Velocity fields around spheres and bubbles investigated by laser-doppler anemometry

An experimental investigation is presented in which the velocity fields around sheres and bubbles moving in a cylinder have been measured by laser-Doppler anemometry (LDA). Instabilities in the flow field at rather low Deborah numbers have been discovered and these instabilities are damped by inertia forces. It is shown that the wall correction factor K is a rapidly decreasing function of the Deborah number. The experimental measurements have been compared with numerical simulations, and on the basis of this comparison it has been possible to identify a time constant and a zero-shear-rate viscosity for the test liquid.     

MHD peristaltic flow of a third order fluid in an asymmetric channel

This study is concerned with peristaltic flow of a magnetohydrodynamic (MHD) fluid in an asymmetric channel. Asymmetry in the flow is induced by waves on the channel walls having different amplitudes and phase. A systematic approach based on an expansion of Deborah number is used for the solution series. Analytic expressions have been developed for the stream function, axial velocity and axial pressure gradient. The pressure rise over a wavelength has been addressed through numerical integration. Particular attention has been given to the effects of Hartman number and Deborah number on the pressure rise over a wavelength and the trapping phenomenon. Several limiting solutions of interest are obtained as the special cases of the presented analysis by taking the appropriate parameter(s) to be zero. Copyright © 2009 John Wiley & Sons, Ltd.     

The behavior of viscoelastic squeeze films subject to normal oscillations, including the effect of fluid inertia

The problem of the squeeze film flow of a viscoelastic fluid between parallel, circular disks is analyzed. The upper disk is subject to small, axial oscillations. Lodge's “rubber-like liquid” is used as the viscoelastic fluid model, and fluid inertia forces are included. An exact solution to the equations of motion is obtained involving in-phase and out-of-phase components of velocity field and load, with respect to the plate velocity. Peculiar resonance phenomena in the load amplitude are exhibited at high Deborah number. At certain combinations of Reynolds number and Deborah number, the in-phase and/or out-of-phase velocity field components may attain an unusual circulating type of motion in which the flow reverses direction across the film. In the low Deborah number limit, and in the low Reynolds number limit, the results of this study reduce to those obtained by other workers.     

Extensional flow of a viscoelastic filament governed by the FENE-CR model

A thin filament model is used to analyze the extensional flow of a viscoelastic thread governed by the FENE-CR model. The problem is solved numerically by finite differences using a third-order upwind scheme in space and a second order Runge-Kutta scheme in time. The behavior of the filament is controlled by the competing effects of surface tension and axial normal stresses which are characterized in terms of three-dimensional groups, the Deborah number De, the extensibility parameter L and the capillary number Ca. Surface tension has a destabilizing effect causing the filament to thin in the mid-section leading to a rupture. On the other hand normal stresses tend to stabilize the filament. If axial normal stresses are sufficiently large the filament deforms almost uniaxially due to strain hardening.     

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.     

Viscoelastic effects in non-Newtonian flows through porous media

An analysis is presented for the flow of polymer solutions through a tube having a periodically varying diameter; this geometry is often used to represent a porous medium. It is found that if the stretch rate is assumed constant, the stress depends not only upon the Deborah number, but also on the ratio of the maximum to the minimum diameter. If the latter dimensionless group is not too large, no shear thickening is predicted to arise irrespective of the value of the Deborah number. These results explain the observed lack of superposition of curves of the product of the friction factor with the Reynolds number plotted against the Deborah number when different porous media are used. In addition, they also, in a qualitative sense, explain the experimentally observed maxima in the plots of the relative pressure drop as a function of the deformation rate.     

Start-up of flow of a FENE-fluid through a 4:1:4 constriction in a tube

The flow of a FENE-fluid through a 4:1:4 constriction in a tube is computed by a split Lagrangian–Eulerian finite element method. In steady flow it is found that the upstream vortex grows with increasing Deborah number, while the down-stream vortex diminishes and disappears. The steady pressure drop decreases with Deborah number unless the finite extensibility L is quite small. Starting from rest at high Deborah number, the upstream vortex grows in two stages, each with their own time scales. A simple model of this growth is proposed.     

On extensibility effects in the cross-slot flow bifurcation

The flow of finite-extensibility models in a two-dimensional planar cross-slot geometry is studied numerically, using a finite-volume method, with a view to quantifying the influences of the level of extensibility, concentration parameter, and sharpness of corners, on the occurrence of the bifurcated flow pattern that is known to exist above a critical Deborah number. The work reported here extends previous studies, in which the viscoelastic flow of upper-convected Maxwell (UCM) and Oldroyd-B fluids (i.e. infinitely extensionable models) in a cross-slot geometry was shown to go through a supercritical instability at a critical value of the Deborah number, by providing further numerical data with controlled accuracy. We map the effects of the L² parameter in two different closures of the finite extendable non-linear elastic (FENE) model (the FENE-CR and FENE-P models), for a channel-intersecting geometry having sharp, “slightly” and “markedly” rounded corners. The results show the phenomenon to be largely controlled by the extensional properties of the constitutive model, with the critical Deborah number for bifurcation tending to be reduced as extensibility increases. In contrast, rounding of the corners exhibits only a marginal influence on the triggering mechanism leading to the pitchfork bifurcation, which seems essentially to be restricted to the central region in the vicinity of the stagnation point.     

Chaining of weakly interacting particles suspended in viscoelastic fluids

An asymptotic theory based on multipole expansions is presented for multiparticle interactions in unbounded, weakly viscoelastic, creeping flows. The theory accounts for non-Newtonian sphere–sphere interactions that are of order O(De(a/R)²)$\text{O}(De{(a/R)}^2)$, where De is the Deborah number, a the sphere radius and R is the sphere–sphere separation. Analytic expressions are derived for the non-Newtonian correction to the multisphere mobility matrix for non-neutrally buoyant sedimenting spheres, and for neutrally buoyant spheres suspended in a shear flow. It is shown that these expressions give rise to particle chaining in simulations of interacting spherical particles.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Chaotic behavior of a single spherical gas bubble surrounded by a Giesekus liquid: A numerical study

In the present work, nonlinear oscillations of a spherical, acoustically driven gas bubble in a Giesekus liquid are examined numerically. A novel approach based on the Gauss–Laguerre quadrature (GLQ) method is implemented to solve the integro-differential equation governing bubble dynamics in a Giesekus liquid. It is shown that, using this robust method, numerical results could be obtained at very high amplitudes and frequencies typical of ultrasound applications. The GLQ method also enabled obtaining results at very high Deborah and Reynolds numbers over prolonged dimensionless times not reported previously. Based on the results obtained in this work, it is concluded that the GLQ method is well suited for bubble dynamics studies in viscoelastic liquids. It is also concluded that the extensional-flow behavior of the liquid surrounding the bubble (as represented by the mobility factor in the Giesekus model) has a strong effect on the chaotic behavior of the bubble, and this is particularly so at high Deborah numbers, high amplitudes and/or high frequencies of the acoustic field. A period-doubling bifurcation structure is predicted to occur for certain values of the mobility factor.     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

Purely elastic flow asymmetries in flow-focusing devices

The flow of a viscoelastic fluid through a microfluidic flow-focusing device is investigated numerically with a finite-volume code using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models. The conceived device is shaped much like a conventional planar “cross-slot” except for comprising three inlets and one exit arm. Strong viscoelastic effects are observed as a consequence of the high deformation rates. In fact, purely elastic instabilities that are entirely absent in the corresponding Newtonian fluid flow are seen to occur as the Deborah number (De) is increased above a critical threshold. From two-dimensional numerical simulations we are able to distinguish two types of instability, one in which the flow becomes asymmetric but remains steady, and a subsequent instability at higher De in which the flow becomes unsteady, oscillating in time. For the UCM model, the effects of the geometric parameters of the device (e.g. the relative width of the entrance branches, WR) and of the ratio of inlet average velocities (VR) on the onset of asymmetry are systematically examined. We observe that for high velocity ratios, the critical Deborah number is independent of VR (e.g. De_c ≈ 0.33 for WR = 1), but depends non-monotonically on the relative width of the entrance branches. Using the PTT model we are able to demonstrate that the extensional viscosity and the corresponding very large stresses are decisive for the onset of the steady-flow asymmetry.     

Stress gradient-induced migration effects in the Taylor–Couette flow of a dilute polymer solution

We present an investigation of the phenomenon of stress-induced polymer migration for dilute polymer solutions in the Taylor–Couette device, consisting of two infinitely long, concentric cylinders rotating at constant angular velocities. The underlying physical model is represented by the dilute limit of a two-fluid Hamiltonian system involving two components: one (the polymer) is viscoelastic and obeys the Oldroyd-B constitutive equation, and the other (the solvent) is viscous Newtonian. The two components are considered to be in thermal, but not mechanical equilibrium, interacting with each other through an isotropic drag coefficient tensor. This allows for stress-induced diffusion of polymer chains. The governing equations consist of the continuity and the momentum equations for the bulk velocity, the constitutive model for the polymer chain conformation tensor and the diffusion equation for the polymer concentration. The diffusion equation contains an extra source term, which is proportional to gradients in the polymer stress, so that polymer concentration gradients can develop even in the absence of externally imposed fluxes in the presence of stress inhomogeneities. The solution to the steady-state purely azimuthal flow is obtained first using a spectral collocation method and an adaptive mesh formulation to track the steep changes of the concentration in the flow domain. The calculations show the development of strong polymer migration towards the inner cylinder with increasing Deborah number (De) in agreement with experimental observations. The migration is enhanced for increasing values of the gap thickness resulting in concentration changes by several orders of magnitude in the area between the inner and outer cylinder walls. The extent of the migration also depends strongly on the ratio of the solvent to the polymer viscosity. In addition to a strongly inhomogeneous polymer concentration, significant deviations from the homogenous flow are also observed in the velocity profile. Next, results are reported from a linear stability analysis around the steady-state solution against axisymmetric disturbances corresponding to various wavenumbers in the axial direction. The calculations show that the steady-state solution remains stable up to moderate values of the Deborah number, explaining why some of the predicted stress-induced migration effects should be experimentally observable. The role of the Peclet number (Pe) on the stability of the system is elucidated.     

The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-ω) formulation

A parallel unstructured finite volume method (FVM) is developed and implemented under a distributed computing environment through the parallel virtual machine (PVM) libraries, and is used to simulate the channel flow of the Oldroyd-B fluid past a circular cylinder. Differing from our previous work 11, 12, a discrete elastic viscous split stress (DEVSS) formulation together with an independent interpolation of the vorticity (DEVSS-ω) is proposed in this paper. This method has almost the same stability behavior as the elastic viscous split stress (EVSS) formulation, and is suitable for complex constitutive models. To further improve the stability at high Deborah numbers, we combine the idea of the discrete adaptive elastic viscous split stress (DAVSS) formulation [7] with the independent interpolation of the vorticity to arrive at the DAVSS-ω method. The numerical implementation is based on the unstructured FVM method and the semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm. The parallelization of the program is implemented by a domain decomposition strategy and using PVM software libraries. The results are compared with those by the EVSS, DEVSS, and the plain Oldroyd-B formulation (without splitting the stress). It is found that the drag coefficient first decreases and then increases with the De number, for a channel half width to cylinder radius ratio of h/R = 2. It is also confirmed that the drag enhancement at high Deborah number is due to the increasing extension effect in the regions near the front and the rear stagnation points.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Instabilities in multi-hole converging flow of viscoelastic fluids

Studies of the onset of instabilities were conducted on single hole and multi-hole contractions using laser speckle visualization. A well characterized elastic fluid was used with constant viscosity of 13.1 Pa · s and elasticity characterized by a longest relaxation time constant of 2.233 s. The onset of instabilities was characterized in terms of the Deborah number and the contraction ratio. Three types of instabilities were observed: pulsing vortices, azimuthally rotating vortices, and swirling vortices. For the single hole contractions the critical Deborah number for instability increased from 4.4 to 5.07 to 5.25 as the contraction ratio increased from 4: 1 to 8: 1 to 12: 1. The magnitude of the instabilities was much greater for the 4: 1 contraction than for the other two contraction ratios. For the multi-hole contraction a square array of nine holes was used and the ratio of the hole diameter to hole spacing was varied. The height of the vortices is very similar for the single hole and multi-hole contractions at low Deborah numbers. At high Deborah numbers the effect of adjacent holes is to reduce the height of the vortices by a factor of three. For the 4: 1 spacing no secondary vortex was observed below a Deborah number of De = 3.7. Secondary vortices occurred for the 8:1 and 10:1 spacing at all Deborah numbers. Unstable pulsing vortices appeared for all spacings at a critical Deborah number around 5.5. Adjacent holes decreased the strength of the unsteady vortex motions. The centerline velocities were measured for the multi-hole contraction at shear rates of 5, 30, and 300 s^–1. The elongational strain rates are similar at a low shear rate of 5 s^–1. As shear rate is increased the onset of stretching occurs closer to the plane of the contraction for the smaller contraction ratios.     

Predictions of the excess pressure drop of Boger fluids through a 2:1:2 contraction–expansion geometry using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamics are used to generate the flow of polymer solutions, specifically of Boger fluids, through a planar 2:1:2 contraction–expansion geometry. The solvent molecules are represented by Lennard–Jones particles, while linear molecules are described by spring-monomers with a finite extensible non-linear elastic spring potential. The equations for Poiseuille flow are solved using a multiple time-scale algorithm extended to non-equilibrium situations. Simulations are performed at constant temperature using Nose–Hoover dynamics. At simulation conditions, changes in concentration show no significant effect on molecular conformation, velocity profiles, and stress fields, while variations in the Deborah number have a strong influence on fluid response. Increasing the magnitude of the Deborah number (De), larger deformation rates are developed in the flow region. For a Deborah number of one, the non-dimensional pressure drop presents values lower than the correspondent Newtonian case. However, for large Deborah numbers, the pressure drop increases above the Newtonian reference. An effective excess pressure drop above the Newtonian value is predicted for Boger fluids along this geometry.     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Mesoscopic dynamics of polymer chains in high strain rate extensional flows

We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De≈0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M=10⁶ Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De≈0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity).