Simon J. Cox

Rheology plays a critical role across science and engineering, affecting sectors from healthcare to energy. In the UK, rheological research supports innovation in both academia and industry. This review, coordinated by the British Society of Rheology (BSR), provides an updated overview of current research directions across the UK and beyond.

Motivated by the treatment of varicose veins with aqueous foams, we determine the velocity profiles and yield surface positions for pressure-driven flows of a Bingham fluid in curved channels, incorporating Navier wall slip to capture the slip behavior of aqueous foams. Assuming a constant pressure gradient – consistent with syringe-driven injection in sclerotherapy – we reproduce a closed-form analytic solution for the flow in a straight channel, and extend this to a uniformly curved channel consisting of a section of an annulus. We find that increasing the slip length in the curved channel moves the unyielded fluid, or plug, towards the inside of the channel and slightly decreases the plug width. Channels with a change in curvature are tackled numerically, with a procedure that is validated against the analytic results. We solve for the flow of a Bingham fluid in the transition region between a straight and a uniformly curved channel and in a non-uniform sinusoidal channel where the wall curvature changes continuously. We find that there is greater yielding of the fluid in curved channels, caused by an increase in the shear stresses. As the slip length increases, there is a reduction in the area of unyielded fluid in regions of high wall curvature. In the context of varicose vein sclerotherapy, our results suggest that slip has both positive and negative consequences: it eliminates the possibility of static regions of foam developing in areas of high curvature in the vein, but also reduces the plug width in general, affecting the efficacy with which the foam displaces blood from the vein.

Aqueous foams are subject to coarsening, whereby gas from the bubbles diffuses through the liquid phase. Gas is preferentially transported from small to large bubbles, resulting in a gradual decrease of the number of bubbles and an increase in the average bubble size. Coarsening foams are expected to approach a scaling state at late times in which their statistical properties are invariant. However, a model predicting the experimentally observed bubble-size distribution in the scaling state of foams with moderate liquid content, as a function of the liquid fraction ϕ, has not yet been developed. To this end, we propose a three-dimensional mean-field bubble growth law for foams without inter-bubble adhesion, validated against bubble-scale simulations, and use it to derive a prediction of the scaling-state bubble-size distribution for any ϕ from zero up to the unjamming transition ϕ_c ≈ 36%. We verify that the derived scaling state is approached from a variety of initial conditions using mean-field simulations implementing the proposed growth law. Comparing our predicted bubble-size distribution with previous simulations and experimental results, we likewise find a large population of small bubbles when ϕ > 0, but there are qualitative differences from prior results which we attribute to the absence of rattlers, i.e. bubbles not pressed into contact with their neighbours, in our model.

Mutually repelling particles spontaneously form ordered clusters when forced into confinement. The clusters may adopt similar spatial arrangements even if the underlying particle interactions are contrastingly different. Here, we demonstrate with both simulations and experiments that it is possible to induce particles of very different types to self-assemble into the same ordered geometric structure by simply regulating the ratio between the repulsive and confining forces. This is the case for both long- and short-ranged potentials. This property is initially explored in systems with two-dimensional circular symmetry and subsequently demonstrated to be valid throughout the transition to one-dimensional structures through continuous elliptical deformations of the confining field. We argue that this feature can be utilized to manipulate the spatial structure of confined particles, thereby paving the way for the design of clusters with specific functionalities.

Microfluidic devices offer unique opportunities to directly observe multiphase flow in porous media. However, as a representation of flow in geological pore networks, conventional microfluidics face several challenges. One is whether steady simultaneous two-phase flow through a two-dimensional network is possible without fluctuating occupancy of the pore constrictions. Flow without fluctuations can occur only if the flow paths of the two phases can cross on the 2D network; this requires that wetting phase can form a bridge across the gap between grains at a pore constriction while non-wetting phase flows through the constriction. We consider the conditions under which this is possible as a function of the local capillary pressure and the geometry of the constriction. Using the Surface Evolver software, we determine conditions for stable interfaces in constricted geometries, the range of capillary pressures at which bridging can occur, and those where the wetting phase would re-invade the constriction to block the flow of the non-wetting phase (“snap-off”). If a constriction is long and either straight or uniformly curved, snap-off occurs at the same capillary pressure as bridging. For constrictions of concave shape, which we represent as constrictions between cylindrical grains, however, we find a range of capillary pressures at which bridging is stable; the range is greater the narrower the diameter of the cylinders (i.e. the more strongly concave the throat) relative to the width of the constriction. For smaller-diameter pillars, the phenomenon of “Roof” snap-off as non-wetting phase invades a downstream pore body, is predicted not to occur.

The collapse of a catenoidal soap film when the rings supporting it are moved beyond a critical separation is a classic problem in interface motion in which there is a balance between surface tension and the inertia of the surrounding air, with film viscosity playing only a minor role. Recently [Goldstein et al., Phys. Rev. E, 2021, 104, 035105], we introduced a variant of this problem in which the catenoid is bisected by a glass plate located in a plane of symmetry perpendicular to the rings, producing two identical hemicatenoids, each with a surface Plateau border (SPB) on the glass plate. Beyond the critical ring separation, the hemicatenoids collapse in a manner qualitatively similar to the bulk problem, but their motion is governed by the frictional forces arising from viscous dissipation in the SPBs. We present numerical studies of a model that includes classical laws in which the frictional force f_v for SPB motion on wet surfaces is of the form f_v ∼ Caⁿ, where Ca is the capillary number. Our experimental data on the temporal evolution of this process confirms the expected value n = 2/3 for mobile surfactants and stress-free interfaces. This study can help explain the fragmentation of bubbles inside very confined geometries such as porous materials or microfluidic devices.

We apply a combination of the Viscous Froth model and a surfactant transfer model [Zaccagnino et al., Phys. Rev. E, 2018, 98, 022801] to predict the rheological response of a two-dimensional dry aqueous foam. The model includes both the effect of friction between the foam and the boundaries of the container and also the dissipative effects on the film interfaces caused by surfactant motion. These dynamics are characterized by two free parameters: the Gibbs elasticity, relating surfactant concentration to interfacial tension, and the mobility of the surfactant molecules on the interfaces. We employ numerical simulations to evaluate the static shear modulus, yield stress and the storage and loss moduli of a foam and investigate the effect of our free parameters on these rheological properties.