Capillary Agglutination Technology

In medical diagnostic tests, including pregnancy testing and tests for typed red blood cells, a small fluid sample is placed at one end of a capillary channel, which has been lined with a dried reagent. If the sample contains the analyte (the substance being tested for) then an agglutination reaction occurs between it and the reagent in the channel, and the agglutinated complexes progressively slow the flow and may even block the channel, partially or completely, so that the flow only reaches the far end very slowly, or not at all. The aim is that this should give a reliable detection of quite low concentrations of analyte in the sample. Platform Diagnostics asked the Study Group to construct a mathematical model of the process, so that, for known binding forces in the agglutination complexes, we can design the channel size and shape, and the fluid viscosity, to maximize the reliable detection of low concentrations. A key question is how the flow time depends on channel size, fluid surface tension and viscosity, (a) in the absence of agglutination, and (b) in the presence of agglutination

Similarity solutions to an averaged model for superconducting vortex motion

Under certain conditions the motion of superconducting vortices is primarily governed by an instability. We consider an averaged model, for this phenomenon, describing the motion of large numbers of such vortices. The model equations are parabolic, and, in one spatial dimension x, take the formH2t = ? (|H3H2x ? H2H3x|H2x), ?xH3t = ? (|H3H2x ? H2H3x|H3x). ?xwhere H2 and H3 are components of the magnetic field in the y and z directions respectively. These equations have an extremely rich group of symmetries and a correspondingly large class of similarity reductions. In this work, we look for non-trivial steady solutions to the model, deduce their stability and use a numerical method to calculate time-dependent solutions. We then apply Lie Group based similarity methods to calculate a complete catalogue of the model’s similarity reductions and use this to investigate a number of its physically important similarity solutions. These describe the short time response of the superconductor as a current or magnetic field is switched on (or off).<br/

Vortex motion in shallow water with varying bottom topography and zero Froude number

The methods of formal matched asymptotics are used to investigate the motion of a vortex in shallow inviscid fluid of varying depth and zero Froude number in the limit as the vortex core radius tends to zero. To leading order the vortex is driven by local gradients in the logarithm of the depth along an isobath (or depth contour). A further term in the vortex velocity is calculated in which effects arising from the global bottom topography, other vortices and the vortex core structure appear. The evolution of the vortex core structure is then calculated. A point-vortex model is formulated which describes the motion of a number of small vortices in terms of the bottom topography, the inviscid flows around the vortices and their evolving core structure. A numerical method for solving this model is discussed and finally some numerical simulations of the motion of vortex pairs over a varying bottom topography are presente

The bifurcation structure of a thin superconducting loop with small variations in its thickness

We study bifurcations between the normal and superconducting states, and between superconducting states with different winding numbers, in a thin loop of superconducting wire, of uniform thickness, to which a magnetic field is applied. We then consider the response of a loop with small thickness variations. We find that close to the transition between normal and superconducting states lies a region where the leading order problem has repeated eigenvalues. This leads to a rich structure of possible behaviours. A weakly nonlinear stability analysis is conducted to determine which of these behaviours occur in practice

Motion by curvature of a three-dimensional filament: similarity solutions

We systematically classify and investigate fully three-dimensional similarity solutions to a system of equations describing the motion of a filament moving in the direction of its principle normal with velocity proportional to its curvature, ν = κn, where n is the principle normal and κ the curvature of the filament. Such formulations are relevant to superconducting vortices and disclinations

Uncertainty of flow in porous media

The problem posed to the Study Group was, in essence, how to estimate the probability distribution of f(x) from the probability distribution of x. Here x is a large vector and f is a complicated function which can be expensive to evaluate. For Schlumberger's applications f is a computer simulator of a hydrocarbon reservoir, and x is a description of the geology of the reservoir, which is uncertain

A one-dimensional model for superconductivity in a thin wire of slowly varying cross-section

Using formal asymptotics, a one–dimensional Ginzburg–Landau model describing superconductivity in a thin wire of arbitrary shape and slowly varying cross-section is derived. The model is valid for all magnetic fields and for temperatures T, such that the thickness of the wire is much less than the coherence length ξT. The model is used to calculate the normal–superconducting transition curves for closed wire loops of different cross-sections, as functions of temperature and the magnetic flux cutting the loop. This shows a periodic dependence on flux, superimposed on a parabolic background

A mathematical model for mechanically-induced deterioration of the binder in lithium-ion electrodes

This study is concerned with modeling detrimental deformations of the binder phase within lithium-ion batteries that occur during cell assembly and usage. A two-dimensional poroviscoelastic model for the mechanical behavior of porous electrodes is formulated and posed on a geometry corresponding to a thin rectangular electrode, with a regular square array of microscopic circular electrode particles, stuck to a rigid base formed by the current collector. Deformation is forced both by (i) electrolyte absorption driven binder swelling, and; (ii) cyclic growth and shrinkage of electrode particles as the battery is charged and discharged. The governing equations are upscaled in order to obtain macroscopic effective-medium equations. A solution to these equations is obtained, in the asymptotic limit that the height of the rectangular electrode is much smaller than its width, that shows the macroscopic deformation is one-dimensional. The confinement of macroscopic deformations to one dimension is used to obtain boundary conditions on the microscopic problem for the deformations in a 'unit cell' centered on a single electrode particle. The resulting microscale problem is solved using numerical (finite element) techniques. The two different forcing mechanisms are found to cause distinctly different patterns of deformation within the microstructure. Swelling of the binder induces stresses that tend to lead to binder delamination from the electrode particle surfaces in a direction parallel to the current collector, whilst cycling causes stresses that tend to lead to delamination orthogonal to that caused by swelling. The differences between the cycling-induced damage in both: (i) anodes and cathodes, and; (ii) fast and slow cycling are discussed. Finally, the model predictions are compared to microscopy images of nickel manganese cobalt oxide cathodes and a qualitative agreement is found.Comment: 25 pages, 11 figure

The evolution of space curves by curvature and torsion

We apply Lie group based similarity methods to the study of a new, and widely relevant, class of objects, namely motions of a space curve. In particular, we consider the motion of a curve evolving with a curvature kappa and torsion tau dependent velocity law. We systematically derive the Lie point symmetries of all such laws of motion and use these to catalogue all their possible similarity reductions. This calculation reveals special classes of law with high degrees of symmetry (and a correspondingly large number of similarity reductions). Of particular note is one class which is invariant under general linear transformations in space. This has potential applications in pattern and signal recognition

Classification of phase transitions in thin structures with small Ginzburg-Landau parameter

Thin superconducting structures are considered. We compute the limit where the thickness and the Ginzburg-Landau parameter tend simultaneously to zero with a preferred scaling. The new equations enable us to divide the parameter space into regimes of first order or second order phase transition. The results are discussed in light of recent experiments