The response of monodisperse materials of different M _w at various flowrates is presented and characterised by Weissenberg numbers for both chain stretch and orientation using a theory for linear entangled polymers. The comparison of stress profiles against Weissenberg number for each process is used to determine whether the PSD pattern observed is independent of M _w and elucidate which relaxation mechanism is dominant in the flow regimes probed. For monodisperse materials, at equivalent chain orientation Weissenberg number (We _τd), different molecular weight materials were seen to exhibit similar steady state PSD patterns independent of We _τR (chain stretch We). Whilst no obvious critical Weissenberg number (We) was found for the onset of increased asymmetry and cusping, it was found to occur in the “orientating flow without chain stretch” regime.     

Purely tangential flow of a PTT-viscoelastic fluid within a concentric annulus

An analytical solution is presented for the steady state, purely tangential flow of a viscoelastic fluid obeying the Phan-Thien–Tanner (PTT) constitutive equation in a concentric annulus with relative rotation of the inner and outer cylinders. The influence on the velocity distribution within the annulus and on fRe of the Weissenberg number, aspect ratio and an elongational parameter are investigated. The results show that the differences between the radial location of the minimum velocity and of the critical angular velocity compared with their Newtonian counterparts increase as the fluid elasticity increases. The results also show that fRe decreases with increasing Weissenberg number, radius ratio and the elongational parameter in the case of inner-cylinder rotation. In contrast, fRe increases with increasing radius ratio when the outer cylinder is rotating while the inner cylinder is at rest.     

On the continuous squeezing flow in a wedge

The paper is concerned with the continuous squeezing flow of Oldroyd-type fluids in a two-dimensional wedge. The flow mimics the lubrication action in a squeezing flow and is important in that there exists a similarity solution for any simple fluid. We are only concerned with Oldroyd-type fluids, however. It is shown by using a parameter continuation method that the Oldroyd-B model has a limiting Weissenberg number. The Phan Thien/Tanner model does not have this limiting Weissenberg number.     

Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation

We present a second-order finite-difference scheme for viscoelastic flows based on a recent reformulation of the constitutive laws as equations for the matrix logarithm of the conformation tensor. We present a simple analysis that clarifies how the passage to logarithmic variables remedies the high-Weissenberg numerical instability. As a stringent test, we simulate an Oldroyd-B fluid in a lid-driven cavity. The scheme is found to be stable at large values of the Weissenberg number. These results support our claim that the high Weissenberg numerical instability may be overcome by the use of logarithmic variables. Remaining issues are rather concerned with accuracy, which degrades with insufficient resolution.     

Effect of purely elastic flow instability on molecular conformation and drag

Linear stability analysis has shown that viscoelastic creeping flow of an Oldroyd-B liquid through a sinusoidal channel is unstable to stationary, wall-localized and short wavelength perturbations [B. Sadanandan, R. Sureshkumar, Global linear stability analysis of non-separated viscoelastic flow through a periodically constricted channel, J. Non-Newtonian Fluid Mech. 122 (2004) 55]. In this work, time-dependent simulations are performed to determine the nonlinear evolution of finite amplitude disturbances in the post-critical flow regime. It is shown that a nonlinear transition, which is facilitated by a supercritical pitchfork bifurcation, establishes a finite amplitude state (FAS) in which the average polymer stretch is highly modulated. The maximum normal stress, observed at the channel nip, can increase by up to approximately 100% when the Weissenberg number, defined as the ratio of the fluid relaxation time to an inverse characteristic shear rate, is increased by only 10% beyond its critical value. This is attributed to the amplification of configurational perturbations by the base flow shear rate, which attains its maximum at the channel nip. The effect of finite chain extensibility on the critical condition and nonlinear instability is investigated using the FENE-CR model. The stabilizing effect of finite extensibility can be expressed through a renormalization of the Weissenberg number by accounting for the screening effect of the nonlinear force law on the transmission of configurational perturbations to polymeric stress. The principal features of the FAS are qualitatively model-independent. The FAS exhibits a small, but numerically perceptible increase in the friction factor as compared to the base flow. The implication of the findings on the experimentally observed flow resistance enhancement phenomenon in viscoelastic creeping flows through converging/diverging geometries is discussed.     

On the high Weissenberg number limit of the upper convected Maxwell fluid

We consider the equations for time-dependent creeping flow of an upper convected Maxwell fluid in the limit of infinite Weissenberg number. We identify a particular class of solutions which is analogous to potential flow and discuss several examples. We also discuss more general solutions for two-dimensional flow.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.