to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

User manual and system documentation of WAVEWATCH III R version 4.18

Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body. Recapitulating investigations of this question, this article states that in 25 years of experience with radar, there have been no known incidents of pilots being affected by radar waves. In the future, however, involvement by some pilots with Vessel Traffic Service shore-based radar could affect pilots somewhat differently from limited exposure to pilot launch radar. Pilots who find themselves in new working conditions close to an emitting source should exercise care all times.

About heat waves

We give a theory of permanent roll waves on a shallow layer of fluid mud which is modelled as a power-law fluid. Based on the long-wave approximation, Karman’s momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow shows that the growth rate of unstable disturbances increases monotonically with the wavenumber, and therefore is insufficient to suggest a preferred wavelength for the roll wave. Nonlinear roll waves are obtained next as periodic shocks connected by smooth profiles with depth increasing monotonically from the rear to the front. Among all wavelengths only those longer than a certain threshold correspond to positive energy loss across the shock, and are physically acceptable. This threshold also implies a minimum discharge, viewed in the moving system, for the roll wave to exist. These facts suggest that a roll wave developed spontaneously from infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. The discontinuity at the wave front is a mathematical shortcoming needing a local requirement. Predictions for the spontaneously developed roll waves in a Newtonian case are compared with available experimental data. Longer roll waves, with dissipation at the discontinuous fronts, cannot be maintained if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However, when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances. Numerical results are presented for the phase speed, wave height and wavenumber, and wave profiles for a representative value of the flow index of fluid mud.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094004064

Roll waves on a shallow layer of mud modelled as a power-law fluid

Publisher Summary This chapter describes some applications of the homogenization theory. The basic idea of the theory of homogenization has been employed for a long time. In the theory of wave propagation over slowly varying media, the familiar ray theory (geometrical optics approximation) is one such example. There, the method of multiple scales is employed to average over the locally periodic waves and find the slow variation of the wave envelope. Seepage through a porous media is one of the first examples to which the method of homogenization was applied. It is a good example to explain the role of physical scales and the mathematical procedure for three-dimensional problems. Furthermore, it can be used to illustrate the derivation of many properties of the constitutive coefficients. Examples of applications are particularly rich in the mechanics of composite media. The obvious example is to determine the relations between stresses and strains in a fiber-reinforced material. The problem of sound propagation through a liquid populated sparsely by bubbles is another interesting application of homogenization theory. The main objective is to find an effective equation for the propagation of sound whose wavelength is much greater than the bubble spacing, which is, in turn, much greater than the bubble radius.

Some Applications of the Homogenization Theory

Water waves over a muddy bed: a two-layer Stokes' boundary layer model

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.

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Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets into the grooves. On comparing with some molecular-dynamics simulation measures, it is pointed out that the applied pressure, which has to be less than the capillary pressure in the superhydrophobic state, can be correlated with the penetration depth of the no-shear interface.

Chiu-On Ng

Papers

Roll waves on a shallow layer of mud modelled as a power-law fluid

Some Applications of the Homogenization Theory

Water waves over a muddy bed: a two-layer Stokes' boundary layer model

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

Stokes shear flow over a grating: Implications for superhydrophobic slip