Hilary Ockendon

Free Boundary Problems

One example of a flow with a free boundary is that of a jet of fluid travelling through a region of constant pressure. There are two typical situations which are shown in Figure. The first is a jet impinging on a fixed wall and the second is a jet emerging from a hole in the wall of a large reservoir. These situations may either be two or three-dimensional, but we can make more analytical progress in the two-dimensional case.

Channel flow with temperature-dependent viscosity and internal viscous dissipation

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.

Resonant sloshing in shallow water

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.

Non-linear wave propagation in a relaxing gas

We consider the propagation of waves of small finite amplitude e in a gas whose internal energy is characterized by two temperatures T (translational) and T i (internal) in the form e = C vf T + C vf T i , and T i is governed by a rate equation dT i / dt = ( T − T i )/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order eδ, the equation satisfied by velocity takes the non-linear form \[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \] where a 1 , a 0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − ( a 2 0 / a 2 1 )), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given. When e is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a 0 U a 1 . If U > a 1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

Nonlinear Surface Waves

We now consider the situation in which the mean depth h0 Of the fluid is comparable with the amplitude of the surface waves but small compared with the lateral scale or wavelength λ of the disturbance. The same assumption was used in discussing Tidal Waves, but in this case we shall not assume that the disturbance is small, and the problem will be essentially nonlinear.

The Fanno model for turbulent compressible flow

The paper considers the derivation and properties of the Fanno model for nearly unidirectional turbulent flow of gas in a tube. The model is relevant to many industrial processes. Approximate solutions are derived and numerically validated for evolving flows of initially small amplitude, and these solutions reveal the prevalence of localized large-time behaviour, which is in contrast to inviscid acoustic theory. The properties of large-amplitude travelling waves are summarized, which are also surprising when compared to those of inviscid theory.

Mathematical modelling of elastoplasticity at high stress

This study describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focused on the increasing ductility that occurs in the over-driven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave.

Dynamic dislocation pile-ups

Abstract The motion of a continuous distribution of dislocations moving under a constant imposed stress towards a small locked dislocation is considered. This problem can be modelled as a non-linear partial differential equation for a complex stress function if a linear stress–velocity law is assumed. If the initial density of the distribution is sufficiently small. the dynamic pile-up that occurs can be described using matched asymptotic expansions. In particular, the force on the locked dislocation is derived as a function of time, It is further shown that the solution remains valid when the velocity in proportional to any positive power of the stress.

Nonlinear Waves in Fluids

We have already encountered several deficiencies in the theories presented in Chapter 4 that indicate the limitations of the linear approximation. In this chapter we will consider three specific nonlinear models, namely, unsteady one-dimensional gas dynamics, two-dimensional steady gas dynamics and shallow water theory.

How to Mitigate Sloshing

Walking across a room carrying a mug of coffee can often lead to spillage. Everyday experience tells us that it is better to walk slowly or not have the mug too full, but it is also well known that carrying coffee in a bucket with a pivoted handle is much less dangerous. Here we show how to construct a mathematical model for sloshing in the very similar problem of a mug on a smooth horizontal table forced to oscillate in one dimension via a spring connection. We find that analyzing this problem using quite simple ideas of mathematical modeling and analysis gives good physical understanding of how to reduce everyday sloshing.