On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.     

Linear stability diagrams for the shear flow of an Oldroyd-B fluid with slip along the fixed wall

We consider the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. Slip is allowed by means of a generic slip equation predicting that the shear stress is a non-monotonic function of the velocity at the wall. The complete one-dimensional stability analysis to one-dimensional disturbances is carried out and the corresponding neutral stability diagrams are constructed. Asymptotic results for large values of the elasticity number and finite element calculations are also presented. The instability regimes are within or coincide with the negative-slope regime of the slip equation. The numerical calculations agree with the linear stability results when the size of the initial perturbation is small. Large perturbations may destabilize a linearly stable steady state, leading to a periodic solution. The period and the amplitude of the periodic solutions increase with elasticity. Received: 19 June 1997 Accepted: 22 September 1997     

Simulation of viscoelastic stagnation flow

A finite element simulation for the steady flow in a planar stagnation die was used to compute the velocity, pressure and stress fields. It is predicted that a region surrounds the stagnation point where the flow approximates a planar extension. This region is circular for the Newtonian liquid and becomes an ellipse for the Maxwell fluid. An isotropic point in the stress field is found for the Newtonian case as well as for the Maxwell fluid. Lubrication of the die wall, modeled as a finite slip, increases the size of extensional flow region by as much as 100%, while causing a migration of the isotropic stress point towards the die wall.A slight increase in the apparent planar extensional viscosity occurs before the numerical scheme fails at deformation rates well below the extensional singularity in the Maxwell model. Slip at the walls does not significantly alter the convergence behavior, which appears to be limited by effects in the entry region. In this region, at a Weissenberg number of 1.16, spatial oscillations in the pressure, deformation rate and stress develop. The stress normal to the main flow direction in the entry region is compressive for all values of the slip coefficient. a Length of hyperbolic region in the stagnation die - A total area for flow in thexy-plane - C die constant;XY = C/h ² at the die wall - da differential area element - ds differential contour length along the boundary - ;A boundary of the flow - h half-width at die entry - I unit tensor - l _i inlet length in the stagnation die - l _o outlet length in the stagnation die - n unit outward normal to the boundary or to a streamline - n _i unit outward normal to a finite element at the boundary - P pressure; normalized byV/h - P _i nodal value of pressure in an element - Q volumetric flow rate through the die - T total stress tensor; normalized byV/h - t unit tangent to the boundary or to a streamline - t _i unit tangent to a finite element at the boundary - V maximum speed at the centerline of inlet or outlet - v velocity vector with components (u, ); normalized byV - v _i nodal velocities in a finite element - v _s velocity of the solid surface; zero in this study - Ws Weissenberg number;V/h - W width of the die normal to thexy-plane - X position vector with components (X, Y) - slip coefficient; normalized byh/ - rate of deformation tensor; normalized byV/h - elongation rate; dimensionless - relaxation time of main fluid - shear viscosity of main fluid - _ex extensional viscosity (_xx — _yy) - finite element shape function for velocity - finite element shape function for pressure and stress - extra stress; normalized byV/h - _i nodal value of extra stress in an element - partial differentiation - gradient operator     

Global stability of equilibrium manifolds,and “peaking” behavior in quadratic differential systems related to viscoelastic models

We study a model inspired by the Oldroyd-B equations for viscoelastic fluids. The objective is to better understand the nonlinear coupling between the stress and velocity fields in viscoelastic flows, and thus gain insight into the reasons that cause the loss of accuracy of numerical computations at high Weissenberg number. We derive a model system by discarding the stress-advection and stress-relaxation terms in the Oldroyd-B model. The reduced (unphysical) model, which bears some resemblance to a viscoelastic solid, only retains the stretching of the stress due to velocity gradients and the induction of velocity by the stress field. Our conjecture is that such a system always evolves toward an equilibrium in which the stress builds up such to cancel the external forces. This conjecture is supported by numerous simulations. We then turn our attention to a finite dimensional model (i.e., a set of ordinary differential equations) that has the same algebraic structure as our model system. Numerical simulations indicate that the finite-dimensional analog has a globally attracting equilibrium manifold. In particular, it is found that subsets of the equilibrium manifold may be unstable, leading to a “peaking” behavior, where trajectories are repelled from the equilibrium manifold at one point, and are eventually attracted to a stable equilibrium point on the same manifold. Generalizations and implications to solutions of the Oldroyd-B model are discussed.     

On the stability of the torsional flow of a class of Oldroyd-type fluids

It is shown that the exact solution of the torsional flow of a class of Oldroyd-type fluids is kinematically similar to that for a Newtonian fluid. Furthermore, it is shown by a linearized stability analysis and by numerical integration, that the basic flow is unstable at high Weissenberg numbers. An Oldroyd fluid which has a negative second-normal stress coefficient is found to be more stable than one with zero (or positive) second-normal stress coefficient in this flow.     

A note on start-up and large amplitude oscillatory shear flow of multimode viscoelastic fluids

Start up of plane Couette flow and large amplitude oscillatory shear flow of single and multimode Maxwell fluids as well as Oldroyd-B fluids have been analyzed by analytical or semi-analytical procedures. The result of our analysis indicates that if a single or a multimode Maxwell fluid has a relaxation time comparable or smaller than the rate of change of force imparted on the fluid, then the fluid response is not singular as Elasticity Number (E ). However, if this is not the case, as E , perturbations of single and multimode Maxwell fluids give rise to highly oscillatory velocity and stress fields. Hence, their behavior is singular in this limit. Moreover, we have observed that transients in velocity and stresses that are caused by propagation of shear waves in Maxwell fluids are damped much more quickly in the presence of faster and faster relaxing modes. In addition, we have shown that the Oldroyd-B model gives rise to results quantitatively similar to multimode Maxwell fluids at times larger than the fastest relaxation time of the multimode Maxwell fluid. This suggests that the effect of fast relaxing modes is equivalent to viscous effects at times larger than the fastest relaxation time of the fluid. Moreover, the analysis of shear wave propagation in multimode Maxwell fluids clearly show that the dynamics of wave propagation are governed by an effective relaxation and viscosity spectra. Finally, no quasi-periodic or chaotic flows were observed as a result of interaction of shear waves in large amplitude oscillatory shear flows for any combination of frequency and amplitudes.     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998     

Linear viscoelastic model for bending and torsional modes in fluid membranes

This paper develops and applies a linear viscoelastic model for bending and torsional modes of fluid membranes, based on the nonlinear Cosserat surface fluid model. The linearized fluid membrane model in spherical and cylindrical geometries is shown to decouple bending and torsional viscoelastic modes. It is found that solutions of the membrane viscoelastic model to small-amplitude oscillatory bending and torsion allows for the measurement of the bending and the torsion viscosity. The model and its potential in characterizing the bending and torsion viscoelasticity of membranes complements the on-going efforts to establish the role of curvature in dissipative process of biological membranes.     

Transient viscoelastic helical flows in pipes of circular and annular cross-section

Start-up helical flows for Oldroyd-B and upper-convected Maxwell fluids are studied in straight pipes of circular and annular cross-section. The differential form of the constitutive equation leads to partial differential equations which are second-order in space and time. Apart from the condition that the fluid is initially at rest another initial condition is required to complete the solution process. By comparing results derived from the integral form of the constitutive equation we show that an appropriate initial condition may be found. Numerical results for start-up rotational flow in pipes of annular cross-section are presented.     

A comparison of stabilisation approaches for finite-volume simulation of viscoelastic fluid flow

In this work we compare different stabilisation approaches currently used in the simulation of viscoelastic fluid flow. These approaches are: the both-sides diffusion, the positive definiteness preserving scheme, the log-conformation tensor representation and the symmetry factorisation of the conformation tensor. The evaluation of these approaches is done regarding their implementation complexity, stability, accuracy, efficiency and applicability to complex problems. Their performances are examined for an Oldroyd-B fluid in the test cases of lid-driven cavity, flow past a cylinder and 4:1 contraction flow. We summarise the situations in which the different approaches can be recommended.     

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π/α with α≥1/2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O(r⁻²) and O(r⁻¹), respectively, where r1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O(r²), their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p₀, both of which are assumed furnished by the flow away from the wedge apex in the r=O(1) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r=O(1) region. Recirculating regions are implicitly assumed to be absent.     

Stability of viscoelastic flow around periodic arrays of cylinders

Low Reynolds number flow of Newtonian and viscoelastic Boger fluids past periodic square arrays of cylinders with a porosity of 0.45 and 0.86 has been studied. Pressure drop measurements along the flow direction as a function of flow rate as well as flow visualization has been performed to investigate the effect of fluid elasticity on stability of this class of flows. It has been shown that below a critical Weissenberg number (We_c), the flow in both porosity cells is a two-dimensional steady flow, however, pressure fluctuations appear above We_c which is 2.95±0.25 for the 0.45 porosity cell and 0.95±0.08 for the higher porosity cell. Specifically, in the low porosity cell as the Weissenberg number is increased above We_c a transition between a steady two-dimensional to a transient three-dimensional flow occurs. However, in the high porosity cell a transition between a steady two-dimensional to a steady three-dimensional flow consisting of periodic cellular structures along the length of the cylinder in the space between the first and the second cylinder occurs while past the second cylinder another transition to a transient three-dimensional flow occurs giving rise to time- dependent cellular structures of various wavelengths along the length of the cylinder. Overall, the experiments indicate that viscoelastic flow past periodic arrays of cylinders of various porosities is susceptible to purely elastic instabilities. Moreover, the instability observed in lower porosity cells where a vortex is present between the cylinders in the base flow is amplifieds spatially, that is energy from the mean flow is continuously transferred to the disturbance flow along the flow direction. This instability gives rise to a rapid increase in flow resistance. In higher porosity cells where a vortex between the cylinders is not present in the base flow, the energy associated with the disturbance flow is not greatly changed along the flow direction past the second cylinder. In addition, it has been shown that in both flow cells the instability is a sensitive function of the relaxation time of the fluid. Hence, the instability in this class of flows is a strong function of the base flow kinematics (i.e., curvature of streamlines near solid surfaces), We and the relaxation time of the fluid.     

Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations

We implement a volume-of-fluid algorithm with a parabolic re-construction of the interface for the calculation of the surface tension force (VOF-PROST). This achieves higher accuracy for drop deformation simulations in comparison with existing VOF methods based on a piecewise linear interface re-construction. The algorithm is formulated for the Giesekus constitutive law. The evolution of a drop suspended in a second liquid and undergoing simple shear is simulated. Numerical results are first checked against two cases in the literature: the small deformation theory for second-order liquids, and an Oldroyd-B extensional flow simulation. We then address the experimental data of Guido et al. (2003) for a Newtonian drop in a viscoelastic matrix liquid. The data deviate from existing theories as the capillary number increases, and reasons for this are explored here with the Oldroyd-B and Giesekus models.     

Application of differential quadrature method to solve entry flow of viscoelastic second‐order fluid

The entry flow of viscoelastic second‐order fluid between two parallel plates is discussed. The governing equations of vorticity and the streamfunction are expanded with respect to a small parameter that characterizes the elasticity of the fluid by means of the standard perturbation method. By using the differential quadrature method with only a few grid points, high‐accurate numerical solutions are obtained. The numerical results show a lot of the features of a viscoelastic second‐order fluid. Copyright © 1999 John Wiley & Sons, Ltd.     

The lowest stability and bifurcation in supercritical Taylor vortices

Here, we numerically investigate the lowest stability and bifurcation boundary of supercritical Taylor vortices in flows with different wavenumbers and for various radius ratios; the radius ratios range from those corresponding to axisymmetrical Taylor vortex flow (TVF) to those corresponding to wavy vortex flow (WVF). The variation in the wavenumber of a supercritical TVF is found to affect the stability of the flow, because the wavenumber of the Taylor vortices remains constant only when the flow is quasi-static. The variation in the wavenumber is examined and found to be significant when the radius ratio is less than 0.7842. The results for TVF are compared with those for the flow during the quasi-static transition from TVF to WVF.     

A predictive model for impact response of viscoelastic polymers in drop tests

We show for the first time that a classical Hookean viscoelastic constitutive law for rubbery materials can predict the impact forces and deflections measured with a commercial drop tester when a mass, or tup with a flat impacting surface is dropped onto a flat pad of commercial impact-absorbing rubber. The viscoelastic properties of the rubber, namely the relaxation times and strengths, are obtained by a standard rheological linear-oscillatory test, and the equation of momentum transfer is then solved, using these measured parameters, assuming a uniaxial deflection of the pad during the impact. Good agreement between measured and predicted forces and deflections is obtained for a series of various drop heights, tup masses, impact areas, and pad thicknesses, as long as the deflection of the pad relative to its thickness is small or modest (<50% or so), and as long as the area of the pad is less than or equal to that of the tup. When the pad area is greater than the tup, forces are higher than predicted, unless an empirical factor is introduced to account for the nonuniaxial stretching of the ring of material that extends outside of the impact area. These results imply that the impact-absorbing properties of a rubbery polymeric material can be assessed by simply examining the material's linear viscoelastic spectrum.