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UCL ›› Department of Mathematics ›› Prof Helen J Wilson ›› Publications ›› Abstracts |

UNDER REVIEW

Taylor Couette instability in disk suspensions

Reference: . Taylor Couette instability in disk suspensions. Physical Review Fluids (submitted).

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.

Paper: 2018-GWa.pdf (536525 bytes).

PUBLISHED OR IN PRESS

A Model for Sphere Suspension Microstructure and Stress

Reference: . Modelling Sphere Suspension Microstructure and Stress. Physical Review E (accepted).

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.

Paper: 2018-GWb.pdf (352463 bytes).

Small- and large-amplitude oscillatory rheometry with bead–spring dumbbells in Stokesian Dynamics to mimic viscoelasticity

Reference: . Small- and large-amplitude oscillatory rheometry with bead–spring dumbbells in Stokesian Dynamics to mimic viscoelasticity. Journal of Non-Newtonian Fluid Mechanics (in press).

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.

Paper: 2018-TW.pdf (1606485 bytes).

Elastic instabilities in pressure-driven channel flow of thixotropic-viscoelasto-plastic fluids

Reference: . Elastic instabilities in pressure-driven channel flow of thixotropic-viscoelasto-plastic fluids. Journal of Non-Newtonian Fluid Mechanics (in press).

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.

Anomalous effect of turning off long-range mobility interactions in Stokesian Dynamics

Reference: . Anomalous effect of turning off long-range mobility interactions in Stokesian Dynamics. Physics of Fluids, vol. 30, 077103, 2018.

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.

"Shear thickening" in non-shear flows: the effect of microstructure

Reference: . Frictional shear thickening in suspensions: The effect of rigid asperities. Journal of Fluid Mechanics, vol. 836, pp. 1-4, 2018.

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.

Frictional shear thickening in suspensions: The effect of rigid asperities

Reference: . Frictional shear thickening in suspensions: The effect of rigid asperities. Physics of Fluids, vol. 29, 121607, 2017.

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.

Paper: 2017-TWb.pdf (2948645 bytes) ; doi: 10.1063/1.4989929. This paper also comes with a scilight by Rachele Hendricks-Stirrup.

Simulations of a heavy ball falling through a sheared suspension

Reference: . Simulations of a heavy ball falling through a sheared suspension. Journal of Engineering Mathematics, vol. 107 (1), pp. 179-200, 2017.

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.

Towards a mechanism for instability in channel flow of highly shear-thinning viscoelastic fluids

Reference: . Towards a mechanism for instability in channel flow of highly shear-thinning viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics, vol. 247, pp. 15-21, 2017.

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.

Shear stress of a monolayer of rough spheres – CORRIGENDUM

Reference: . Shear stress of a monolayer of rough spheres – CORRIGENDUM. Journal of Fluid Mechanics, vol. 814, pp. 614-617, 2017.

Abstract: We have recently discovered that the implicit assumption that *N*_{2} = 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 *N*_{1} is now observed to change sign and become positive for concentrated systems.

The fluid dynamics of the chocolate fountain

Reference: . The fluid dynamics of the chocolate fountain. European Journal of Physics, vol. 37, 015803, 2016.

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.

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.

Linear instability of a highly shear-thinning fluid in channel flow

Reference: . Linear instability of a highly shear-thinning fluid in channel flow. Journal of Non-Newtonian Fluid Mechanics, vol. 223, pp. 200-208, 2015.

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.

We compare with microchannel experiments (Bodiguel

Stokes flow past three spheres

Reference: . Stokes flow past three spheres. Journal of Computational Physics, vol. 245, pp. 302-316, 2013.

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.

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.

Paper: 2013-W.pdf (179796 bytes)) or from UCL Discovery or Elsevier version;
doi: 10.1016/j.jcp.2013.03.020.

Rolie-Poly fluid flowing through constrictions: Two distinct instabilities

Reference: . Rolie-Poly fluid flowing through constrictions: Two distinct instabilities. Journal of Non-Newtonian Fluid Mechanics, vol. 195, pp. 77-87, 2013.

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.

Open mathematical problems regarding non-Newtonian fluids

Reference: . Open mathematical problems regarding non-Newtonian fluids. Nonlinearity, vol. 25(3), pp. R45, 2012.

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.

Shear banding and interfacial instability in planar Poiseuille flow

Reference: . Shear banding and interfacial instability in planar Poiseuille flow. Journal of Non-Newtonian Fluid Mechanics, 165, pp. 196-202, 2010.

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.

Molecular physics of a polymer engineering instability: experiments and computation

Reference: . Molecular physics of a polymer engineering instability: experiments and computation. Physical Review E, 77(5), 050801, 2008.

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.

A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows

Reference: . A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows. Journal of Non-Newtonian Fluid Mechanics, vol. 155, pp. 1-14, 2008.

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 *MU*ltifrontal *M*assively *P*arallel *S*olver (*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.

A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements

Reference: . A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements. Journal of Non-Newtonian Fluid Mechanics, vol. 147, pp. 79-91, 2007.

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.

Instabilities and constitutive modelling

Reference: . Instabilities and constitutive modelling. Philosophical Transactions of the Royal Society A, vol. 364, pp. 3267-3283, 2006.

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.

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.

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson-Segalman fluids

Reference: . Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson-Segalman fluids. Journal of Non-Newtonian Fluid Mechanics, vol. 138, pp. 181-196, 2006.

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.

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

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

We discuss the relevance of this work to recent experimental observations of complex dynamics seen in shear-banded flows.

Competition and interaction of polydisperse bubbles in polymer foams

Reference: . Competition and interaction of polydisperse bubbles in polymer foams. Journal of Non-Newtonian Fluid Mechanics, vol. 137, pp. 60-71, 2006.

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.

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.

Bubble growth in a two-dimensional viscoelastic foam

Reference: . Bubble growth in a two-dimensional viscoelastic foam. Journal of Non-Newtonian Fluid Mechanics, vol. 137, pp. 46-59, 2006.

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.

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.

An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow

Reference: . An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow. Journal of Fluid Mechanics, vol. 534, pp. 97-114, 2005.

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.

Iterative convergence of passage-time densities in semi-Markov performance models

Reference: . Iterative convergence of passage-time densities in semi-Markov performance models. Performance Evaluation, vol. 60, pp. 237-254, 2005.

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.

Hypergraph-based Parallel Computation of Passage Time Densities in Large Semi-Markov Models

Reference: . Hypergraph-based Parallel Computation of Passage Time Densities in Large Semi-Markov Models. Journal of Linear Algebra and Applications, vol. 386(C), pp. 311-334, 2004.

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.

Solid-solid contacts due to surface roughness and their effects on suspension behaviour

Reference: . Solid-solid contacts due to surface roughness and their effects on suspension behaviour. Philosophical Transactions: Mathematical, Physical & Engineering Sciences, vol. 361, pp. 871-894, 2003.

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.

Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams

Reference: . Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams. Journal of Non-Newtonian Fluid Mechanics, vol. 114, pp. 83-107, 2003.

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.

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.

Shear stress of a monolayer of rough spheres

Reference: . Shear stress of a monolayer of rough spheres. Journal of Fluid Mechanics, vol. 452, pp. 425-441, 2002.

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 (*N*_{1}) at order *c²* in particle area concentration. The magnitude of *N*_{1} increases with increasing roughness height in the dilute limit but the trend reverses for more concentrated systems. *N*_{1} 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.

Paper: [ original version (343393 bytes) | corrigendum (8384 bytes) ]; [ Original doi: 10.1017/S0022112001006838 | corrigendum doi: 10.1017/jfm.2017.29 ].

The viscosity of a dilute suspension of rough spheres

Reference: . The viscosity of a dilute suspension of rough spheres. Journal of Fluid Mechanics, vol. 421, pp. 339-367, 2000.

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*.

Paper: [original version (555941 bytes) | errata (61076 bytes) | corrected version (267680 bytes) ]; doi: 10.1017/S0022112000001695.

Aggregation of charged particles under electrophoresis or gravity at arbitrary Péclet numbers

Reference: . Aggregation of charged particles under electrophoresis or gravity at arbitrary Péclet numbers. Journal of Colloid and Interface Science, vol. 221, pp. 87-103, 2000.

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.

Instability of channel flow of a shear-thinning White-Metzner fluid

Reference: . Instability of channel flow of a shear-thinning White-Metzner fluid. Journal of Non-Newtonian Fluid Mechanics, vol. 87, pp. 75-96, 1999.

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.

The mechanism of this elastic shear-thinning instability is discussed.

Instability of channel flows of elastic liquids having continuously stratified properties

Reference: . Instability of channel flows of elastic liquids having continuously stratified properties. Journal of Non-Newtonian Fluid Mechanics, vol. 85, pp. 273-298, 1999.

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.

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

Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids

Reference: . Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. Journal of Non-Newtonian Fluid Mechanics, vol. 80, pp. 251-268, 1999.

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.

Short wave instability of co-extruded elastic liquids with matched viscosities

Reference: . Short wave instability of co-extruded elastic liquids with matched viscosities. Journal of Non-Newtonian Fluid Mechanics, vol. 72, pp. 237-251, 1997.

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.

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.

PHD THESIS

Shear Flow Instabilities in Viscoelastic Fluids

Reference: . Shear Flow Instabilities in Viscoelastic Fluids. PhD Thesis, University of Cambridge, 1998.

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.

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 (

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.

Thesis: PhDThesis.pdf (1771028 bytes)

Last updated: Fri 7 September 2018