A viscous fluid is confined between two smooth horizontal walls, in a vertical channel. The upper wall may move with constant speed, but the lower wall is stationary and a portion of it is heated. A plume of heated fluid develops, and may also be swept downstream by the motion of the upper wall. When the heating effect is small and the upper plate does not move, a closed-form solution for the temperature profile is presented. A numerical spectral method is then presented, and allows highly accurate nonlinear solutions to be obtained, for the temperature and the fluid motion. These are compared against the closed-form solution in the linearized case, and the effects of nonlinearity on temperature and velocity are revealed. The results also show that periodic plume shedding from the heated region can occur in the nonlinear case.

Double-masking may be used to reduce the transmission of a virus. If additionally the masks are compressible, with different permeabilities and behaviour under compression, then it may be possible to design a mask that allows for easy breathing under normal breathing conditions, but is relatively impermeable under coughing or sneezing conditions. Such a mask could be both comfortable to wear and effective. We obtain analytical solutions for the steady-state flow-through behaviour of such a double mask under flow-out conditions. The results show that the reduction in permeability required to produce a relatively impermeable mask under high flux expulsion (sneezing) conditions could be achieved using either a single filter compressible mask or two filters with different poroelastic parameters. The parameters can be more easily adjusted using a double mask. For both single- and double-mask cases, there is an abrupt cut off, whereby through-flux levels reduce from a maximum value to zero as pressure drop levels increase beyond a critical value. Additionally, in the double-mask case, there exists a second steady-state solution for particular parameter ranges. This second solution is unlikely to occur under normal circumstances.

Thin spray-on liners (TSLs) have been found to be effective for structurally supporting the walls of mining tunnels and thus reducing the occurrence of rock bursts, an effect primarily due to the penetration of cracks by the liner. Surface tension effects are thus important. However, TSLs are also used to simply stabilize rock surfaces, for example, to prevent rock fall, and in this context crack penetration is desirable but not necessary, and the tensile and shearing strength and adhesive properties of the liner determine its effectiveness. We examine the effectiveness of nonpenetrating TSLs in a global lined tunnel and in a local rock support context. In the tunnel context, we examine the effect of the liner on the stress distribution in a tunnel subjected to a geological or mining event. We show that the liner has little effect on stresses in the surrounding rock and that tensile stresses in the rock surface are transmitted across the liner, so that failure is likely to be due to liner rupture or detachment from the surface. In the local rock support context, loose rock movements are shown to be better achieved using a liner with small Young’s modulus, but high rupture strength.

In this paper, we investigate the thermal evolution in a one-dimensional bagasse stockpile. The mathematical model involves four unknowns: the temperature, oxygen content, liquid water content and water vapour content. We first nondimensionalize the model to identify dominant terms and so simplify the system. We then calculate solutions for the approximate and full system. It is shown that under certain conditions spontaneous combustion will occur. Most importantly, we show that spontaneous combustion can be avoided by sequential building. To be specific, in a situation where, say, a $4.7\,$m stockpile can spontaneously combust, we could construct a $3\,$m pile and then some days later add another $1.7\,$m to produce a stable $4.7\,$m pile.

We provide an analytic solution of the Rössler equations based on the asymptotic limit $c\to \infty $ and we show in this limit that the solution takes the form of multiple pulses, similar to “burst” firing of neurons. We are able to derive an approximate Poincaré map for the solutions, which compares reasonably with a numerically derived map.

Sending microwaves through bauxite ore allows almost continuous measurement of moisture content during offload by conveyor belt from a ship. Data and results from a microwave analyser were brought to a European Study Group with Industry at the University of Limerick, with the over-arching question of whether the results are accurate enough. The analyser equipment uses linear regression against phase shifts and signal attenuation to infer moisture content in real time. Simple initial modelling conducted during the Study Group supports this use of linear regression for phase shift data. However, that work also revealed striking and puzzling differences between model and attenuation data.

We present an improved model that allows for multiple reflections of travelling microwaves within the bauxite and in the air above it. Our new model uses four differential equations to describe how electric fields change with distance in each of four layers. By solving these equations and taking reflections into account, we can accurately predict what the receiving antenna will pick up.

Our new solution provides much-improved matches to data from the microwave analyser, and indicates the deleterious effects of reflections. Modelled signal strength behaviour features a highly undesirable noninvertible dependence on bauxite mixture permittivity.

Practical measures that might be expected to reduce the effects of microwave reflections and improve the accuracy of microwave analyser results are suggested based on our improved model solution. This modelling approach and these results are anticipated to extend to the analysis of moisture content during transport on conveyor belts of other ores, slurries, coal, grains and pharmaceutical powders, especially when the depth of the conveyed material is variable.

To compute the maximum speed threshold for helicopters, we model the lift produced by the rotor blades. Using this model, we derive limits for each method traditionally used to alleviate dissymmetry of lift. Additionally, we find the minimum rotor angular velocity required to produce a prescribed lift at a given forward velocity. We derive conditions on the coefficient of lift for helicopter airfoils that maintain altitude. Further considerations are also made with regard to the properties of the air and its effect on helicopter dynamics.

Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

The effects of apparatus-induced dispersion on nonuniform, density-dependent flow in a cylindrical soil column were investigated using a finite-element model. To validate the model, the results with an analytical solution and laboratory column test data were analysed. The model simulations confirmed that flow nonuniformities induced by the apparatus are dissipated within the column when the distance to the apparatus outlet exceeds $3R/2$, where R represents the radius of the cylindrical column. Furthermore, the simulations revealed that convergent flow in the vicinity of the outlet introduces additional hydrodynamic dispersion in the soil column apparatus. However, this effect is minimal in the region where the column height exceeds $3R/2$. Additionally, it is found that an increase in the solution density gradient during the solute breakthrough period led to a decrease in flow velocity, which stabilized the flow and ultimately reduced dispersive mixing. Overall, this study provides insights into the behaviour of apparatus-induced dispersion in nonuniform, density-dependent flow within a cylindrical soil column, shedding light on the dynamics and mitigation of flow nonuniformities and dispersive mixing phenomena.