Helen J. Wilson

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

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 (Gorodstov and Leonov, J. Appl. Math. Mech. (PMM) 31 (1967) 310) 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.

The viscosity of a dilute suspension of rough spheres

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 decreases 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.

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

Abstract The effects of fluid viscoelasticity 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 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 (De) 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 each 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 fluid model is calculated from Rubinstein et al. [Dynamic scaling for polymer gelation, in: F. Tanaka, M. Doi, T. Ohta (Eds.), Space–Time Organisation in Macromolecular Fluids, Springer, Berlin, 1989, pp. 66–74], where the relaxation spectrum of a molecule is obtained from percolation theory and Rouse dynamics. We discretise this linear spectrum and, by 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.

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 regime. 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.

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

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→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 [S.M. Fielding, Linear instability of planar shear banded flow, Phys. Rev. Lett. 95 (2005) 134501]. 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. [E.J. Hinch, O.J. Harris, J.M. Rallison, The instability mechanism for two elastic liquids being coextruded, J. Non-Newtonian Fluid Mech. 43 (1992) 311–324]; 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.

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

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.

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

Abstract The stability of channel flow of coextruded elastic liquids having matched viscosities ties 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 restabilised 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.

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

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 c(2) 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(aPe(-1)) boundary layer and demonstrating that diffusion reduces the magnitude of the contact effect but does not qualitatively alter it.

Stokes flow past three spheres

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.

Instability of channel flows of elastic liquids having continuously stratified properties

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.