An expansion scheme is developed to describe the steady axisymmetric flow of a thin tubular liquid film of varying radius; the necessary small parameter is provided by the ratio between the characteristic film thickness and the characteristic tube radius. The co-ordinate system used is an orthogonal one based on the fluid interface and the fluid streamlines. The differential equations that arise thus treat the metric as an unknown set of variables. The method is restricted to situations dominated by viscous forces. Reference is made to numerical solutions that have been obtained in connexion with an industrial polymer-film-blowing process.

The presence of the edge of a half plane in a turbulent fluid results in a large increase in the noise generated by that fluid at low Mach numbers. The parameter which is important is the product $2k\overline{r}_0$, where $\overline{r}_0$ is the distance of the centre of an eddy from the edge. Eddies which satisfy the inequality $2k\overline{r}_0\ll 1$ have the sound output of the quadrupoles associated with the fluid motion in a plane normal to the edge increased by the factor $(k\overline{r}_0)^{-3}$. There is no enhancement of the sound from the longitudinal quadrupoles with axes parallel to the edge: the rr, θr and θθ quadrupoles are the dominant sound sources. The far field sound intensity induced by these sources depends upon the fifth power of a typical fluid velocity. The intensity has a directional dependence on cos2 ½θ if the half plane is rigid and sin2 ½θ if it is a pressure release surface, θ = 0 being a direction in the half plane.

If the eddies are far from the edge so that $(k\overline{r}_0)^{\frac{1}{2}} \gg 1$ then the farfield sound has the same features as would be predicted by geometrical acoustics. The edge does not produce any significant sound amplification.

The spin-up of a homogeneous rotating fluid bounded at the top and/or bottom by a permeable medium has been proposed by Bretherton & Spiegel (1968) as a model for the spin-up in natural flows where turbulent processes transmit the direct effect of the boundaries deeper into the fluid than does the laminar Ekman layer. The theoretical analysis for the spin-up of a laterally unbounded fluid bounded by a permeable medium below is presented here. In addition, an experimental study of the process is presented. Theory and experiment agree reasonably well with a maximum difference of about 8% in the predicted and measured spin-up times. The effects of the side-wall boundary have been studied theoretically by Howard (1969). Experimental observations in the side-wall boundary layer confirm qualitatively the results of Howard's theory.

A thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid. The plate is assumed to be so wide that the resulting equations may be regarded as two-dimensional. In addition, a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate. The resulting linearized boundary-value problem is solved in closed form for the velocity potential everywhere in the fluid and on the plate. Expressions are derived for the first- and second-order forces and moments on the plate, and for the wave amplitudes at a large distance either side of the plate. Numerical results are obtained for the case of the plate held fixed in an incident wave-train. It is shown how these results, in the special case when the plate intersects the free surface, agree, with one exception, with results obtained by Ursell (1947) and Haskind (1959) for this problem.

The problem of magnetohydrodynamic free convection of an electrically conducting fluid in a strong cross field is investigated. It is solved by using a singular perturbation technique. The solutions presented cover the range of Prandtl numbers from zero to order one. This includes both the important cases of liquid metals and ionized gases. A general examination is given of the role of the important parameters: Hartmann, Grashof and Prandtl numbers of the problem. This provides clear insight into its singular character and yields the correct expansion parameters. The boundary-layer approximations are derived from the complete Navier-Stokes and energy equations. The conditions for these approximations to be valid will be explicitly stated. Attention is given to ‘power law’ wall-temperatures and magnetic fields, and an assessment is given of the range of application.

The title flow has been studied by measuring the drag force on, and by observing the flow field around, a sphere rising through a large, rotating tank of water. Long, almost stagnant, regions are formed up- and downstream within the shadow of the sphere and are surrounded by a thin annular region within which the velocity is larger than the mean velocity of the approach flow. Several regions are found within which vortex-jump phenomena occur and it is concluded that such features exert a controlling influence over the dynamics of the observed flow field.

A concentrated electric current enters a region of fluid through an interface. It is shown that the magnetic field due to the current in general gives rise to rotational magnetic forces which must cause motions of the fluid. In particular the paper solves the axisymmetric non-linear problem in which a concentrated current enters a semi-infinite region of inviscid conducting fluid of constant density, inducing an inwards flow along the wall and a jet away from the wall opposite the current source. The case treated is the practically realistic one in which the effective magnetic Reynolds number is small and the current flows isotropically from the source. The first-order perturbation of this current distribution by electromagnetic induction is also calculated.

An analytical solution is possible because the non-linear equation of motion happens to be a linear equation in the square of the Stokes stream function. The motion is analytically related to viscous jet flows discussed by Slezkin, Landau and Squire.

An analysis is presented for the interaction of capillary and gravity waves in a liquid layer of finite depth. The method of multiple scales is used to obtain a third-order expansion uniformly valid for all times. Although this expansion is valid for a wide range of wave-numbers, it breaks down at two critical wave-numbers if the liquid depth is larger than √3/kc, kc = (ρg/T)½, where g is the gravitational acceleration, and ρ and T are the liquid density and surface tension, respectively. For a deep liquid, the singularities are at kc/√2 and kc/√3 respectively, as found by Wilton (1915), and Pierson & Fife (1961).

A second-order expansion valid for wave-numbers near the first critical value (corresponding to a wavelength of 2·44 cm in deep water) is obtained. This expansion shows that two different wave profiles could exist at or near the first critical wave-number. One of these profiles is gravity-like while the other is capillary-like.

The two-dimensional flow of an incompressible viscous fluid past a circular cylinder, symmetrically placed in a uniform shear field, is considered both theoretically and experimentally for small values of the shear Reynolds number. A series of angular rotational speeds is covered, each giving rise to a fundamentally different flow pattern. It is shown first that the Stokes solution to this problem is not entirely consistent everywhere with the linear shear boundary condition which presumably exists far from the body. Using the method of inner and outer expansions, this solution is then improved by properly taking into account the first-order effects of the inertia terms, but, surprisingly, the streamline structure in the outer region is still found to depart from that of the uniform shear sufficiently far away from the object.

In spite of the somewhat bizarre nature of the theoretical solution far from the cylinder, experimental studies clearly show, however, that it accurately represents the actual flow within the inner region over a wide range of cylinder rotation rates.

This paper presents the results of a set of detailed experimental measurements on the resonant interaction of capillary-gravity waves for a case in which the entire propagation is in one direction. The influence of viscous attenuation is accounted for in the analysis. The measurements trace the entire spatial variation, or modulation envelope, of the amplitudes of the interacting modes from their inception near a wave-maker to their ultimate extinction through viscous dissipation, in excellent agreement with the theory. This is an unambiguous demonstration that at resonance and for the initial conditions specified at the wave-maker, a wave of uniform profile cannot exist.

This paper describes theoretical and experimental studies for induction velometry in a shock wave. The induced potential gradient profiles across the shock is obtained with an assumption that the velocity of the charged particles is given by Mott-Smith's result. The calculated result of the induced potential gradient shows that it is not proportional to the velocity of the charged particles, since eddy currents are also induced by the existence of non-uniform velocity through the shock. The induced potential gradient is measured across the shock produced by a shock holder. This result is compared with the theoretical result. The experimental result shows that the charged particles follow the Rankine-Hugoniot relation in the velocity.

The stability of viscous time-dependent rotational Couette flow, induced by an impulsively started inner cylinder was experimentally investigated. The ratio of the radius of the inner cylinder to that of the outer cylinder was $\frac{1}{10}$. The space between the cylinders was filled with distilled water. The time-dependent instabilities were observed by means of dye injection. They appeared as a series of disks more or less evenly spaced along the inner cylinder. The spacing and growth of these instabilities were recorded using a motion picture camera. From the motion pictures the critical time (which is the time from the impulsive start to the first onset of instability) and spacing of the instabilities were experimentally determined, and a marginal stability curve (Reynolds number versus critical time) was constructed.

Heat transfer from a cylinder placed symmetrically in a constant shear field is considered experimentally for low values of the shear Reynolds number, but for Péclet numbers Pe as large as 2000. With the cylinder held stationary, the experimentally obtained Nusselt number Nu is found to be in excellent agreement with the theoretically derived asymptotic expression, $Nu = 1.641 Pe^{\frac{1}{3}} (Pe \rightarrow \infty)$, for Péclet numbers in excess of 100. In contrast, with the cylinder rotating at a speed corresponding to zero torque, the Nusselt number becomes effectively independent of the Péclet number for Pe > 70. This surprising behaviour, predicted theoretically by Frankel & Acrivos (1968), results from the presence of a region consisting entirely of closed streamlines which surrounds the freely rotating cylinder.

The phenomenon under investigation occurs in various flow systems characterized by being dispersive to small-amplitude waves, but such that phase and group velocities approach the same finite limit at extreme wavelengths: for example, water in an open channel, density-stratified fluid flowing between horizontal boundaries, and rotating fluid contained in a tube. It is well known in each of these examples that, when a solid body is moved steadily at a subcritical velocity (i.e. less than the long-wave limit) relative to the undisturbed fluid, the body experiences resistance accountable to the continual development of a pattern of waves on the leeward side. But in these circumstances there is a second effect, upstream influence, consisting of a disturbance in the form of a uniform long wave that extends to a steadily increasing distance ahead of the body; and this has for some time been the subject of a controversy, particularly with regard to the question whether or not it is always present. The aim of the paper is to establish the principle that this second effect is inevitably coexistent with the first, being an essential component of the mechanism of wave resistance.

Perturbation series are developed and mathematically justified, using a straightforward perturbation formalism (that is more widely applicable than those given in standard textbooks), for the case of the two-dimensional inviscid Orr-Sommerfeld-like eigenvalue problem describing quasi-geostrophic wave instabilities of parallel flows in rotating stratified fluids.

The results are first used to examine the instability properties of the perturbed Eady problem, in which the zonal velocity profile has the form u = z + μu1(y, z) where, formally, μ [Lt ] 1. The connexion between baroclinic instability theories with and without short wave cutoffs is clarified. In particular, it is established rigorously that there is instability at short wavelengths in all cases for which such instability would be expected from the ‘critical layer’ argument of Bretherton. (Therefore the apparently conflicting results obtained earlier by Pedlosky are in error.)

For the class of profiles of form u = z + μu1(y) it is then shown from an examination of the O(μ) eigenfunction correction why, under certain conditions, growing baroclinic waves will always produce a counter-gradient horizontal eddy flux of zonal momentum tending to reinforce the horizontal shear of such profiles. Finally, by computing a sufficient number of the higher corrections, this first-order result is shown to remain true, and its relationship to the actual rate of change of the mean flow is also displayed, for a particular jet-like form of profile with finite horizontal shear. The latter detailed results may help to explain at least one interesting feature of the mean flow found in a recent numerical solution for the wave régime in a heated rotating annulus.

The capillary instability of vertical liquid jets of different viscosities have been examined by imposing audio-frequency disturbances. Real time sequences of photographs allow a direct measurement of growth rates of disturbances of various wavelengths. Results show that in general non-linear effects dominate the growth processes. This is in agreement with Yuen's analysis. The growth rate of the difference between the neck and the swell, however, agrees well with the linearized analysis of Rayleigh and Chandrasekhar. The non-linear effect causes a liquid jet to disintegrate into drops with ligaments in between. The sizes of the ligaments decrease with increasing wave-number. The subsequent roll up of the ligament into droplet, the eventual coalescing of the droplet with the main drop and drop oscillation have also been studied.

The main thrust of this work is to treat the initial convective phase of a fluid heated from below as a statistical initial value problem. The advantage of the approach is that it allows a continuous bandwidth of modes to be represented in the initial spectrum. We show that if the initial disturbance field is small and has a sufficiently smooth spectrum, then a natural statistical selection process chooses from the initial disorder a perfectly ordered field of single rolls. The scale of this roll is the scale corresponding to the most critical wave-number obtained from the linear stability problem. We relate this solution to the optimal solution which would be obtained by the upper bound procedures of Howard, Malkus and Busse. Moreover, we show in addition, that if the initial disturbance field is weighted in favour of a particular single roll whose scale is close to critical, the final solution reflects the initial condition providing a certain stability criterion is met. In the two-dimensional case we analyze, this turns out to be the Eckhaus stability condition previously obtained by a discrete multimodal analysis.

Subcritically stable motion of long gravity waves of finite amplitude in a liquid layer flowing down an inclined plane is shown to be impossible. However, super-critically stable wave régimes for such flows are found and curves of constant wave amplitude in such régimes are obtained. The mechanism of non-linear stability is investigated by considering the energy transfer between the mean flow and the disturbances. The results obtained show that the mechanism of stability in a parallel flow with a free surface is quite different from that in a parallel flow without a free surface.

The objective of the paper is to formulate a physically realistic definition of shock impedance. The definition obtained, expression (15), is consistent with the acoustic limit, with the Polachek & Seeger result for head-on shock incidence, and with the various reflexion limits. It seems to be applicable to all known regular and irregular refractions, with the possible exception of a refraction containing a Guderley patch. It is shown that one result of increasing impedance mismatch is to produce precursor and postcursor waves. It is thought the impedance mismatch caused by the jets of supersonic aircraft might be used to reduce the sonic boom overpressure.

Measurements are presented of the reflected wave field produced by a plane oblique shock wave impinging on a turbulent boundary layer at an initial Mach number of 2·5. The outgoing waves are either a single shock, with the same deflexion as the incident shock, or a shock of approximately 10° deflexion followed by a region of compression in which is embedded an expansion fan having the same turning as the incident shock. The transition between these two types of wave field was not studied, but it is fairly abrupt and appears to be closely linked to the onset of boundary-layer separation. The observed wave systems broadly agree with the suggestions of a number of previous workers, but not with a recent theoretical treatment. Surface-pressure measurements and oil flow photographs are used to determine the onset of separation, and from these it is found that the overall pressure rise associated with incipient separation is rather smaller than previous work would suggest.