Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results.

The scaled boundary finite element method (SBFEM) is a semi-analytical computational method initially developed in the 1990s. It has been widely applied in the fields of solid mechanics, oceanic, geotechnical, hydraulic, electromagnetic and acoustic engineering problems. Most of the published work on SBFEM has focused on its theoretical development and practical applications, but, so far, no explicit discussion on the numerical stability and accuracy of its solution has been systematically documented. However, for a reliable engineering application, the inherent numerical problems associated with SBFEM solution procedures require thorough analysis in terms of its causes and the corresponding remedies. This study investigates the numerical performance of SBFEM with respect to matrix manipulation techniques and their properties. Some illustrative examples are given to identify reasons for possible numerical difficulties, and corresponding solution schemes are proposed to overcome these problems.

We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

The lower part of the spectrum of the Helmholtz equation for a heterogeneous system in a finite region in $d$ dimensions, where the solutions to the corresponding homogeneous system are known, can be systematically approximated by means of iterative methods. These methods only require the specification of an arbitrary ansatz and converge to the desired solution, regardless of the strength of the inhomogeneities, provided the ansatz has a finite overlap with it. In this paper, different boundary conditions at the borders of the domain are assumed, and some applications are used to illustrate the methods.

We investigate two routing problems that arise when order pickers traverse an aisle in a warehouse. The routing problems can be viewed as Euclidean travelling salesman problems with points on two parallel lines. We show that if the order picker traverses only a section of the aisle and then returns, then an optimal solution can be found in linear time, and if the order picker traverses the entire aisle, then an optimal solution can be found in quadratic time. Moreover, we show how to approximate the routing cost in linear time by computing a minimum spanning tree for the points on the parallel lines.

In a glass furnace solid batches of material are fed into a chamber and radiation heating applied. An individual batch is melted over the course of several minutes to form molten glass. A travelling front within the batch designates the progress of the melting, a process characterized by multiple radiation reflections. This results in an effective conductivity within the melting zone that is significantly larger than that in the unmelted batch. Approximations based on these disparate conductivities enable accurate explicit expressions for the almost constant melting front speed and the associated temperature profile to be derived. Our results compare favourably with existing numerical simulations of the process, with the advantage of being both analytic and relatively simple. These predictions may be useful in suggesting how a furnace might be most effectively controlled under varying batch conditions, as well as ensuring the quality of the glass sheets produced.

We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.