Oscillatory flow over a circular cylinder, or part-cylinder, placed on a plane boundary, when the Strouhal and streaming Reynolds numbers are large, is considered. The solution is developed in matching inner and outer boundary layers. A steady streaming motion in the outer layer can lead to a net flow away from the cylinder along the plane boundary. A simple experiment substantiates this prediction, and the implications for bed-scouring are examined.

We study a system of two second-order differential equations with cubic nonlinearities which model a film of superconductor material subjected to a tangential magnetic field. We verify some recent conjectures of one of the authors about multiplicity of solutions. We show that for an appropriate range of parameter values the relevant boundary value problem has at least two symmetric solutions. It is also proved that a second range of parameters exists for which there are three symmetric solutions.

In this paper we examine the movement of hard contact lenses on the eye. In so doing, we take into account hydrodynamic forces underneath the lens, as well as surface tension forces at the lens periphery. This involves solving for the free surface of the tear film away from the lens in order to determine the magnitudes of the pressure and surface tension forces on the lens. The analysis, which assumes quasi-steady motion, is carried out in both two and three dimensions.

In this paper we study the question of existence and uniqueness of solutions to the equation

which occurs in the modelling of hard contact lenses. Using ‘shooting’ arguments, we show that, depending on the parameters in the problem, there are cases where there is precisely one solution, others where there is more than one, and others where there are none.

We study the singular limit of the dimensionless phase-field equations

We consider two cases: either the space dimension is 1 and then ɛ tends to zero; or the solutions are radially symmetric and then both ɛ and α tend to zero. It turns out that, in the first case, the limiting functions solve the Stefan problem with kinetic undercooling, provided the initial temperature is small compared to the surface tension and the latent heat. In the second case, the limiting functions satisfy the Stefan problem coupled with the Gibbs–Thomson law for the melting temperature. We show, in addition, that the multiplicity of the interface is always one, in a sense to be explained at the end of § 1. As main tool we use energy type estimates, and prove that the formal first-order asymptotic expansion with respect to ɛ in fact gives an approximation of the exact solution. Our results hold without smoothness assumptions on the limiting Stefan problem.

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

M. Escobedo and R. E. Grundy Vol. 7 (1996), pp. 395–416; The publisher apologises for an oversight which resulted in an error in the above paper. The following three figures were inadvertently published in an incomplete form, with some of the labelling omitted.