Small amplitude forced horizontal or vertical oscillations of a container of liquid with a free surface may give rise to motions in the liquid on a scale much greater than the forcing amplitude. Three such situations are analysed and, in those cases where the response is still small compared with the dimensions of the container, explicit asymptotic solutions for the liquid motion are obtained.

Many of the topics in inviscid fluid dynamics are not only vitally important mechanisms in everyday life but they are also readily observable without any need for instrumentation. It is therefore stimulating when the mathematics that emerges when these phenomena are modelled is novel and suggestive of alternative methodologies. This book provides senior undergraduates who are already familiar with inviscid fluid dynamics with some of the basic facts about the modelling and analysis of viscous flows. It clearly presents the salient physical ideas and the mathematical ramifications with exercises designed to be an integral part of the text. By showing the basic theoretical framework which has developed as a result of the study of viscous flows, the book should be ideal reading for students of applied mathematics who should then be able to delve further into the subject and be well placed to exploit mathematical ideas throughout the whole of applied science.

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The ordinary differential equation \[ {\textstyle\frac{1}{3}}\kappa^2(g^{\prime\prime}+g) - \lambda g - {\textstyle\frac{3}{2}}g^2 + \frac{2}{\pi} \cos t = -\frac{3}{2}\int_{-\pi}^{\pi}g^2\,{\rm d}t, \] which represents forced water waves on shallow water near resonance, is considered when the dispersion κ is small. Asymptotic methods are used to show that there are multiple solutions with period 2π for a given value of the detuning parameter λ. The effects of dissipation are also considered.

The problem of an expansion wave propagating into a saturated magma is solved and used as a model of a vulcanian eruption. In our model for an explosive eruption we assume that a drop in pressure leads to the exsolution of magmatic volatiles.Initially the exsolved vapours create bubbles in the magma. We model the subsequent part of the exsolution process in which a foam is created; this is believed to be an essential feature of explosive volcanic eruptions. The foam also has the advantage that it can be modelled as a mixture or 'pseudo-gas' without slippage between the phases. Eventually the foam breaks up and becomes volcanic ash.Assuming that the exsolution of vapour is given by Henry's law, that the temperature is constant, that the magma and vapour have equal velocities, and neglecting wall friction and gravitational effects, an analytic solution for pressure, velocity, and vapour fraction is obtained for the expanding mixture in a constant area duct. The exit velocity u for the mixture is where @o is the original mass fraction of dissolved vapour, R is the gas constant for the vapour, the constant temperature, and p o / p the pressure ratio across the expansion. With +o = 1 per cent, To = 1000"K, and p o / p = 100 we find uo = 300 m s-', consistent with observations for vulcanian eruptions.

This paper uses asymptotic methods to analyse the flow in a narrow channel of a fluid with temperature-dependent viscosity and internal viscous dissipation. When the Nahme–Griffith number is large we show how the flow evolves from Poiseuille flow with a uniform temperature distribution to a plug flow with hot boundary layers on the walls. An asymptotic solution is obtained for the flow in the region of transition from Poiseuille to plug flow and an explicit equation is derived for the pressure gradient in terms of the local downstream co-ordinate in this transition region.

An asymptotic description is given of Newtonian fluid flow in a channel which is suddenly heated or cooled. The viscosity is assumed to be purely a function of temperature. The asymptotic approximation is that the downstream viscosity at the channel wall differs by an order of magnitude from that in the upstream flow. Although we make the drastic assumption that viscous dissipation is negligible, we can analyse flows where the viscosity depends either algebraically or exponentially on the temperature.

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