Geoffrey McKay

Onset of bouyancy-driven convection in superposed reacting fluid and porous layers

A linear stability analysis is applied to the onset of bouyancy-driven convection in a horizontal layer of reacting fluid overlying a porous region saturated with the same fluid. The fluid is assumed to be undergoing zero-order exothermic reactions in both regions. At the interface between fluid and porous layers the boundary conditions proposed by Nield [1] are employed; these include the empirical slip condition suggested by Beavers and Joseph [2]. Predictions for the onset of convection and critical wavenumbers are obtained from the analysis by the collocation method and solution of the resulting generalized eigenvalue problem. The effect of variable fluid/porous layer depth ratio, Frank–Kamenetskii number or thermal boundary conditions on the onset of fluid motions is studied, and ignition values of the Frank–Kamenetskii number for the system are calculated.

Bistable surface anchoring and hysteresis of pitch jumps in a planar cholesteric liquid crystal

AbstractThe relationship between bistable surface anchoring and the pitch jump process is examined for a planar cholesteric liquid crystal. Introducing a high-order, azimuthal surface anchoring potential into a simple model to describe a cholesteric, we derive an expression for the director twist as the natural pitch of the liquid crystal is allowed to vary. Writing the energy in terms of the surface twist, we are able to determine the twists which minimize the total energy of the system. We demonstrate how a pitch jump is related to an energy exchange from one branch of metastable states to another. We then discuss how the co-existence of energy minima and their associated solution branches may help explain the thermal hysteresis observed experimentally in cholesterics in the neighbourhood of a pitch jump. The presence of a higher-order surface energy term can expand the range of anchoring strengths in which pitch jumps are possible. We also investigate the influence of bidirectional surface anchoring on the behaviour of the total energy. Intermediate quarter-turn pitch jumps can occur, depending on the relative strength of the high-order anchoring term, and these can have a significant effect on the system hysteresis.

Elongational flow and the wet spinning process

Abstract A study is carried out to investigate fibre formation in the wet spinning process. Elongational and shear flow measurements on solutions of polyurethane in dimethylformamide (DMF) have been examined. To simulate changes which take place during wet spinning, water has been added to these solutions and their properties re-examined before solidification took place. The results are discussed with the aim of understanding the complicated rheological changes which occur during wet spinning. The work shows that as non-solvent diffuses into the liquid filament the polymer solution becomes more visous, more elastic and the elongational viscosity increases.

Director Orientation of a Twisted Nematic Under the Influence of an In-Plane Magnetic Field

ABSTRACT We consider an incompressible nematic liquid crystal that exhibits an equilibrium twist profile across the cell due to in-plane but opposing anchoring directions at the plates. We examine the director orientation under the influence of a magnetic field applied in a direction parallel to the plates. Characteristic switching times are determined as the angle of twist on the plates and the direction of the applied field are varied. Furthermore, extending classical analyses we discuss Freedericksz transitions and critical field strengths.

Energy considerations for parabolic cyclides in SmA liquid crystals

ABSTRACT For Dupin cyclides in the SmA liquid crystal phase, Kleman and Lavrentovich [1] found that increasing the saddle-splay constant K¯ caused the minimum of the total elastic energy to occur at a decreased value of the eccentricity e, close to unity. We extend this work to the parabolic cyclides and present an analytical expression for the total elastic energy that is finite over a suitable range of values for the latus rectum, conventionally denoted in this context by −4ℓ (>0). The length of the latus rectum characterises the parabolic focal conic structure analogously to the eccentricity of the Dupin cyclide focal conics. We demonstrate that the total energy is minimised at a particular value of ℓ. It is further observed that the usual saddle-splay elastic term acts independently of ℓ and that varying the value of K¯ does not affect the actual value of ℓ at which the minimum of the elastic energy occurs.

Competing polarization effects in a homeotropically aligned smectic liquid crystal

We examine the equilibrium configurations of a thin SmA* liquid crystal film with its layers parallel to a lower substrate and a free upper surface. The molecules can exhibit variations in molecular tilt or twist around the smectic cone which are influenced by an in-plane electric field. We examine the competing effects of flexoelectricity and spontaneous polarization on the molecular tilt and twist. Our analysis is based upon a continuum theory for smectics which allows variation in the smectic layer spacing and the molecular tilt. For a fixed voltage the system can exhibit qualitatively different types of equilibrium profiles, depending on the relative sizes of the spontaneous polarization and flexoelectric contributions to the bulk energy. Energy and stability calculations allow us to predict physically relevant molecular configurations. These structures are closely related to synclinic and anticlinic arrangements in liquid crystal freestanding films.

Measuring the daily stepping activity of people with transtibial amputation using the ActivPAL™ activity monitor

ABSTRACT This study compared the general activity during 1 week and detailed activity during a 24-hr period of 48 established unilateral transtibial prostheses users. Activity was measured by instrumenting their prescribed prosthesis, which they have been using for a minimum of 6 months, with the ActivPAL™ activity monitor. Half (n = 24) were fitted with a prosthesis with total surface bearing (TSB) pressure-cast sockets (“hands-off” group), and the other half (n = 24) had been wearing prostheses with hand-cast patellar tendon-bearing (PTB) sockets (“hands-on” group). As a prerequisite, the long-term reliability of the ActivPAL activity monitor was assessed, and it was found to exhibit a high level of consistency between devices (intraclass correlation coefficient [ICC] of 0.997 during the 24-hr period). The monitors were used to examine the activity levels of the two groups of transtibial prostheses users wearing their own prosthesis. The results indicated that both subject groups were active throughout the day, walking a mean of more than 8000 steps. No statistically significant difference in daily stepping activity was seen between the two groups (p = 0.173). Despite differences in prosthetic socket design, the daily activity profiles of both subject groups were similar.

Decoupling of the Ericksen–Leslie equations

We examine the hydrodynamic behaviour of a nematic liquid crystal confined between two parallel electrodes and subject to a non-uniform electric field across the layer. By decoupling the Ericksen–Leslie equations for the nematic, we are able to derive a dynamic equation for the director orientation which is not explicitly dependent on the fluid velocity, rather a non-local term contains the effects of flow. The flow velocity and electric potential are determined subsequently from the calculated director profile. Our numerical scheme enables us to predict an effective rotational viscosity which takes account of fluid shear viscosities and allows us to establish the behaviour in special symmetry cases. Significantly, our calculations also demonstrate how kickback can be avoided, and the switch-off time optimised, by restricting the applied voltage to an appropriate range.

Effects of weak anchoring on C1 and C2 chevron structures

ABSTRACT We present a theoretical study of the effect of weak anchoring on the transition between C1 and C2 chevron structures in smectic C liquid crystals. We employ a continuum theory which allows for variable cone, azimuthal and layer tilt angles. Equilibrium profiles for the director cone and azimuthal angles in the C1 and C2 states are calculated from the standard Euler-Lagrange minimisation of the total energy of the system. By comparing the total energies of the C1 and C2 states we can determine the globally stable chevron profile and calculate the critical temperature for the C1–C2 transition, which depends on anchoring strength and pretilt angle variations.

In-plane switching of a homeotropically aligned, thin smectic C* liquid crystal

Most early theoretical studies of smectic liquid crystals exclude variations in the interlayer spacing or changes in molecular tilt with respect to the smectic layer normal, e.g. Leslie et al . [1] Recently McKay and Leslie [2] presented a theory for smectics which does allow for variations in tilt. With this theory they modelled a smectic C liquid crystal confined in a cell, its layers coplanar with the boundary plates, but subject to strong anchoring incompatible with the smectic C tilt. Subsequently Mazzulla and Sambles [3] found good agreement between theoretical predictions using the theory and their experimental observations. We present a model similar to [2] which also allows a twist in the molecule profile across the thickness of the sample. We consider a thin sample of Sm C* liquid crystal at a temperature well below the Sm A-Sm C* phase transition. A twist profile is induced by incorporating an in-plane electric field (e.g. Oh-e and Kondo [4] ) into the model.