Antony N. Beris

Creeping motion of a sphere through a Bingham plastic

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Y g below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Y g is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Y g there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Direct numerical simulation of the turbulent channel flow of a polymer solution

In this work, we present from first principles a direct numerical simulation (DNS) of a fully turbulent channel flow of a dilute polymer solution. The polymer chains are modeled as finitely extensible and elastic dumbbells. The simulation algorithm is based on a semi-implicit, time-splitting technique which uses spectral approximations in the spatial coordinates. The computations are carried out on a CRAY T3D parallel computer. The simulations are carried out under fully turbulent conditions albeit, due to computational constraints, not at as high Reynolds number as that usually encountered in polymer-induced drag reduction experiments. In order to compensate for the lower Reynolds number, we simulate more elastic fluids than the ones encountered in drag reduction experiments resulting in Weissenberg numbers (a dimensionless number characterizing the flow elasticity) of similar magnitude. The simulations show that the polymer induces several changes in the turbulent flow characteristics, all of them consi...

Transient phenomena in thixotropic systems

Abstract A nonlinear rheological model which accounts for the time-dependent elastic, viscous and yielding phenomena is developed in order to describe the flow behavior of thixotropic materials which exhibit yield stress. A key feature of the formulation is a smooth transition from an ‘elastically’ dominated response to a ‘viscous’ response without a discontinuity in the stress–strain curve. The model is phenomenological and is based on the kinetic processes responsible for structural changes within the thixotropic material. As such, it can predict thixotropic effects, such as stress overshoot during start-up of a steady shear flow and stress relaxation after cessation of flow. Thus this model extends a previously proposed viscoplastic model [J. Rheol. 34 (1991) 647] to include thixotropy. An analysis and comparison to experimental data involving oscillatory shear flow are provided to evaluate the accuracy of the model and to estimate the model parameters in a prototype concentrated suspension. The experiments were conducted using a series of concentrated suspensions of silicon particles and silicon carbide whiskers in polyethylene. The data obtained with this experimental system indicated much better agreement between the theory and experiments that obtained in earlier work.

The Cox–Merz rule extended: A rheological model for concentrated suspensions and other materials with a yield stress

A nonlinear rheological model combining elastic, viscous, and yielding phenomena is developed in order to describe the rheological behavior of materials which exhibit a yield stress. A key feature of the formulation is the incorporation of a recoverable strain; it has a maximum value equal to the critical strain at which the transition from an elastic solid‐like response to a viscous shear thinning response occurs. An analysis is presented to enable determination of all the model parameters solely from dynamic measurements which are easily accessible experimentally. A rigorous correlation, analogous in form to the Cox–Merz rule, is shown to exist between the steady shear viscosity and the complex dynamic viscosity in terms of a newly defined ‘‘effective shear rate.’’ Experimental data obtained for a 70 volu2009% suspension of silicon particles in polyethylene indicate agreement with theoretical predictions for both the dynamic and steady shear behavior.

Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction : effect of the variation of rheological parameters

Abstract In this work, we present the results from direct numerical simulations of the fully turbulent channel flow of a polymer solution. Using constitutive equations derived from kinetic and network theories, in particular the FEN E-P and the Giesekus models, we predict drag reduction for a variety of rheological parameters, extending substantially previous calculations, [Sureshkumar et al., Phys. Fluids, 9 (1997) 743-755]. The simulation algorithm is based on a semi-implicit, time-splitting technique which uses spectral approximations in the spatial domain. The computations were carried out on a CRAY T3E-900 parallel supercomputer, under fully turbulent conditions. In this work, we demonstrate the existence of a critical range of the Weissenberg number, where the onset of drag reduction occurs, which is independent of the model and also remains the same as the chain extensibility is increased. By allowing for higher extensibility of the polymer chains, we also observed an almost triple in magnitude increase in drag reduction from previous and reported results. The simulations show that the polymer induces several changes in the turbulent flow characteristics, all of them consistent with available experimental results. In addition to decreased fluctuations in the streamwise vorticity and increased streak spacing, we have seen changes, such as the increase of the slope of the logarithmic layer asymptote for the mean velocity profile, which are consistent with high magnitude of drag reduction, as well as with the behaviour of more concentrated systems. This is more consistent with the use of the Giesekus model, which is well suited for concentrated systems, suggesting that there is potential with that model for capturing quite subtle changes in the structure of the turbulent flow field. Results for different contributions of molecular extensibility, L, and solvent viscosity ratio, β, indicate that for the FENE-P model the phenomena are determined almost exclusively by the extensional viscosity and Weissenberg number. However, results obtained with the Giesekus model, for the same extensional viscosity, demonstrate a further drag reducing effect which can be attributed to the non-zero second normal stress coefficient. All results point to a mechanism for drag reduction where a partial inhibition of eddies within the buffer layer by the macromolecules. The simulation results are consistent with the hypothesis that one of the prerequisites for the phenomenon of drag reduction is sufficiently enhanced extensional viscosity, corresponding to the level of intensity and duration of extensional rates typically encountered during the turbulent flow, as has been proposed by various investigators in the past.

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.

Calculations of steady-state viscoelastic flow in an undulating tube

Abstract The flow of a viscoelastic fluid in an undulating tube is of importance both for modeling the flow of polymeric fluids through porous media and for testing numerical techniques for the simulation of viscoelastic flows. This geometry represents one of the simplest pore models for the consideration of the converging-diverging nature of the flow field in applications ranging from enhanced oil recovery to manufacturing of advanced composites using resin transfer molding. For an upper convected Maxwell fluid, a mixed pseudospectral/finite difference method (PSFD) is developed, employing the coordinate system provided by the streamlines and the lines orthogonal to them. For an Oldroyd-B fluid, a modification of the PSFD method is used, implemented in a stretched cylindrical coordinate system (PCFD method). For both the Maxwell and the Oldroyd-B fluid, solutions converged with mesh refinement are obtained up to high Deborah numbers. The results obtained using the Maxwell fluid show no increase in the flow resistance with increasing flow elasticity, in reference to the purely viscous flow. The results for the Oldroyd-B fluid show a very small increase in the flow resistance. These findings are in disagreement with experimental data reported in the literature, which have been obtained with Boger fluids. For the Maxwell fluid, a perturbation analysis valid for small amplitudes of undulation shows that extremely steep boundary layers are developed with increasing elasticity in the flow. The boundary layers are developed both near the wall and the centerline of the tube and their resolution becomes a key element for any potentially successful numerical method. In addition, their presence explains, at least partially, the failure of previous attempts by other investigators to obtain converged solutions at high values of elasticity.

Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm

Abstract An algorithm to compute the extremal eigenvalues of a non-Hermitian matrix, based on Arnoldi-orthogonalization, is employed in the linear stability analysis of the viscoelastic Poiseuille flow at high Reynolds numbers. It is shown that this algorithm is both computationally efficient and accurate in reproducing the most unstable modes in the pseudo-spectrally discretized eigenspectrum of the original problem. The Upper Convected Maxwell (UCM), Oldroyd-B and Chilcott-Rallison models are considered for the linear stability analysis of the high Reynolds number viscoelastic Poiseuille flow. Results for the UCM model show a large destabilization of the flow compared to the Newtonian limit, even for low values of flow elasticity, ϵ = We / Re , of the order 10 −3 , realized at high Reynolds numbers for O (1) We. These results agree qualitatively with those reported by Porteus and Denn, although some quantitative differences exist, especially at high values of elasticity. Furthermore, it is shown that the number of spectral modes necessary to obtain converged results increases substantially as the flow elasticity is increased. A comparison of the linear stability characteristics of the Oldroyd-B and the UCM models has revealed that the presence of a non-zero solvent viscosity has a pronounced stabilizing effect on the flow. Further stabilization occurs through the introduction of a finite molecular extensibility in the Chilcott-Rallison model.

Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow

The budgets of the Reynolds stress, turbulent kinetic energy and streamwise enstrophy are evaluated through direct numerical simulations for the turbulent channel flow of a viscoelastic polymer solution modeled with the Finitely Extensible Nonlinear Elastic with the Peterlin approximation (FENE-P) constitutive equation. The influence of viscoelasticity on the budgets is examined through a comparison of the Newtonian and the viscoelastic budgets obtained for the same constant pressure drop across the channel. It is observed that as the extensional viscosity of the polymer solution increases there is a consistent decrease in the production of Reynolds stress in all components, as well as in the other terms in the budgets. In particular, the effect of the flow elasticity, which is associated with the reduction in the intensity of the velocity-pressure gradient correlations, potentially leads to a redistribution of the turbulent kinetic energy among the streamwise, the wall-normal and the spanwise directions....

Spectral/finite-element calculations of the flow of a maxwell fluid between eccentric rotating cylinders

Abstract The steady-state, two-dimensional creeping flow of an Upper-Convected Maxwell fluid between two eccentric cylinders, with the inner one rotating, is computed using a spectral/finite-element method (SFEM). The SFEM is designed to alleviate the numerical oscillations caused by excessive dispersion error in previous finite-element calculations and to resolve the stress boundary-layers that exist for high elasticity, as measured by the Deborah number De . Calculations for cylinders with low eccentricity (ϵ = 0.1) converged to oscillation-free solutions for De ≈ 90, extending the domain of convergence over traditional finite-element methods by a factor of thirty. The results are confirmed by extensive refinement of the discretization. At high De , steep radial boundary layers form in the stress, which match closely with those predicted by asymptotic analysis. Calculations at higher eccentricity require extreme refinement of the discretization to resolve the variations in the stress field in both the radial and azimuthal directions associated with the existence of the recirculation region. Results for ϵ = 0.4 show that the recirculation region present for the Newtonian fluid ( De = 0) shrinks and then grows with increasing De . Calculations for ϵ = 0.4 are terminated by a limit point near DeL ≈ 7.24 for the finest discretization used. The Fourier series approximations are not convergent for this mesh, so the limit point must be considered to be an artifact of the discretization.