This study explores the dynamics of a simple mechanical oscillator involving a magnet on a spring constrained to an axis; this magnet is additionally subject to the attractive force from a second magnet, which is placed on a parallel offset axis. The moments of both magnets remain aligned. The dynamics of the first magnet is first analysed in isolation for an unforced situation in which the second magnet is static and its position is taken as a parameter. We find codimension-1 saddle-node bifurcations, as well as a codimension-2 cusp bifurcation. The system has a region of bistability which increases in size with increasing force ratio. Next, the parametrically forced situation is considered, in which the second magnet moves sinusoidally. A comprehensive analysis of the forced oscillator behaviour is presented from the dynamical-systems standpoint. The solutions are shown to include periodic, quasiperiodic and chaotic trajectories. Resonances are shown to exist and the effect of weak damping is explored. Layered stroboscopic maps are used to produce cross-sections of the chaotic attractor as the parametric forcing frequency is varied. The strange attractor is found to disappear for a narrow window of forcing frequencies near the natural frequency of the spring.

The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

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.

Recent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here.

We consider fully three-dimensional time-dependent outflow from a source into a surrounding fluid of different density. The source is distributed over a sphere of finite radius. The nonlinear problem is formulated using a spectral approach in which two streamfunctions and the density are represented as a Fourier-type series with time-dependent coefficients that must be calculated. Linearized theories are also discussed and an approximate stability condition for early stages in the outflow is derived. Nonlinear solutions are presented and different outflow shapes adopted by the fluid interface are investigated.

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour.

When open-cut mines are eventually abandoned, they leave a large hole with sloping sides. The hole fills with rain water, and there is also contaminated run-off from surrounding land, that moves through the rock and eventually through the sloping sides of the abandoned mine. This paper considers a two-dimensional unsteady model motivated by this leaching flow through the rock and into the rain-water reservoir. The stability of the interface between the two fluids is analysed in the inviscid limit. A viscous Boussinesq model is also presented, and a closed-form solution is presented to this problem, after it has been linearized in a manner consistent with Boussinesq theory. That solution suggests that the interfacial zone is effectively neutrally stable as it evolves in time. However, an asymptotic theory in the interfacial region shows the interface to be unstable. In addition, the nonlinear Boussinesq model is solved using a spectral method. Interfacial travelling waves and roll-up are observed and discussed, and compared against the predictions of asymptotic Boussinesq theory.

As a possible model for fluid turbulence, a Reiner–Rivlin-type equation is used to study Poiseuille–Couette flow of a viscous fluid in a rotating cylindrical pipe. The equations of motion are derived in cylindrical coordinates, and small-amplitude perturbations are considered in full generality, involving all three velocity components. A new matrix-based numerical technique is proposed for the linearized problem, from which the stability is determined using a generalized eigenvalue approach. New results are obtained in this cylindrical geometry, which confirm and generalize the predictions of previous recent studies. A possible mechanism for the transition to turbulent flow is discussed.

Large-scale low-pressure systems in the atmosphere are occasionally observed to possess Kelvin–Helmholtz fingers, and an example is shown in this paper. However, these structures are hundreds of kilometres long, so that they are necessarily affected strongly by nonlinearity. They are evidently unstable and are observed to dissipate after a few days.

A model for this phenomenon is presented here, based on the usual $f$-plane equations of meteorology, assuming an atmosphere governed by the ideal gas law. Large-amplitude perturbations are accounted for, by retaining the equations in their nonlinear forms, and these are then solved numerically using a spectral method. Finger formation is modelled as an initial perturbation to the $n$th Fourier mode, and the numerical results show that the fingers grow in time, developing structures that depend on the particular mode. Results are presented and discussed, and are also compared with the predictions of the ${\it\beta}$-plane theory, in which changes of the Coriolis acceleration with latitude are included. An idealized vortex in the northern hemisphere is considered, but the results are at least in qualitative agreement with an observation of such systems in the southern hemisphere.

Fluid outflow is considered, from a binary system of two point sources. The sources inject fluid of a lower density than the surrounding medium, and there is a sharp interface separating the two fluids. The overall geometry is taken to be axisymmetric around the line joining the two sources. Numerical solutions are presented for the shape of the interface in unsteady flow, and compared with a linearized solution based on small deformation of the interface from its initial spherical configuration. In addition, a novel spectral method is developed for the solution of the Boussinesq viscous flow problem, accounting exactly for the presence of the two sources and modelling the interface as a narrow region in which fluid mixing is possible. Bipolar outflow jets are seen to be possible.

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.

The Kelvin–Helmholtz flow is a shearing instability that occurs at the interface between two fluids moving with different speeds. Here, the two fluids are each of finite depth, but are highly viscous. Consequently, their motion is caused by the horizontal speeds of the two walls above and below each fluid layer. The motion of the fluids is assumed to be governed by the Stokes approximation for slow viscous flow, and the fluid motion is thus responsible for movement of the interface between them. A linearized solution is presented, from which the decay rate and the group speed of the wave system may be obtained. The nonlinear equations are solved using a novel spectral representation for the streamfunctions in each of the two fluid layers, and the exact boundary conditions are applied at the unknown interface location. Results are presented for the wave profiles, and the behaviour of the curvature of the interface is discussed. These results are compared to the Boussinesq–Stokes approximation which is also solved by a novel spectral technique, and agreement between the results supports the numerical calculations.

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

Fluid turbulence is often modelled using equations derived from the Navier–Stokes equations, perhaps with some semi-heuristic closure model for the turbulent viscosity. This paper considers a possible alternative hypothesis. It is argued that regarding turbulence as a manifestation of non-Newtonian behaviour may be a viewpoint of at least comparable validity. For a general description of nonlinear viscosity in a Stokes fluid, it is shown that the flow patterns are indistinguishable from those predicted by the Navier–Stokes equation in one- or two-dimensional geometry, but that fully three-dimensional flows differ markedly. The stability of linearized plane Poiseuille flow to three-dimensional disturbances is then considered, in a Tollmien–Schlichting formulation. It is demonstrated that the flow may become unstable at significantly lower Reynolds numbers than those expected from Navier–Stokes theory. Although similar results are known in sections of the rheological literature, the present work attempts to advance the philosophical viewpoint that turbulence might always be regarded as a non-Newtonian effect, to a degree that is dependent only on the particular fluid in question. Such an approach could give a more satisfactory account of the underlying physics.

In models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

A previous study by Forbes (ANZIAM J., vol. 53, 2011, pp. 87–121) has argued that, when a light fluid is injected from a point source into a heavier ambient fluid, the interface between them is most unstable to perturbations at the lowest spherical mode. This means that, regardless of initial conditions, the outflow from a point source eventually becomes a one-sided jet. However, two-sided (bi-polar) outflows are nevertheless often observed in astrophysics, in apparent contradiction to this prediction. While there are many possible explanations for this fact, the present paper considers the effect of a straining flow in the ambient fluid. In addition, solid-body rotation in the inner fluid is also accounted for, in a Boussinesq viscous model. It is shown analytically that there are circumstances under which straining flow alone is sufficient to convert the one-sided jet into a genuine bi-polar outflow, in linearized theory. This is confirmed in a numerical solution of a viscous model of the flow, based on a spectral solution technique that accounts for nonlinear effects. Rotation can also generate flows that are two-sided, and this is likewise revealed through an asymptotic analysis and numerical solutions of the nonlinear equations.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

Selective withdrawal of a two-layer fluid is considered. The fluid layers are weakly compressible, miscible and viscous and therefore flow rotationally. The lower, denser fluid flows with constant velocity out through one or more drain holes in the bottom of a rectangular tank. The drain is opened impulsively and the subsequent draw-down produces waves in the interface which travel outward to the edges of the tank and are reflected back with a $18{0}^{\circ } $ change of phase. The points on the interface that have the highest absolute gradient form regions of high vorticity in the tank, enabling mixing of the fluids. An inviscid linearized interface is computed and compared to contour plots of density for the viscous solution. The two agree closely at early times in the withdrawal process, but as time increases, nonlinear and viscous effects take over. The time at which the lighter fluid starts to flow out of the tank is dependent on the number of drains, their width, and the fluid flow rate and density, and is investigated here.

The vertical rise of a round plume of light fluid through a surrounding heavier fluid is considered. An inviscid model is analysed in which the boundary of the plume is taken to be a sharp interface. An efficient spectral method is used to solve this nonlinear free-boundary problem, and shows that the plume narrows as it rises. A generalized condition is also introduced at the boundary, and allows the ambient fluid to be entrained into the rising plume. In this case, the fluid plume first narrows then widens as it rises. These features are confirmed by an asymptotic analysis. A viscous model of the same situation is also proposed, based on a Boussinesq approximation. It qualitatively confirms the widening of the plume due to entrainment of the ambient fluid, but also shows the presence of vortex rings around the interface of the rising plume.