Oldroyd-B Constitutive Equation Research Articles

On algebraic extra-stress models for the simulation of viscoelastic flows

A methodology is presented for the formulation of an algebraic extra-stress model derivable from the Oldroyd-B constitutive equation for a viscoelastic fluid. An implicit algebraic tensor relation is derived directly from the differential constitutive relation by involving a slow variation condition on the evolution of the deviatoric part of the extra-stress tensor. An explicit solution to this algebraic constitutive equation is then obtained using representation bases developed previously for the turbulent flow of Newtonian fluids. A series of validation flows are then studied to assess the accuracy and range of applicability of the algebraic extra-stress model.

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.

Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids

The linear stability of plane Poiseuille flows of two and three-symmetrical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equations and hence the stability is controlled by the viscous and elastic stratifications and the layer thicknesses. For the three symmetrical-layer Poiseuille flow, competition between varicose (symmetrical) and sinuous (antisymmetrical) mode is considered. In both cases (two and three symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous and elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille flow is also stable with respect to moderate wavelength disturbances and the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity profile means a positive jump of shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important influence of the thickness ratio on the critical wavenumber is found for three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio).

Observations of viscoelastic instabilities in recirculation flows of Boger fluids

A purely elastic flow instability in the recirculation flow of a driven half-cavity or blocked channel is discovered and examined both experimentally and numerically. Video image analysis is used to determine the critical Weissenberg number and characteristic wavenumber of the instability while laser sheet imaging provides information about the structure and location of the secondary flow. To understand the basic flow and ultimately the flow instabilities. we numerically solve the combined momentum conservation and constitutive equations describing the flow of elastic fluids in the block flow geometry. Local rates of polymer stretch are estimated with a weakly elastic expansion using the Oldroyd-B constitutive equation for the polymer stress. Numerical solutions for the kinematics and polymer stretch are also obtained using the Chilcott-Rallison FENE model to describe the polymeric stress. These results are discussed in the context of known purely elastic instability mechanisms in order to better understand the stability of general recirculation flows.

Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows

In this work, we investigate the effect of the introduction of a stress diffusive term into the classical Oldroyd-B constitutive equation on the numerical stability of time-dependent viscoelastic flow calculations. The channel Poiseuille flow at Re ⪢ 1 and O(1) We is chosen as a test problem. Through a linear stability analysis, we demonstrate that the introduction of a small amount of (dimensionless) diffusivity, typically of the order 10 −3, does not affect the critical eigenmodes of the viscoelastic Orr-Sommerfeld problem appreciably. However, a diffusive term of that magnitude is shown to have a significant influence on the singular eigenmodes of the classical Oldroyd-B model, associated with the continuum spectra. A finite amplitude perturbation is constructed as a linear superposition of the eigenvectors corresponding to the most unstable eigenvalues of the problem. This is superimposed on the steady Poiseuille flow solution to provide the initial conditions for time-dependent simulations. The numerical algorithm involves a fully spectral spatial discretization and a semi-implicit second order integration in time. For the Oldroyd-B fluid, depending on the magnitude of the initial perturbation, numerical instabilities set in at relatively short times while the components of the conformation tensor increase monotonically in magnitude with time. Introduction of a diffusive term into this model is shown to stabilize the calculations remarkably, and for a three-dimensional simulation with Re = 5000 and We = 1, no instabilities were observed even at very large times. The effect of the magnitude of the diffusivity on the stability and the flow dynamics is addressed through a direct comparison of the results with those obtained for the Oldroyd-B model.

Viscoelastic free surface flows: thin film hydrodynamics of Hele-Shaw and dip coating flows

The effect of elasticity on the meniscus shape and film thickness for the free boundary creeping flow created by injecting air to a Hele-Shaw cell initially filled with a viscoelastic fluid is studied theoretically. The theory is developed with the assumptions that the displaced viscoelastic fluid wets the walls and that capillary number Ca and the local Weissenberg number We are both small. The transition region between the advancing meniscus and the entrained film is where the fluid rheology has its greatest effect. The Oldroyd-B constitutive equation is used to model the viscoelastic fluid. The theory is formulated as a double expansion in Ca 1 3 and We Ca 1 3 . According to our asymptotic analysis, as the fluid becomes more viscoelastic, the film thickness decreases and the pressure drop at the meniscus tip increases. A detailed analysis shows that the dominant mechanisms are the resistance to stream-wise strain, tending to lower the film thickness, and the buildup of shear stress, tending to raise the film thickness, with the former being the numerically larger of the two. Our theory leads to the prediction that as viscoelasticity comes to dominate the shear resistance, the film thickness will scale with U 4 3 . The effects of shear thinning and normal stress thinning are analyzed by adapting an approximate model obtained by retaining only the dominant terms in the force balance. Our Hele-Shaw cell theory is extended to dip coating and soap film forming flows. Our theoretical results pertaining to the effects of viscoelasticity on the thickness of the film are in qualitative agreement with recent experimental data.

Self-similar spiral instabilities in elastic flows between a cone and a plate

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of aBernoulli Spiral.Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduction of a radial coordinate transformation yields an accurate description of the logarithmic spiral instabilities observed experimentally, and substitution into the linearized disturbance equations leads to a separable eigenvalue problem. Experiments and calculations for two different elastic fluids and for a range of cone angles and Deborah numbers are presented to systematically explore the effects of geometric and rheological variations on the spiral instability. Excellent quantitative agreement is obtained between the predicted and measured wavenumber, wave speed and spiral mode of the elastic instability. The Oldroyd-B model correctly predicts the non-axisymmetric form of the spiral instability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott–Rallison model is shown to be essential in describing the variation of the stability boundaries with increasing shear rate.

Viscoelastic free surface flows: spin coating and dynamic contact lines

The effect of elasticity on the spreading of a spinning drop of fluid is investigated in the context of lubrication theory. It is shown that the Oldroyd-B constitutive equation permits a solution in which the free surface of the central part of the drop thins uniformly in space. Perturbation results for small effects of elasticity indicate an increased thinning rate of the free surface compared to Newtonian results for the central part of the spinning drop, and that this enhanced thinning rate persists only over a few characteristic relaxation times. Elastic effects in the capillary region near the moving contact line are also investigated by perturbation theory for small elasticity. Two methods for resolving the contact line singularity are chosen: matching the free surface profile to a precursor film of thickness b, and introducing slip at the spinning plate. For the precursor film model, the free surface correction changes character from a net enhancement of the capillary ridge near the contact line for large b, to a negative correction over most of the profile for small b. With the slip model, the free surface correction gives a net enhancement of the capillary ridge for all values of the slip parameter α. The difference between the models for thin precursor films or slight slip is explained by examining the manner in which the stress relaxes near the contact line. The results suggest that viscoelastic contact line dynamics may be more sensitive to the local molecular physics than the Newtonian counterparts.

Viscoelastic Taylor–Couette flow: bifurcation analysis in the presence of symmetries

The morphogenesis of secondary vortices is investigated for the axisymmetric flow of a viscoelastic fluid, confined between two independently rotating, infinitely long cylinders, in the region near the onset of instability of the purely azimuthal Couette flow. The Oldroyd-B constitutive equation is used to model viscoelasticity. Three characteristic regions in the parameter space, corresponding to three distinct solution families have been investigated where the onset of instability is due primarily to inertia, both inertia and elasticity, and exclusively elasticity, respectively. The secondary flow corresponds to a steady Taylor vortex in the first case, but to a time-periodic one when elasticity becomes important (Hopf bifurcation). Degenerate Hopf bifurcation theory in the presence of symmetries ( O (2) × S 1 ) has been used to show the existence of two different time-periodic solution families, each following either one of two possible patterns, the rotating wave or the standing wave. Through a computer-aided nonlinear analysis, all of the steady and time-periodic bifurcating solutions are shown to be supercritical, implying that one and only one is stable. These results are consistent with the conclusions of time-dependent numerical simulations which have demonstrated an exchange of stabilities from the rotating to the standing wave pattern emerging after the bifurcation, as the elasticity of the fluid increases.

Continuum and molecular interpretation of ideal elastic fluids

Measurements of the storage modulus as a function of frequency and the steady shear viscosity and first normal stress difference as a function of shear rate are presented for ideal elastic fluids constructed by dissolving a high molecular weight polyisobutylene in a solvent composed of kerosene and a low molecular weight poly(1-butene). The data, presented for four concentrations and at two temperatures, are extrapolated to zero concentration to determine the intrinsic viscosity and intrinsic first normal stress coefficient. The intrinsic quantities are discussed in terms of molecular theories which include variable hydrodynamic interaction and excluded volume effects. While the intrinsic quantities show significant deviation from ideal elastic fluid behaviour in a molecular sense, they are well represented when viewed from a continuum point of view. The steady and dynamic shear properties are well predicted with the Oldroyd-B constitutive equation when the relaxation time of the solvent is included.

Viscoelastic flow in an undulating tube. Part II. Effects of high elasticity, large amplitude of undulation and inertia

A mixed pseudospectral/finite difference technique is used in the simulation of steady-state viscoelastic flow through an undulating tube. The viscoelastic fluid behavior is modelled by the Oldroyd-B constitutive equation. It is shown that for creeping flow, even at high values of elasticity and large amplitude values, the flow resistance does not differ substantially from the Newtonian value. However, when inertia is present, the flow resistance can increase appreciably. Moreover, it is shown that viscoelasticity can significantly affect the shape and the size of the recirculation vortex at large enough amplitude values. The mixed pseudospectral/finite difference numerical method has succeeded in providing accurate solutions (converged with mesh refinement), provided that the size of the numerical discretinization was fine enough, depending on the flow parameters investigated. Even when inertia was present, under conditions when the steady-state equations change type, oscillation-free solutions were obtained. These calculations have confirmed the presence of standing waves of the vorticity in the radial direction predicted by Ahrens, Yoo and Joseph, J. Non-Newtonian Fluid Mech., 24 (1987) 67.

Thinning of a viscoelastic film

The film-thinning step of bubble coalescence is modeled assuming a planar film using the Oldroyd-B constitutive equation. Radially mobile and immobile bubble surfaces are studied including van der Waals' froces. When bubble surfaces are mobile, fluids with small elastic properties increase film thinning rates, while fluids with large elastic properties reduce thinning rates. Viscoelastic fluids are less stable to van der Waals' forces than Newtonian fluids of identical properties, van der Waals' forces cause the films to rapidly rupture once a critical film thickness is obtained. Predictions are compared with experimental data.

A transition occurring in ideal elastic liquids during shear flow

Dilute solutions of high molecular weight polyisobutylene dissolved in kerosene and low molecular weight polybutene have previously been reported to behave as ideal elastic liquids (“Boger fluids”). We report here rheological properties for similar solutions, having, however, higher molecular weights for the polyisobutylene. At low shear rates, these solutions exhibit the expected Boger-type rheological behavior, and approximately obey the Oldroyd-B constitutive equation. However, above a critical shear rate that depends upon molecular weight, prolonged shearing in a cone-and-plate or parallel-plate rheometer induces a time-dependent increase in the solution viscosity and elasticity. We find that the dependence of this transition on the Weissenberg number and the gap width or cone angle is consistent with a viscoelastic instability predicted by Phan-Thien for Oldroyd-B fluids. This instability appears to be of some generality for Boger fluids, since we have also observed it in a new monodisperse Boger fluid (polystyrene in low molecular weight polystyrene and dioctyl phthalate). Furthermore, this transition may have previously been observed (though not identified) by Jackson et al., using the Boger fluid polyacrylamide in maltose.