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.

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $.

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.

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.