Abstract: The Oldroyd-B model has been used extensively to predict a host of instabilities in shearing flows of viscoelastic fluids, often realized experimentally using polymer solutions. The present review, written on the occasion of the birth centenary of James Oldroyd, provides an overview of instabilities found across major classes of shearing flows. These comprise (i) the canonical rectilinear shearing flows including plane Couette, plane and pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines such as those in the Taylor-Couette, cone-and-plate and parallel-plate geometries; (iii) non-viscometric shearing flows with an underlying extensional flow topology such as the flow in a cross-slot device; and (iv) multilayer shearing flows. While the underlying focus in all these cases is on results obtained using the Oldroyd-B model, we also discuss their relation to the actual instability, and as to how the shortcomings of the Oldroyd-B model may be overcome by the use of more realistic constitutive models. All the three commonly used tools of stability analysis, viz., modal linear stability, nonmodal stability, and weakly nonlinear stability analyses are discussed, with supporting evidence from experiments and numerical simulations as appropriate. Despite only accounting for a shear-rate-independent viscosity and first normal stress coefficient, the Oldroyd-B model is able to qualitatively predict the majority of instabilities in the aforementioned shearing flows. The review also highlights, where appropriate, open questions in the area of viscoelastic stability.

Abstract: Shear thinning is fundamental to a broad range of particle suspensions, both in nature and in industrial applications. Yet the mechanisms governing it remain unclear. In particular, the distinct, and often competing, roles of the interparticle, particle-fluid interactions and the particle surface morphology need clarity. By using silica particles with different morphology, surface functional groups and suspending media, here we reveal two different shear thinning mechanisms, controlled either by frictional or adhesion forces between particles. Smooth glass sphere suspensions in a polar medium (glycerol), where particles interact strongly with the solvent, showed no shear thinning even at high volume fractions (φ≥0.5), while rough silica particles, with similar size distribution, induced shear thinning behaviour at φ values of 0.25 and above. The latter is attributed to the increased frictional contacts in rough and irregular particles. Considering surface irregularity as elastically-deformable asperities enabled us to estimate the critical load above which two neighbouring rough particles experience frictional contacts giving rise to shear thinning. In contrast, in a non-polar (mineral oil) solvent, with which the particles do not interact strongly, both glass spheres and the rough silicas showed a pronounced shear thinning response and yield stress behaviour at volume fractions as low as 2% v/v. The rheology of these suspensions is dictated by the adhesion forces between the particles that lead to the formation of large agglomerates, which break down under increasing shear. The evolution of the sheared suspensions microstructure was captured using an optical shearing cell, which also enabled us to quantify the particle agglomeration characteristics using an aggregation index. To demonstrate the generality of our approach, we modified the surface chemistry of the glass spheres by introducing hydrophobic groups (e.g. a fluorosilane or palmitic acid) to inhibit inter-particle interactions and improve the dispersion of the otherwise inherently hydrophilic glass spheres in mineral oil; as expected, this suppressed the shear thinning behaviour of the suspensions. The present results clearly elucidate alternative design routes to controlling the suspension rheology, whether to promote or suppress shear thinning, offering new insights for the manufacturing and manipulation of complex particle suspensions.

Abstract: Using the well-known hydrodynamic theory for dilute suspensions of spheroids, we have previously predicted the destabilisation of Taylor Couette flow, due to anisotropic viscous stresses induced by suspended disk-shaped particles [J. Gillissen and H. Wilson, Phys. Rev. Fluids 3, 113903 (2018)]. Here we provide experimental evidence for the destabilisation mechanism using suspensions of mica flakes. As a function of the mica concentration, there is good qualitative agreement between the experiment and the theory in the concentration dependence of the critical speed for instability onset and of the axial wavelength of the corresponding Taylor vortices. Quantitative differences are attributed to hydrodynamic interactions between the disks, which we account for in the theory in an ad-hoc fashion using rotary diffusion.

Abstract: We consider a suspension of solid spheres in a viscoelastic fluid matrix. We use the second-order fluid model to capture the first effects of viscoelasticity in the suspension, and a mean-field cell model to express the influence of the solid particles. We create a semi-analytical constitutive equation for the whole suspension, a second-order fluid with modified material parameters. We find that, in simple shear, the ratio of first normal stress difference to shear stress is independent of volume fraction, but that the second normal stress difference can change sign as the volume fraction of solids increases. Our model is valid for all flows for which the second-order fluid expansion is applicable.

Abstract: We carry out a linear stability analysis of the generalised BMP model, which incorporates the shear-banding phenomenon into flows of thixotropic-viscoelasto-plastic fluids: common behaviours observed in wormlike micellar solutions. We introduce a new dimensionless parameter governing shear-banding, and find that when a flow is unstable in the absence of shear-banding, this parameter has a stabilising effect. Above a critical value of the shear-banding parameter, shear bands appear in the flow gradient direction. Here two different modes of instability are observed: a bulk mode (which is the same as that seen in the absence of shear banding) and an interfacial mode. We found that the former dominate over the latter at low values of the shear-banding parameter, but the interfacial instability becomes dominant at very high values of the parameter. In particular, the interfacial modes are more unstable in flows where the low shear band has almost constant viscosity. The two modes of instability depend on different material timescales: interfacial modes become more dangerous if the interface location is near the channel wall, which is seen when the structural relaxation time is much larger than that of plasticity, while bulk modes (as seen in previous work) are highly unstable if the timescale of viscoelasticity is much longer than both those of plasticity and thixotropy.

Abstract: A scaling relationship is developed for the asymptotic approach of the suspension viscosity of adhesive, rigid and non-Brownian particles to that of non-adhesive particles in the limit of high shear-rate where the agglomerates are broken down completely. The relationship is fitted to experimental data, for spherical particles, irregularly shaped particles and biconcave, discoid-shaped red blood cells. The relationship allows us to extract a proxy for the inter-particle adhesion force which, for the case of the blood cells, is in qualitative agreement with alternative force measurements in the literature.

Abstract: We recently developed a tensorial constitutive model for dense, shear-thickening particle suspensions that combines rate-independent microstructural evolution with a stress-dependent jamming threshold. This gives a good qualitative account for reversing flows, although it quantitatively overestimates structural anisotropy [J. J. J. Gillissen et al., Phys. Rev. Lett. 123 (21), 214504 (2019)]. Here we use the model to predict the unjamming effect of superposed transverse oscillations on a steady shear flow in the thickened regime [N. Y. C. Lin et al., Proc. Nat. Acad. Sci. USA 113, 10774 (2016)]. The model successfully reproduces the oscillation-mediated viscosity drop observed experimentally. We compare the time-dependent components of the stress and microstructure tensors to discrete-element simulations. Although the model correctly captures the main qualitative behaviour, it generally over-predicts the microstructural anisotropy in steady shear, and it under-predicts the number of particle contacts in oscillating shear. It also does not fully capture the correct variation in phase angle between the transverse component of the microstructure and the shear rate oscillations, as the amplitude of the latter is increased. These discrepancies suggest avenues for future improvements to the model.

Abstract: We develop a tensorial constitutive model for dense, shear-thickening particle suspensions subjected to time-dependent flow. Our model combines a recently proposed evolution equation for the suspension microstructure in rate-independent materials with ideas developed previously to explain the steady flow of shear-thickening ones, whereby friction proliferates among compressive contacts at large particle stresses. We apply our model to shear reversal, and find good qualitative agreement with particle-level, discrete-element simulations whose results we also present.

Abstract: We employ a rheological theory to show that circular Taylor-Couette flow of a suspension of non-Brownian spheres is less stable than that of a Newtonian fluid, at equal effective viscosity. The destabilization is related to the preferred orientation of the separation vector of the closely interacting spheres, in the compressive direction of the base flow. The results agree qualitatively with experimental observations from the literature.

Abstract: BACKGROUND: The rheology of shear thickening fluids is well characterized for many physical applications, however the literature surrounding biologically or cryobiologically compatible shear thickening fluids is less well understood.
OBJECTIVE: This study examined fluids consisting of corn-derived hydroxyethyl starch with a variety of sugars and cryoprotectants to characterize their shear-rate viscosity relationship. The objective was to establish if cryobiologically relevant materials could be used to afford biologics protection through shear-thickening.
RESULTS: Fluids consisting of 50% hydroxyethyl starch by weight exhibited shear thickening with a variety of cryoprotectants. Lowering the temperature of the fluid both reduced critical shear rates and enhanced thickening magnitude. Starch derived from corn, wheat, and rice all exhibited non-Newtonian shear-dependent viscosity behaviour at 50% by weight in water. Between the starch sources however, the shear-rate viscosity relationship varied widely, with wheat-derived starch shear thinning, and the remaining starches forming shear thickening fluids. Different starch sources had different baseline viscosities, critical shear rates, and rates of viscosity increase.
CONCLUSIONS: This study established that shear thickening is compatible with cryobiologically relevant agents, particularly so at lower temperatures. This forms the basis for harnessing these phenomena in biological processes such as cryopreservation.

Abstract: We present a tensorial theory for the microstructure and the stress in shear rate invariant particle suspensions that includes hydrodynamic and normal but not tangential hard sphere interaction forces. The theory predicts that hydrodynamic forces produce a negligible first normal stress difference, while contact forces produce a positive first normal stress difference. The theory thereby provides a rationale for seemingly contradicting experimental observations in the literature. In addition, the theory captures the experimentally observed time dependence of the shear stress after shear reversal.

Abstract: We study the stability of dilute suspensions of spheroids in Taylor Couette flow. We focus on axisymmetric perturbations and on the limiting cases of thin disks and long rods. It is found that in the non-Brownian limit, the rods have a negligible affect on the stability, while the disks are destabilising. The instability is driven by a tilting of the disks, which allows them to transfer energy from the base flow into azimuthal velocity fluctuations. The resulting instability mode has a wavelength, which is smaller than the unstable Newtonian mode. These findings agree qualitatively with experiments from the literature.

Abstract: We develop a model for the microstructure and the stress, in dense suspensions of non-Brownian, perfectly smooth spheres at vanishing particle Reynolds number. These quantities are defined in terms of the second-order moment a of the distribution function of the separation vector between hydrodynamically interacting particles. We show, from first principles, that the evolution equation of a contains a source term, that accounts for the association and the dissociation of interacting particle pairs. The model provides a microscopic explanation for typical non-Newtonian behaviour, observed in experiments in the literature, including normal stress differences in steady shear flow, as well as time-dependent stress after abruptly reversed shear flow and during oscillating shear flow.

Abstract: In many areas of suspension mechanics, such as filled polymer fluids or household products such as toothpaste, the suspending fluid itself is inherently non-Newtonian and may exhibit viscoelastic properties. In this paper, we extend the Stokesian Dynamics formalism to incorporate a simple model of viscoelasticity by using small spheres as "beads" in a bead–spring dumbbell (such as is found in the derivation of Oldroyd and FENE constitutive models for dilute polymer solutions). Various different spring laws are then tested in both small-amplitude and large-amplitude oscillatory shear, and their rheological behaviour is compared to continuum constitutive models.

Abstract: We study the stability of pressure-driven channel flow of a thixotropic-viscoelasto-plastic fluid. Several recent experiments have shown that channel flows of shear-thinning polymer solutions can be linearly unstable even at very low Reynolds numbers. We use the Bautista–Manero–Puig model (BMP) to attempt to capture the physics of these instabilities. We obtain an analytic solution for the steady-state velocity profile, dependent on our fluid parameters, that is able to predict a large variety of base states. We derive dimensionless groups to compare the effects of viscoelasticity, thixotropy and plasticity on the flow stability. We find that sinuous perturbations are slightly more unstable than varicose modes, and we identify the values of our model parameters for which the instability has its strongest effects. We conclude that dominant thixotropy can stabilise the flow, but instability occurs when the characteristic time scale of viscoelasticity is much longer than both those of plasticity and thixotropy. The most dangerous situation is a plug flow with an apparent yield surface just within the channel, and we find that the growth rate of the instability scales with the rate of thixotropic structure recovery.

Abstract: In Stokesian Dynamics, particles are assumed to interact in two ways: through long-range mobility interactions and through short-range lubrication interactions. To speed up computations, in shear-driven concentrated suspensions, often found in rheometric contexts, it is common to consider only lubrication. We show that, although this approximation may provide acceptable results in shear-driven, periodic suspensions, for bidisperse suspensions where the particles are exposed to an external force, it can produce physically unreasonable results. We suggest that this problem could be mitigated by a careful choice of particle pairs on which lubrication interactions should be included.

Abstract: The bizarre behaviour of a cornstarch suspension (sometimes called oobleck) is well known to all of us who have led public engagement events. At the right solids fraction, it flows smoothly at slow speeds, but can be shattered with a quick spoon movement; if you prepare a large enough sample you can run across the surface (but if you stand still, you will sink). In rheology circles this phenomenon is known as shear thickening, though the flows described above are not necessarily shear-dominated. In recent years there has been a proliferation of research on the mechanism behind true shear thickening, using both experiments and numerical simulations of shear flows. The understanding of the underlying mechanism is improving markedly. But the paper 'Microstructure and thickening of dense suspensions under extensional and shear flows' (Seto, Giusteri & Martinello, J. Fluid Mech. 825, 2017, R3) is the first to consider more general flows. We have, for the first time, simulations of thickening in extensional flows, which are a far better description of oobleck with a runner on top – and can begin to quantify the difference between the idealised shear thickening and the extension thickening that happens in practice.

Abstract: We study non-Brownian suspensions under steady shear flow. In concentrated suspensions, we are trying to reproduce the shear thickening phenomenon seen in, for example, cornstarch. We investigate the effect of different frictional contact models. When contact acts to impose a fixed minimum separation between particles, there is a strict upper bound to the viscosity predicted by simulations. We deduce that soft or compressible contacts are a critical component of the strong shear thickening seen in experiments.

Abstract: In recent experiments, Blanc et al. dropped a heavy sphere through a concentrated suspension of smaller, neutrally buoyant particles. They found that the application of a lateral oscillatory shear flow caused the heavy ball to fall faster on average; and that for highly concentrated suspensions, at certain moments of the cycle of shear oscillation, the heavy ball moves upwards. We use Stokesian Dynamics to model these experiments and other related scenarios. We show how the motion of the heavy particle and the microstructure of the suspension depend on two key dimensionless parameters: the frequency of the oscillations (relative to a typical settling time) and the strength of repulsive interparticle forces, relative to the buoyancy-adjusted weight of the heavy ball. We offer a mechanism which describes some of the observed behaviour: the formation and breakup of vertical repulsion chains.

Abstract: We consider the linear stability of channel flow of a shear-thinning viscoelastic fluid, replicating a instability recently discovered in experimental and theoretical work. We have extended the fluid model to allow for an inelastic shear-thinning stress component, and find that this additional contribution always has a stabilising influence on the instability. We conclude that, while shear-thinning is critical to the instability, the mechanism is primarily elastic.

Abstract: We have recently discovered that the implicit assumption that N₂ = 0 for a monolayer suspension in the paper by Wilson & Davis (2002) was an error: the repercussions of this error are corrected below. For dilute systems, we neglected to calculate the second normal stress difference, which is negative. In concentrated systems, the viscosity is rather larger than reported (although the trends remain the same), and the second normal stress difference is negative; but the first normal stress difference N₁ is now observed to change sign and become positive for concentrated systems.

Abstract: We consider the fluid dynamics of the chocolate fountain. Molten chocolate is a mildly shear-thinning non-Newtonian fluid. Dividing the flow into three main geometries—the pumped flow up the centre, the film flow over each dome, and the freely-falling curtain flow between the domes—we generate a wide-ranging study of Newtonian and non-Newtonian fluid mechanics. The central pumped flow is a benchmark to elucidate the effects of shear-thinning. The dome flow can be modelled as a thin-film flow with the leading-order effects being a simple balance of gravity and viscosity. Finally, the curtain flow is analytically intractable but matches onto the existing theory of water bells (both inviscid and viscous).
In pipe flow, Newtonian fluids exhibit a parabolic velocity profile; shear-thinning makes the profile more blunted. In thin-film flow over the dome, gravitational and viscous effects balance and the dome shape is not important beyond the local slope. We find that the chocolate thins and slows down as it travels down the dome. Finally, in the curtain flow, we predict the shape of the falling sheet for an inviscid fluid, and compare this with the literature to predict the shape for a viscous fluid, having shown that viscous forces are too great to ignore. We also find that the primary effect driving the shape of the curtain (which falls inwards towards the axis of the fountain) is surface tension.
We find that the three domains provide excellent introductions to non-Newtonian mechanics, the important mathematical technique of scaling, and how to manipulate existing data to make our own predictions. We also find that the topic generates interest among the public in our engagement work.

Abstract: We study pressure-driven channel flow of a simple viscoelastic fluid whose elastic modulus and relaxation time are both power-law functions of shear-rate. We find that a known linear instability for the case of constant elastic modulus (Wilson & Rallison 1999) persists and indeed becomes more dangerous when the elastic modulus is allowed to vary. The most unstable scenario is a highly shear-thinning relaxation time with a slightly shear-thinning elastic modulus, and typical unstable perturbations have a wavelength comparable with the channel width. Inertia is mildly destabilising.
We compare with microchannel experiments (Bodiguel et al. 2015), and find qualitative agreement on the critical flow rate for instability; however, because of the artificial nature of the power-law viscosity, we have excluded the sinuous modes of instability which are seen in experiment.

Abstract: In this paper we present a numerical method to calculate the dynamics of three spheres in a quiescent viscous fluid. The method is based on Lamb's solution to Stokes flow and the method of reflections, and is arbitrarily accurate given sufficient computer memory and time. It is more accurate than multipole methods, but much less efficient. Although it is too numerically intensive to be suitable for more than three spheres, it can easily handle spheres of different sizes. We find no convergence difficulties provided we study mobility problems, rather than resistance problems.
After validating against the existing literature, we make a direct comparison with Stokesian Dynamics (SD), and find that the largest errors in SD occur at a sphere separation around 0.1 radius. Finally, we present results for an example system having different-sized spheres.

Abstract: Elastic instabilities of entangled polymer melts are common in industrial processes but the physics responsible is not well understood. We present a numerical linear stability study of a molecular-based constitutive model which grants us physical insight into the underlying mechanics involved. Two constriction flows are considered – one shear dominated, the other extension dominated – and two distinct instabilities are found. The influence of the molecular structure and the behaviour of the polymer dynamics are investigated and in both cases chain relaxation and orientation play a crucial role. This suggests a molecular-based physical interpretation of the underlying mechanisms responsible for flow instabilities.

Abstract: We present three open problems in the mathematical modelling of the flow of non-Newtonian fluids. The first problem is rather long standing: a discontinuity in the dependence of the rise velocity of a gas bubble on its volume. This is very well characterised experimentally but not, so far, reproduced either numerically or analytically. The other two are both instabilities. The first is observed experimentally but never fully predicted analytically or numerically. In the second instability, numerical studies reproduce the experimental observations but there is as yet no analytical or semi-analytical prediction of the linear instability which must be present.

Abstract: Motivated by the need for a theoretical study in a planar geometry that can easily be implemented experimentally, we study the pressure driven Poiseuille flow of a shear banding fluid. After discussing the "basic states" predicted by a one-dimensional calculation that assumes a flat interface between the bands, we proceed to demonstrate such an interface to be unstable with respect to the growth of undulations along it. We give results for the growth rate and wavevector of the most unstable mode that grows initially, as well as for the ultimate flow patterns to which the instability leads. We discuss the relevance of our predictions to the present state of the experimental literature concerning interfacial instabilities of shear banded flows, in both conventional rheometers and microfluidic channels.

Abstract: Entangled polymer melts exhibit a variety of flow instabilities that limit production rates in industrial applications. We present both experimental and computational findings, using flow of monodisperse linear polystyrenes in a contraction--expansion geometry, which illustrate the formation and development of one such flow instability. This viscoelastic disturbance is first observed at the slit outlet and subsequently produces large-scale fluid motions upstream. A numerical linear stability study using the molecular structure based Rolie-Poly model confirms the instability and identifies important parameters within the model, which gives physical insight into the underlying mechanism. Chain stretch was found to play a critical role in the instability mechanism, which partially explains the effectiveness of introducing a low-molecular weight tail into a polymer blend to increase its processability.

Abstract: A parallel unstructured finite volume method is presented for analysis of the stability of two-dimensional steady Oldroyd-B fluid flows to small amplitude three-dimensional perturbations. A semi-staggered dilation-free finite volume discretization with Newton's method is used to compute steady base flows. The linear stability problem is treated as a generalized eigenvalue problem (GEVP) in which the rightmost eigenvalue determines the stability of the base flow. The rightmost eigenvalues associated with the most dangerous eigenfunctions are computed through the use of the shift-invert Arnoldi method. To avoid fine meshing in the regions where the flow variables are changing slowly, a local mesh refinement technique is used in order to increase numerical accuracy with a lower computational cost. The CUBIT mesh generation environment has been used to refine the quadrilateral meshes locally. In order to achieve higher performance on parallel machines the algebraic systems of equations resulting from the steady and the generalized eigenvalue problems (GEVP) have been solved by implementing the MUltifrontal Massively Parallel Solver (MUMPS). The proposed method is applied to the linear stability analysis of the flow of an Oldroyd-B fluid past a linear periodic array of circular cylinders in a channel and a linear array of circular half cylinders placed on channel walls. Two different leading eigenfunctions are identified for close and wide cylinder spacing for the periodic array of cylinders. The numerical results indicate good agreement with the numerical and experimental results available in the literature.

Abstract: The dilation-free semi-staggered finite volume method presented in Int. J. Numer. Meth. Fluids 49 (2005) 959-974 has been extended for the numerical solution of viscoelastic fluid flows on all-quadrilateral(2D)/hexahedral(3D) meshes. The velocity components are defined at element node points, while the pressure term and the extra stress tensor are defined at element centroids. The continuity equation is satisfied exactly within each element. An upwind least square method is employed for the calculation of the extra stresses at control volume faces in order to maintain stability for hyperbolic constitutive equations. The time stepping algorithm used decouples the calculation of the extra stresses from the evaluation of the velocity and pressure fields by solving a generalised Stokes problem. The resulting linear systems are solved using the GMRES method provided by the PETSc library with an ILU(k) preconditioner obtained from the HYPRE library. We apply the method to both two and three dimensional flow of an Oldroyd-B fluid past a confined circular cylinder in a channel with blockage ratio 0.5.

Abstract: The plastics industry today sees huge wastage through product defects caused by unstable flows during the manufacturing process. In addition, many production lines are throughput-limited by a flow speed threshold above which the process becomes unstable. It is therefore critically important to understand the mechanisms behind these instabilities.
In order to investigate the flow of a molten plastic, the first step is a model of the liquid itself, a relation between its current stress and its flow history called a constitutive relation. These are derived in many ways, and tested on several benchmark flows, but rarely is the stability of the model used as a criterion for selection. The relationship between the constitutive model and the stability properties of even simple flows is not yet well understood: we show that in one case a small change to the model, which does not affect the steady flow behaviour, entirely removes a known instability; in another, a change which makes a qualitative difference to the steady flow makes only tiny changes to the stability.
The long-term vision of this research is to quantify exactly what are the important properties of a constitutive relation as far as stability is concerned. If we could understand that, then not only could very simple stability experiments be used to choose the best constitutive models for a particular material, but our ability to predict and avoid wasteful industrial instabilities would be vastly improved.

Abstract: We consider the linear stability of shear banded planar Couette flow of the Johnson-Segalman fluid, with and without the addition of stress diffusion to regularise the equations. In particular, we investigate the linear stability of an initially one-dimensional "base" flow, with a flat interface between the bands, to two-dimensional perturbations representing undulations along the interface. We demonstrate analytically that, for the linear stability problem, the limit in which diffusion tends to zero is mathematically equivalent to a pure (non-diffusive) Johnson-Segalman model with a material interface between the shear bands, provided the wavelength of perturbations being considered is long relative to the (short) diffusion lengthscale.
For no diffusion, we find that the flow is unstable to long waves for almost all arrangements of the two shear bands. In particular, for any set of fluid parameters and shear stress there is some arrangement of shear bands that shows this instability. Typically the stable arrangements of bands are those in which one of the two bands is very thin. Weak diffusion provides a small stabilising effect, rendering extremely long waves marginally stable. However, the basic long-wave instability mechanism is not affected by this, and where there would be instability as wavenumber k tends to 0 in the absence of diffusion, we observe instability for moderate to long waves even with diffusion.
This paper is the first full analytical investigation into an instability first documented in the numerical study of [Fielding, PRL95, 134501 (2005)]. Authors prior to that work have either happened to choose parameters where long waves are stable or used slightly different constitutive equations and Poiseuille flow, for which the parameters for instability appear to be much more restricted.
We identify two driving terms that can cause instability: one, a jump in N1, as reported previously by Hinch et al. [Hinch, Harris & Rallison, JNNFM 43, 311-324 (1992)]; and the second, a discontinuity in shear rate. The mechanism for instability from the second of these is not thoroughly understood.
We discuss the relevance of this work to recent experimental observations of complex dynamics seen in shear-banded flows.

Abstract: The effects of interactions between bubbles of different sizes during bubble growth in a polymeric foam are investigated. Two models are used: a two-dimensional simulation in which both the effects of gas diffusion through the polymer and bubble interactions through fluid stresses are included, and a three-dimensional model in which bubbles are assumed to interact only through direct competition for gas, and diffusion of gas into the bubbles is instantaneous.
In the two-dimensional model, two different bubble sizes are used in a hexagonal array. For slow gas diffusion, the additional polymer stresses have little effect on the final bubble size distribution. For faster gas diffusion the growth occurs in two phases, just as was found in earlier work for isolated bubbles: an initial rapid viscous phase and a later phase controlled by the rate of polymer relaxation. In this later phase, polymers in the windows between neighbouring bubbles become highly stretched and these regions of high stress determine the dynamics of the growth.
In the three-dimensional model we consider the effects of rheology on a pair of different-sized spherical bubbles, interacting only through competition for available gas. Viscoelastic effects result in a wider distribution of bubble volumes than would be found for a Newtonian fluid.

Abstract: The effects of viscoelasticity on the expansion of gas bubbles arranged in a hexagonal array in a polymeric fluid are investigated. The expansion is driven by the diffusion of a soluble gas from the liquid phase, and the rate of expansion is controlled by a combination of gas diffusion, fluid rheology and surface tension.
In the diffusion limited case, the initial growth rate is slow due to small surface area, whereas at high diffusivity initial growth is rapid and resisted only by background solvent viscosity. In this high Deborah number limit, we see a two stage expansion in which there is an initial rapid expansion up to the size at which the elastic stresses balance the pressure difference. Beyond this time the bubble expansion is controlled by the relaxation of the polymer. We also illustrate how viscoelasticity affects the shape of the bubble.
In addition to a full finite element calculation of the two-dimensional flow, two one-dimensional approximations valid in the limits of small and large gas area fractions are presented. We show that these approximations give accurate predictions of the evolution of the bubble area, but give less accurate predictions of the bubble shape.

Abstract: The effect of particle-particle contact on the stress of a suspension of small spheres in plane strain flow is investigated. We provide an analytic form for the particle pair distribution function in the case of no Brownian motion, and calculate the viscosity and normal stress difference based on this. We show that the viscosity is reduced by contact, and a normal stress difference induced, both at order c² for small particle volume concentration c. In addition, we investigate the effect of a small amount of diffusion on the structure of the distribution function, giving a self-consistent form for the density in the O(a/Pe) boundary layer identified by Brady & Morris [JFM 348, 103-139 (1997)] and demonstrating that diffusion reduces the magnitude of the contact effect but does not qualitatively alter it.

Abstract: Passage-time densities are important for the detailed performance analysis of distributed computer and communicating systems. We provide a proof and demonstration of a practical iterative algorithm for extracting complete passage-time densities from expressive semi-Markov systems. We end by showing its application to a distributed web-server cluster model of 15.9 million states.

Abstract: Passage time densities and quantiles are important performance and quality of service metrics, but their numerical derivation is, in general, computationally expensive. We present an iterative algorithm for the calculation of passage time densities in semi-Markov models, along with a theoretical analysis and empirical measurement of its convergence behaviour. In order to implement the algorithm efficiently in parallel, we use hypergraph partitioning to minimise communication between processors and to balance workloads. This enables the analysis of models with very large state spaces which could not be held within the memory of a single machine. We produce passage time densities and quantiles for very large semi-Markov models with over 15 million states and validate the results against simulation.

Abstract: Solid-solid contacts due to microscopic surface roughness in viscous fluids were examined by observing the translational and rotational behaviours of a suspended sphere falling past a lighter sphere or down an inclined surface. In both cases, a roll-slip behaviour was observed, with the gravitational forces balanced by not only hydrodynamic forces but also normal and tangential solid-solid contact forces. Moreover, the nominal separation between the surfaces due to microscopic surface roughness elements is not constant but instead varies due to multiple roughness scales. By inverting the system, so that the heavy sphere fell away from the lighter sphere or the plane, it was found that the average nominal separation increases with increasing angle of inclination of the plane or the surface of the lighter sphere from horizontal; the larger asperities lift the sphere up from the opposing surface and then gravity at large angles of inclination is too weak to pull the sphere back down to the opposing surface before another large asperity is encountered. The existence of microscopic surface roughness and solid-solid contacts is shown to modify the rheological properties of suspensions. For example, the presence of compressive, but not tensile, contact forces removes the reversibility of sphere-sphere interactions and breaks the symmetry of the particle trajectories. As a result, suspensions of rough spheres exhibit normal stress differences that are absent for smooth spheres. For the conditions studied, surface roughness reduces the effective viscosity of a suspension by limiting the lubrication resistance during near-contact motion, and it also modifies the suspension microstructure and hydrodynamic diffusivity.

Abstract: The effects of fluid viscoselasticity on the expansion of gas bubbles in polymer foams for the cases of reactive and non-reactive polymers are investigated. For non-reactive polymers, bubble expansion is controlled by a combination of gas diffusion and fluid rheology. In the diffusion limited case, the initial growth rate is small due to small surface area, whereas at high diffusivity initial growth is rapid and resisted only by background solvent viscosity. In this high Deborah number limit, we see a two stage expansion in which there is an initial rapid expansion up to the size at which the elastic stresses balance the pressure difference. Beyond this time the bubble expansion is controlled by the relaxation of the polymer.
In the model for reactive polymer systems the polymer molecules begin as a mono-disperse distribution of a single reacting species. As the reaction progresses molecules bond to form increasingly large, branched, structures with a spectrum of relaxation modes, which gel to form a viscoelastic solid. Throughout this process gas is produced as a by-product of the reaction. The linear spectrum for this model is calculated from Rubinstein, Colby and Gillmor's "Dynamics Scaling For Polymer Gelation" in "Space-time Organisation in Macromolecular Fluids" (editors F Tanaka, M Doi and T Ohta) pages 66-74, where the relaxation spectrum of a molecule is obtained from percolation theory and Rouse dynamics. We discretize this linear spectrum and, treating each mode as a mode in a multimode Oldroyd B fluid obtain a model for the non-linear rheology. Using this model we describe how the production of gas, diffusion of gas through the liquid, and evolution of the largest molecule are coupled to bubble expansion and stress evolution. Thus we illustrate how the rate of gas production, coupled to the rate of gas diffusion, affects the bubble size within a foam.

Abstract: We consider viscous shear flow of a monolayer of solid spheres and discuss the effect that microscopic particle surface roughness has on the stress in the suspension. We consider effects both within and outside the dilute régime. Away from jamming concentrations, the viscosity is lowered by surface roughness, and for dilute suspensions it is insensitive to friction between the particles. Outside the dilute region, the viscosity increases with increasing friction coefficient. For a dilute system, roughness causes a negative first normal stress difference (N₁) at order c² in particle area concentration. The magnitude of N₁ increases with increasing roughness height in the dilute limit but the trend reverses for more concentrated systems. N₁ is largely insensitive to interparticle friction. The dilute results are in accord with the three-dimensional results of our earlier work (Wilson & Davis 2000), but with a correction to the sign of the tangential friction force.

Abstract: We consider the flow of a dilute suspension of equisized solid spheres in a viscous fluid. The viscosity of such a suspension is dependent on the volume fraction, c, of solid particles. If the particles are perfectly smooth, then solid spheres will not come into contact, because lubrication forces resist their approach. In this paper, however, we consider particles with microscopic surface asperities such that they are able to make contact. For straining motions we calculate the O(c²) coefficient of the resultant viscosity, due to pairwise interactions. For shearing motions (for which the viscosity is undetermined because of closed orbits on which the probability distribution is unknown) we calculate the c² contribution to the normal stresses N1 and N2. The viscosity in strain is shown to be slightly lower than that for perfectly smooth spheres, though the increase in the O(c) term caused by the increased effective radius due to surface asperities will counteract this decrease. The viscosity increases with increasing contact friction coefficient. The normal stresses N1 and N2 are zero if the surface roughness height is less than a critical value of 0.000211 times the particle radius, and then become negative as the roughness height is increased above this value. N1 is larger in magnitude than N2.

Abstract: Collision efficiencies are considered for colloidal suspensions of solid spheres moving in a viscous fluid under the influence of electrophoresis or gravity, Brownian motion, and electrostatic and van der Waals forces. The results are compared to those for convection (electrophoresis or gravity) and diffusion (Brownian motion) acting independently. The collision efficiency increases by many orders of magnitude over that predicted by simply adding diffusive and convective efficiencies in a specific parameter régime. This régime occurs when there is a large energy barrier in the interparticle potential, causing a stable region of parameter space if there is no diffusion. Brownian motion alone will only cause small amounts of aggregation under these conditions. However, for electric fields or buoyancy effects which are only slightly too weak to allow particles to overcome the potential barrier, the addition of weak Brownian motion to a system with convection can cause significant numbers of particles to overcome the energy barrier and aggregate.

Abstract: We consider the inertialess planar channel flow of a White-Metzner (WM) fluid having a power-law viscosity with exponent n. The case n=1 corresponds to an Upper Convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength 1/k. We find numerically that if n < nc ~ 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to nc, this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds nc, all disturbances are linearly stable. We consider asymptotically both the long wave limit which is stable for all n, and the short wave limit for which waves grow or decay at a finite rate independent of k for each n.
The mechanism of this elastic shear-thinning instability is discussed.

Abstract: This paper investigates inertialess channel flow of elastic liquids having continuously stratified constitutive properties. We find that an Oldroyd-B fluid having a sufficiently rapid normal stress variation shows instability. The mechanism is the same as for the two-fluid co-extrusion instability that arises when elasticity varies discontinuously. We find, using numerical and asymptotic methods, that this mechanism is opposed by convective effects, so that as the scale over which the elastic properties vary is increased, the growth rate is reduced, and finally disappears. A physical explanation for the stabilisation is given.
Regarding an Oldroyd-B fluid as a suspension of Hookean dumbbells, we show that a sufficiently steep variation in dumbbell concentration (with attendant rapid changes in both viscosity and elasticity) will provide an instability of the same kind.
Finally we show that Lagrangian convection of material properties (either polymer concentration or relaxation time) is crucial to the instability mechanism. A White-Metzner fluid having identical velocity and stress profiles in a channel flow is found to be stable. The implications for extrudate distortion, and constitutive modelling are briefly discussed.

Abstract: We provide a mathematical analysis of the spectrum of the linear stability problem for one and two layer channel flows of the upper-convected Maxwell (UCM) and Oldroyd-B fluids at zero Reynolds number. For plane Couette flow of the UCM fluid, it has long been known (Gorodtsov & Leonov 1967) that, for any given streamwise wave number, there are two eigenvalues in addition to a continuous spectrum. In the presence of an interface, there are seven discrete eigenvalues. In this paper, we investigate how this structure of the spectrum changes when the flow is changed to include a Poiseuille component, and as the model is changed from the UCM to the more general Oldroyd-B. For a single-layer UCM fluid, we find that the number of discrete eigenvalues changes from two in Couette flow to six in Poiseuille flow. The six modes are given in closed form in the long wave limit. For plane Couette flow of the Oldroyd-B fluid, we solve the differential equations in closed form. There is an additional continuous spectrum and a family of discrete modes. The number of these discrete modes increases indefinitely as the retardation time approaches zero. We analyze the behavior of the eigenvalues in this limit.

Abstract: The stability of channel flow of coextruded elastic liquids having matched viscosities but a jump in elastic properties is studied. Inertia and surface tension are neglected. A short wave disturbance is found, confined near the interface, whose growth rate is independent of wavelength. For dilute Oldroyd-B fluids this disturbance is unstable for any non-zero jump in normal stresses, and has a maximum growth rate for intermediate levels of elasticity in the two fluids. When one or other fluid is highly elastic the growth rate falls. In the concentrated limit (a UCM fluid), the disturbance is unstable only for a finite range of normal stress jumps, and is restabilized if one fluid is much more elastic than the other.
Because the short wave disturbance is localised, the results apply for any steadily sheared interface across which the normal stress jumps. The results are confirmed for moderate parameter values by means of a full numerical solution for a three-layer planar flow.

Abstract: The dissertation is concerned with the stability of channel flows of viscoelastic fluids. The content is primarily theoretical. The dissertation begins with a review of instabilities observed in experiments and then attempts to elucidate possible mechanisms using linear stability theory.
The first section considers a previously known interfacial instability in coextrusion flows, whose mechanism is purely elastic. This instability is investigated in different parameter régimes for an Oldroyd-B fluid.
The next section generalises the study to a continuously stratified fluid, and finds that a class of models with rapid variation in their elastic properties will also show the instability. These results are confirmed both numerically and using asymptotic methods. The fundamental mechanism of this "coextrusion" instability is the same as for the interfacial instability above.
The next part concerns a shear-thinning White-Metzner fluid (i.e. a viscoelastic fluid having a relaxation time that is an instantaneous function of the local shear-rate). Evidence for another instability is found where the degree of thinning in the shear viscosity is high. The mechanism for this instability is fundamentally different from that in coextrusion.
In the final section of the dissertation a study of two fluids of different constitutive types but identical base-state velocity and stress profiles shows that the criterion for the `coextrusion' instability depends on properties of the model itself.
The flows in question are relevant to the practical problem of extrusion of polymeric liquids. The two instabilities found may provide mechanisms for experimental observations of helical distortions of extrudates. The demonstration that the constitutive type of a model has a crucial effect on its stability may have implications for future constitutive modelling in this field.