According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

When a semi-infinite flat plate is immersed parallel to an unbounded, plane, steady, asymmetric, constant shear flow of an incompressible viscous fluid, an interaction occurs between the surface-generated vorticity and the external vorticity. A physical assumption was made in a previous paper (Mark 1962) concerning this problem that the pressure field in a thin layer adjacent to the top-side of the plate may be accurately approximated by the undisturbed constant-pressure field—that which exists before the insertion of the plate into the flow. This means that the vorticity interaction is assumed to have no effect on the undisturbed pressure field. That this is a valid first approximation far downstream along the plate where the interaction is intense is given rigorous support in this paper.

The flow below the plate is examined on a heuristic basis. It is found that there is a strong possibility for the flow to separate from the lower surface near the leading edge of the plate. However, far downstream the flow settles down to a Stokes-type flow near the plate.

The paper presents a new analytical and experimental study of transonic flow around spheres. The results of the analytical study, which employs the method of orthogonal collocation for simultaneous solution of the momentum equations, the equation of continuity and the energy equation, are compared with hitherto unpublished measurements obtained on spheres of various sizes (1·02, 2·54 and 3·81 cm in diameter) in air, in dry steam and in wet steam with free-stream Mach numbers in the transonic range (0·58 < M∞ < 0·97). The relationship θ h = 91·78 + 8·59 M∞ between the attached-shock angle and the free-stream Mach number was obtained by fitting the theoretical pressure distributions to the experimental ones.

The flow field in the nose region of a blunt body in hypersonic flow is studied by considering the transport of vorticity and enthalpy. The entire region between body and shock is considered to be viscous, not necessarily thin in comparison with the nose radius of the body and to be of slowly varying density. The (given) post-shock vorticity need not be small and the density ratio ρ∞/ρs may either be small or near unity, the analysis being valid asymptotically at both limits.

It is found that the vorticity equation may be uncoupled from the total enthalpy equation if μ√ρ is constant. While the equations are not expected to be necessarily restricted to the immediate vicinity of the stagnation line, only there can the solution be written down explicitly; elsewhere, numerical integration is required.

The results of Ursell (1964) are confirmed by proving that there are no additional contributions to Ursell's integral from singularities of the integrand at infinity. The method consists of proving that the asymptotic expansion of a force coefficient Δ(N) is uniformly valid in a finite sector of the complex N-plane. This in turn requires that the kernel of an integral equation remains small in this sector.

A numerical method is developed for computing steady supercritical flow about an ellipse at zero angle of attack. The flow is assumed to be two-dimensional, inviscid, isentropic and irrotational. The free-stream Mach number lies in the high subsonic range so that a shock wave occurs locally near the body. The full potential equations are solved by Telenin's method and the ‘method of lines’. Smooth interpolating functions are assumed for the unknown flow variables in selected co-ordinate directions. The resulting set of ordinary differential equations is then integrated away from or along the body depending upon whether the flow is smooth or discontinuous. Jump conditions of the governing equations are applied across the shock wave so that it is perfectly sharp. A doublet solution for flow past a closed body is used as the farfield boundary condition. Supercritical flow calculations have been performed for ellipses with thickness ratio of 0·2 and 0·4 at various free-stream Mach numbers. The present results are compared with the shock-capturing method, and good agreement is obtained.

A class of nonlinear boundary layers in a hydromagnetic flow under differential rotation is analysed. The function of these layers is to provide smooth transition from the conditions within the electrically conducting container to those in the region far away from the boundary through the flow regime. The structure of the double-decker boundary layer depends on the diffusivity of the fluid, the conductance of the rigid boundary and the relative strength of the applied magnetic field. The method of multiple scales is used to obtain uniformly valid solutions with the conductance of the container varying from zero to infinity. It is found that even a small differential rotation (ε → 0) can induce perturbations of order ε½ or of order unity in the field functions.

This paper treats the steady inertialess flow of an incompressible viscous fluid through an infinite rectangular duct rotating rapidly about an axis (the y axis) perpendicular to its centre-line (the x axis). The prototype considered has parallel sides at z = ± 1 for all x, parallel top and bottom at y = ± a for x < 0 and straight diverging top and bottom at y = ± (a + bx) for x > 0. An earlier paper (Walker 1975) presented solutions for b = ±(1), for which the flow in the diverging part (x > 0) is carried by a thin, highvelocity sheet jet adjacent to the side at z = 1, the flow elsewhere in this part being essentially stagnant. The present paper considers the evolution of the flow as the divergence decreases from O(1) to zero, the flow being fully developed for b = 0. This evolution involves four intermediate stages depending upon the relationship between b and E, the (small) Ekman number. In each successive stage, the flow-carrying side layer in the diverging part becomes thicker, until in the fourth stage, it spans the duct, so that none of the fluid is stagnant.