Jonathan J. Wylie

Rheology of suspensions with high particle inertia and moderate fluid inertia

We consider the averaged flow properties of a suspension in which the Reynolds number based on the particle diameter is finite so that the inertia of the fluid phase is important. When the inertia of the particles is sufficiently large, their trajectories, between successive particle collisions, are only weakly affected by the interstitial fluid. If the particle collisions are nearly elastic the particle velocity distribution is close to an isotropic Maxwellian. The rheological properties of the suspension can then be determined using kinetic theory, provided that one knows the granular temperature (energy contained in the particle velocity fluctuations). This energy results from a balance of the shear work with the loss due to the viscous dissipation in the interstitial fluid and the dissipation due to inelastic collisions. We use lattice-Boltzmann simulations to calculate the viscous dissipation as a function of particle volume fraction and Reynolds number (based on the particle diameter and granular temperature). The Reynolds stress induced in the interstitial fluid by the random motion of the particles is also determined. We also consider the case where the interstitial fluid is moving relative to the particles, as would occur if the particles experienced an external body force. Owing to the nonlinearity of the equations of motion for the interstitial fluid, there is a coupling between the viscous dissipation caused by the fluctuating motion of the particles and the drag associated with a mean relative motion of the two phases, and this coupling is explored by computing the dissipation and mean drag for a range of values of the Reynolds numbers based on the mean relative velocity and the granular temperature.

Particle clustering due to hydrodynamic interactions

Dynamic simulations of an isotropic suspension of particles in a viscous gas are performed. The energy in the suspension decays with time as a result of viscous dissipation in the gas. The rate of viscous dissipation in sufficiently energetic suspensions, those with a high Stokes number, is consistent with the theory of Sangani et al. [J. Fluid Mech. 313, 309–341 (1996)] for homogeneous hard-sphere suspensions. As the suspension loses energy, the dissipation rate decreases dramatically and the particles cluster as indicated by the presence of many more near neighbors than would be found in a hard sphere distribution.

A Mathematical Model of the Metabolic and Perfusion Effects on Cortical Spreading Depression

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism—aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium–ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD—vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization—found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences

Let (X i ) ι =1 be a possibly nonstationary sequence such that L(X i ) = P n if i ≤ n6 and L(X i ) = Q n if > nθ, where 0 < θ < 1 is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions F. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the 1/n rate is achieved for a wide class of processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.

The effects of temperature-dependent viscosity on flow in a cooled channel with application to basaltic fissure eruptions

A theoretical description is given of pressure-driven viscous flow of an initially hot fluid through a planar channel with cold walls. The viscosity of the fluid is assumed to be a function only of its temperature. If the viscosity variations caused by the cooling of the fluid are sufficiently large then the relationship between the pressure drop and the flow rate is non-monotonic and there can be more than one steady flow for a given pressure drop. The linear stability of steady flows to two-dimensional and three-dimensional disturbances is calculated. The region of instability to two-dimensional disturbances corresponds exactly to those flows in which an increase in flow rate leads to a decrease in pressure drop. At higher viscosity contrasts some flows are most unstable to three-dimensional (fingering) instabilities analogous, but not identical, to Saffman-Taylor fingering. A cross-channel-averaged model is derived and used to investigate the finite-amplitude evolution.

Thermal instability in drawing viscous threads

We consider the stretching of a thin viscous thread, whose viscosity depends on temperature, that is heated by a radiative heat source. The thread is fed into an apparatus at a fixed speed and stretched by imposing a higher pulling speed at a fixed downstream location. We show that thermal effects lead to the surprising result that steady states exist for which the force required to stretch the thread can decrease when the pulling speed is increased. By considering the nature of the solutions, we show that a simple physical mechanism underlies this counterintuitive behaviour. We study the stability of steady-state solutions and show that a complicated sequence of bifurcations can arise. In particular, both oscillatory and non-oscillatory instabilities can occur in small isolated windows of the imposed pulling speed.

Extensional flows with viscous heating

In this paper we investigate the role played by viscous heating in extensional flows of viscous threads with temperature-dependent viscosity. We show that there exists an interesting interplay between the effects of viscous heating, which accelerates thinning, and inertia, which prevents pinch-off. We first consider steady drawing of a thread that is fed through a fixed aperture at given speed and pulled with a constant force at a fixed downstream location. For pulling forces above a critical value, we show that inertialess solutions cannot exist and inertia is crucial in controlling the dynamics. We also consider the unsteady stretching of a thread that is fixed at one end and pulled with a constant force at the other end. In contrast to the case in which inertia is neglected, the thread will always pinch at the end where the force is applied. Our results show that viscous heating can have a profound effect on the final shape and total extension at pinching.

Drag-induced particle segregation with vibrating boundaries

We consider a system composed of two different types of particles that have different radii, but equal density. Both particles experience gravity and a linear drag force from the interstitial fluid. They are excited by a boundary that vibrates with high frequency and adds sufficient energy that the particles near the boundary become highly dilated. For moderate energy input rates we show that a single large particle introduced into a large number of small particles will rapidly move to a fixed height and remain approximately stationary. In particular, the large particle will never come into contact with the vibrating base. If there are a large number of large particles, then this behavior leads to a very distinct segregation in which the large particles are sandwiched between two layers of small particles. We show that this behavior occurs as a direct result of non-equilibrium effects and develop a simple phenomenological model that gives good predictions of the height at which the sandwiched layer occurs.

On the Formation of Glass Microelectrodes

Glass microelectrodes are used widely in experimental studies of the electrophysiology of biological cells and their membranes. However, the pulling of these electrodes remains an art, based on trial and error. Following Huang et al. [SIAM J. Appl. Math., 63 (2003), pp. 1499–1519], we derive a one‐dimensional model for the stretching of a hollow glass tube that is being radiatively heated. Our framework allows us to consider two commonly used puller designs, that is, horizontal (constant force) and vertical (variable force) pullers. We derive explicit solutions and use these solutions to identify the principal factors that control the final shape of the microelectrodes. The design implications for pullers also are discussed.

Mathematical approaches to modeling of cortical spreading depression

Migraine with aura (MwA) is a debilitating disease that afflicts about 25%-30% of migraine sufferers. During MwA, a visual illusion propagates in the visual field, then disappears, and is followed by a sustained headache. MwA was conjectured by Lashley to be related to some neurological phenomenon. A few years later, Leão observed electrophysiological waves in the brain that are now known as cortical spreading depression (CSD). CSD waves were soon conjectured to be the neurological phenomenon underlying MwA that had been suggested by Lashley. However, the confirmation of the link between MwA and CSD was not made until 2001 by Hadjikhani et al. [Proc. Natl. Acad. Sci. U.S.A. 98, 4687-4692 (2001)] using functional MRI techniques. Despite the fact that CSD has been studied continuously since its discovery in 1944, our detailed understandings of the interactions between the mechanisms underlying CSD waves have remained elusive. The connection between MwA and CSD makes the understanding of CSD even more compelling and urgent. In addition to all of the information gleaned from the many experimental studies on CSD since its discovery, mathematical modeling studies provide a general and in some sense more precise alternative method for exploring a variety of mechanisms, which may be important to develop a comprehensive picture of the diverse mechanisms leading to CSD wave instigation and propagation. Some of the mechanisms that are believed to be important include ion diffusion, membrane ionic currents, osmotic effects, spatial buffering, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. Discrete and continuum models of CSD consist of coupled nonlinear differential equations for the ion concentrations. In this review of the current quantitative understanding of CSD, we focus on these modeling paradigms and various mechanisms that are felt to be important for CSD.