We theoretically and experimentally study nematic liquid crystal equilibria within shallow rectangular wells. We model the wells within a two-dimensional Oseen-Frank framework, with strong tangent anchoring, and obtain explicit analytical expressions for the director fields and energies of the 'diagonal' and 'rotated' solutions reported in the literature. These expressions separate the leading-order defect energies from the bulk distortion energy for both families of solutions. The continuum Oseen-Frank study is complemented by a microscopic mean-field approach. We numerically minimize the mean-field functional, including the effects of weak anchoring, variable order and random initial conditions. In particular, these simulations suggest the existence of higher-energy metastable states with internal defects. We compare our theoretical results to experimental director profiles, obtained using two types of filamentous virus particles, wild-type fd-virus and a modified stiffer variant (Y21M), which display nematic ordering in rectangular chambers, as found by confocal scanning laser microscopy. We combine our analytical energy expressions with experimentally recorded frequencies of the different equilibrium states to obtain explicit estimates for the extrapolation length, defined to be the ratio of the nematic elastic constant to the anchoring coefficient, of the fd-virus.

We study planar nematic equilibria on a two-dimensional annulus with strong and weak tangent anchoring, in the Oseen-Frank theoretical framework. We analyze a radially invariant defect-free state and compute analytic stability criteria for this state in terms of the elastic anisotropy, annular aspect ratio, and anchoring strength. In the strong anchoring case, we define and characterize a new spiral-like equilibrium which emerges as the defectfree state loses stability. In the weak anchoring case, we compute stability diagrams that quantify the response of the defect-free state to radial and azimuthal perturbations. We study sector equilibria on sectors of an annulus, including the effects of weak anchoring and elastic anisotropy, giving novel insights into the correlation between preferred numbers of boundary defects and the geometry. We numerically demonstrate that these sector configurations can approximate experimentally observed equilibria with boundary defects.

This review article is a consolidated but not exhaustive account of recent modelling and numerical work on nematic-filled square or cuboid shaped wells with planar degenerate boundary conditions. This seemingly simple geometry can be modelled with a simplistic Oseen-Frank approach or a more sophisticated twodimensional and three-dimensional Landau-de Gennes approach. We discuss these approaches, reconcile the findings and in doing so, elucidate the complex interplay between material properties, temperature, geometry and boundary conditions in both equilibrium and non-equilibrium phenomena. We largely focus on static equilibria with some discussion on metastable or transient states of experimental relevance.

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The Study Group investigated the possibility of a small standalone deadreckoning system making use of cheaply available accelerometer, gyroscope and magnetic field sensors, similar to those included in many modern smart phones. The Study Group captured data using an Android device strapped to a skateboard to simulate the type of movements a diver might make underwater. Different modes of movement were evident from filtered versions of the sensor outputs, so indicating that a pedometry-based solution ought to be feasible. The Study Group then formulated and investigated the feasibility of a generic dead-reckoning system. Although sensors provide more data than is strictly necessary, significant errors arise from imperfect calibration and from noise for which the Study Group derived estimates of the resulting drift in position over time. The accuracy of practical numerical integration schemes in the context of rotating frames was investigated, and a Kalman filter was used to reduce error in the orientation data by combining accelerometer and gyroscopic data.

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We study the azimuthal defect-free nematic state on a two-dimensional annulus within a simplified and reduced two-dimensional Landau-de Gennes model for nematic liquid crystals. We perform a detailed asymptotic analysis of the instabilities of the defect-free state in terms of a dimensionless material and temperature-dependent variable and the annular aspect ratio. The asymptotic analysis is accompanied by a rigorous local stability result, again in terms of a dimensionless material and temperature-dependent parameter and annular aspect ratio. In contrast to Oseen-Frank predictions, the defect-free state can be unstable in this model, with elastic isotropy and strong anchoring, for a range of macroscopically relevant annular aspect ratios.