In this paper our concern is with solutions w(z; a, fJ) of the fourth Painleve equation (PIV), where a and {J are arbitrary real parameters. It is known that PIV admits a variety of solution types and here we classify and characterise these. Using Backlund transformations we describe a novel method for efficiently generating new solutions of PIV from known ones. Almost all the established Backlund transformations involve differentiation of solutions and since all but a very few solutions of PIV are given by extremely complicated formulae, those transformations which require differentiation in this way are very awkward to implement in practice. Depending on the values of the parameters a and {J, PIV can admit solutions which may either be expressed as the ratio of two polynomials in z, or can be related to the complementary error or parabolic cylinder functions; in fact, all exact solutions of PIV are thought to fall in one of these three hierarchies. We show how, given a few initial solutions, it is possible to use the structures of the hierarchies to obtain many other solutions. In our approach we derive a nonlinear superposition formula which relates three solutions of PI V; the principal attraction is that the process involves only algebraic manipulations so that, in particular, no differentiation is required. We investigate the properties of our computed solutions and illustrate that they have a large number of physical applications.

The linear stability of the Stokes layer generated by an oscillating flat plate is investigated using Floquet theory. The results obtained include the behaviour of the growth rate of the disturbances, part of the corresponding neutral curve and the structure of neutrally stable disturbances. Previously unknown properties of the growth rate cause the neutral curve to have a complicated geometry: the majority of the marginal curve is defined by waves propagating relative to the basic flow and the curve is smooth in character, but for certain very narrow bands of wavenumbers it was found that stationary modes are the first to become unstable. This phenomenon has the consequence that the underlying smooth neutral curve is punctuated by thin finger-like features. The structure of the eigenfunctions showed that the neutrally stable disturbances tend to grow most rapidly just after the wall velocity passes through zero.

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The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as 'axisymmetrization', is studied asymptotically and numerically. The vortex is perturbed at t = 0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t −λ with λ = 1 + (n 2 + 8) 1/2 , where n is the azimuthal wavenumber of the perturbation and n > 1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982). IntroductionIn fluid flow at high Reynolds number there is a tendency for vorticity to aggregate to form coherent vortices, both for planar flows (e.g. McWilliams 1984McWilliams , 1990Benzi et al. 1986;Brachet et al. 1988) and in three dimensions (e.g. Kuo & Corrsin 1971, 1972Siggia 1981;Kerr 1985;She, Jackson & Orszag 1990;Vincent & Meneguzzi 1991). Such concentrations of vorticity are associated with differential rotation of the fluid, both inside and outside the vortex. This causes stretching of fluid elements and means that fluctuations of vorticity (or a passive scalar) are wound up into characteristic spiral structures, and so are driven to small scales. Our goal is to discuss some of the consequences of differential rotation and this winding-up process and its implications for the behaviour of vorticity and scalars in coherent planar vortices.We consider an idealized problem in which a smooth axisymmetric vortex with vorticity Ω = Ω 0 (r) and associated stream function Ψ = Ψ 0 (r) is perturbed so as to gain small non-axisymmetric components, and we study their subsequent evolution. If this event occurs at t = 0 then for t > 0 the total vorticity and stream function may be written