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.     

Purely elastic interfacial instabilities in superposed flow of polymeric fluids

Purely elastic interfacial stability of superposed plane Poiseuille flow of polymeric liquids has been investigated utilizing both asymptotic and numerical techniques. It is shown that these instabilities are caused by an unfavorable jump in the first normal stress difference across the fluid interface. To determine the significance of these instabilities in finite experimental geometries, a comparison between the maximum growth rates of purely elastic instabilities with instabilities driven primarily by a viscosity or a combined viscosity and elasticity difference is made. Based on this comparison, it is shown that purely elastic interfacial instabilities can play a major role in superposed flow of polymeric liquids in finite experimental geometries.     

On viscoelastic effects in non-Newtonian steady flows through porous media

This paper presents a mathematical model for describing approximately the viscoelastic effects in non-Newtonian steady flows through a porous medium. The rheological behaviour of power law fluids is considered in the Maxwell model of elastic behaviour of the fluids. The equations governing the steady flow through porous media are derived and an analytical solution of these equations in the case of a simple flow system is obtained. The conditions for which the viscoelastic effects may become observable from the pressure distribution measurements are shown and expressed in terms of some dimensionless groups. These have been found to be relevant in the evaluation of viscoelastic effects in the steady flow through porous media.     

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发,推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程,采用改进的有限差分法,分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

Inviscid instabilities in two-dimensional spatially periodic flows

The instability of two-dimensional vortex arrays with hexagonal symmetry is investigated with respect to inviscid perturbations. A numerical treatment in the framework of Floquet theory provides critical parameters for instability onset and the spatial structure of unstable modes which are in agreement with experiments in electromagnetically driven flows conducting fluids. The stability of point vortex lattices is briefly discussed.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

Efficient simulation of nonlinear viscoelastic fluid flows

This paper presents a new and efficient method for computing the flow of a non-Newtonian fluid. The approach is based on two independent concepts:Time-dependent simulation of viscoelastic flow: A new decoupled algorithm, presented in P. Saramito, Simulation numérique d'ecoulements de fluides viscoélastiques par éléments finis incompressibles et une méthode de directions alternées; applications, Thèse de l'Institut National Polytechnique de Grenoble, 1990 and P. Saramito, Numerical simulation of viscoelastic fluid flows using incompressible finite element method and a θ-method, Math. Modelling Num. Anal., 35 (1994) 1–35, enables us to split the major difficulties of this problem, and to solve it more efficiently. Moreover, this scheme is of order two in time, and can be used to obtain stationary flows in an efficient way.Conservative finite element method: this method combines the incompressible Raviart Thomas element, the discontinuous Lesaint-Raviart element, and a finite volume element method. It satisfies exactly the mass conservation law, and leads to an optimal size for the nonlinear system in terms of the total degree of freedom versus the mesh size.We apply our numerical procedure to the Phan-Thien-Tanner model with a classical benchmark: the four to one abrupt contraction. The numerical solutions exhibit good behavior, especially near the singularity, in the vicinity of the re-entrant corner. The numerical experiments present the main features of such flows: vortex development and overshooting of the velocity profile along the axis of symmetry in the entry region.     

A simple method for simulating viscoelastic fluid flows in slit channel

A low-cost semi-analysis finite element technique, named the finite piece method (FPM) is presented in this article. It aims to solve three-dimensional (3D) viscoelastic slit flows. The viscoelastic stress of the fluid is modelled using an K-BKZ integral constitutive equation of the Wagner type. Picard iteration is used to solve non-linear equations. The FPM is tested on flow problems in both planar and contraction channels. The accuracy of the method is assessed by comparing flow distributions and pressure with results obtained by 3D finite element method (FEM). It shows that the solution accuracy is excellent and a substantial amount of computing time and memory requirement can be saved.     

Stability and chaotic motion in columns of nonlinear viscoelastic material

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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.     

Boundary element analysis of three-dimensional mixing flow of Newtonian and viscoelastic fluids

The boundary element method (BEM) is implemented for the simulation of three-dimensional transient flows of typical relevance to mixing. Creeping Newtonian and viscoelastic fluids of the Maxwell type are examined. A boundary-only formulation in the time domain is proposed for linear viscoelastic flows. Special emphasis is placed on cavity flows involving simple- and multiple-connected moving domains. The BEM becomes particularly suited in multiple-connected flows, where part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity or barrel) is at rest. In this case, conventional methods, such as the finite element method (FEM), generally require remeshing or mesh refinement of the three-dimensional fluid volume as the flow evolves and the domain of computation changes with time. The BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non-linear periodic dynamical behavior with time for fluids obeying linear constitutive equations. © 1998 John Wiley & Sons, Ltd.     

Computation of viscoelastic cable coating flows

A viscoelastic analysis is presented for model tube tooling, draw-down and combined geometry flows encountered in the cable coating industries. The work investigates the development of stress fields and studies the effect of varying entry flow stress boundary conditions. The analysis takes into account tube tooling and draw-down flow sections individually, and in combination. The flow behaviour of cable-coating grade low density polyethylene is studied assuming a viscoelastic, isothermal flow, and employing a Taylor–Petrov–Galerkin finite element scheme with an exponential Phan-Thien–Tanner constitutive model. © 1998 John Wiley & Sons, Ltd.     

Homogeneous and heterogeneous distributed cluster processing for two‐ and three‐dimensional viscoelastic flows

A finite‐element study of two‐ and three‐dimensional incompressible viscoelastic flows in a planar lid‐driven cavity and concentric rotating cylinders is presented. The hardware platforms consist of both homogeneous and heterogeneous clusters of workstations. A semi‐implicit time‐stepping Taylor–Galerkin scheme is employed using the message passing mechanism provided by the Parallel Virtual Machine libraries. DEC‐alpha, Intel Solaris and AMD‐K7(Athlon) Linux clusters are utilized. Parallel results are compared against single processor (sequentially) solutions, using the parallelism paradigm of domain decomposition. Communication is effectively masked and practically ideal, linear speed‐up with the number of processors is realized. Copyright © 2002 John Wiley & Sons, Ltd.     

Spectral element method for viscoelastic flows in a planar contraction channel

A new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic fluid flows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four‐to‐one contraction channel for a fluid governed by an Oldroyd‐B constitutive equation. The behaviour of the Oldroyd‐B fluids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Secondary instabilities of the in-phase synchronized wakes past two circular cylinders in side-by-side arrangement

In the present study we investigate the secondary instability of the in-phase synchronized vortex shedding from two side-by-side circular cylinders at low Reynolds numbers. Two distinct Floquet modes become unstable for different values of the Reynolds number and of the non-dimensional gap spacing, leading to the onset of the well-known flip-flop instability of the two cylinder wakes. In both cases the two-dimensional Floquet analysis reveals that at very low Reynolds numbers, a pair of complex-conjugate multipliers crosses the unit circle, showing the same frequency as the biased gap-flow flip-over. In the past literature this behaviour has been often ascribed to a bistability of the flow. On the contrary, the present DNS and stability results provide evidence that at low Reynolds numbers, the flip-flopping behaviour originates from a Neimark–Sacker bifurcation of the in-phase shedding cycle.     

Solution behavior of the falling sphere problem in viscoelastic flows

Thehp-finite element method and pseudo arc-length method are combined to track solution behavior of the steady flow of a sphere falling in a tube filled with viscoelastic fluids. The computations is proved to be convergent and stable by a posteriori error analyses; the solutions smooth and with extremely steep stress gradient are obtained by using appropriate high-order interpolation distributions. The commonly used drag coefficient is proved not a reliable indicator for the approximation error. The solution curves of the upper convected Maxwell (UCM) and the Oldroyd-B fluids clearly indicate that there exist limiting points, parameterized by the Deborah number, where the magnitude of the solutions and the axial normal stress gradients behind the rear stagnation rise up rapidly, maybe go to infinity. The limiting points can be simply removed by employing the modified Chilcott and Rallison (MCR) fluid, which replaces the infinitely extendable, linear Gaussian chain with the finitely extendable nonlinear elastic spring (the FENE chain). Therefore the limiting behavior is caused by the physical model; it is not a numerical artifact. The project supported by the National Natural Science Foundation of China (20274041) and the Cooperative Research Center for Polymers, Australia     

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.     

Numerical research on the coherent structure in the viscoelastic second-order mixing layers

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