Two distinct kinds of transition have been identified in Couette flow between concentric rotating cylinders. The first, which will be called transition by spectral evolution, is characteristic of the motion when the inner cylinder has a larger angular velocity than the outer one. As the speed increases, a succession of secondary modes is excited; the first is the Taylor motion (periodic in the axial direction), and the second is a pattern of travelling waves (periodic in the circumferential direction). Higher modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This kind of transition may be viewed as a cascade process in which energy is transferred by non-linear interactions through a discrete spectrum to progressively higher frequencies in a two-dimensional wave-number space. At sufficiently large Reynolds numbers the discrete spectrum changes gradually and reversibly to a continuous one by broadening of the initially sharp spectral lines.

These periodic flows are not uniquely determined by the Reynolds number. For the case of the inner cylinder rotating and the outer cylinder at rest, as many as 20 or 25 different states (each state being defined by the number of Taylor cells and the number of tangential waves) have been observed at a given speed. As the speed changes, theso states replace each other in a repeatable but irreversible pattern of transitions; vortices appear or disappear in pairs, and waves are added or subtracted. More than 70 such transitions have been found in the speed range up to about 10 times the first critical speed. Regardless of the state, however, the angular velocity of the tangential waves is nearly constant at 0.34 times the angular velocity of the inner cylinder.

The second kind of transition, which will be called catastrophic transition, is characteristic of the motion when the outer cylinder has a larger angular velocity than the inner one. At a fixed Reynolds number, the fluid is divided into distinct regions of laminar and turbulent flow, and these regions are separated by interfacial surfaces which may be propagating in either direction. Under some conditions the turbulent regions may appear and disappear in a random way; under other conditions they may form quite regular patterns. One common pattern of particular interest is a spiral band of turbulence which rotates at very nearly the mean angular velocity of the two walls without any change in shape except possibly an occasional shift from a right-hand to a left-hand pattern. One example of this spiral turbulence is being studied in some detail in an attempt to clarify the role played in transition by interfaces and intermittency.

The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field.

The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number (M), large positive and negative velocities of order MVc are induced, where Vc is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.

The stability of steady, plane, one-dimensional, trans-Alfvénic shocks to small normal disturbances (i.e. those in which the perturbed quantities are functions only of time and the distance from the plane of the shock wave) is discussed. The magnetic diffusivity of the ambient gas is taken to be very much greater than each of the viscous diffusivities and the thermal diffusivity.

It is confirmed in detail that all plane-polarized trans-Alfvénic shocks, except the 2–3 type (which is the only type that has no steady-state structure), are unstable to disturbances in those components of the magnetic field and velocity which are transverse to the plane of polarization. An incident Alfvénic wave, consisting of a weak, diffusing current-sheet would initially cause the shock profiles of these transverse quantities to grow linearly with time, while outside this shock region steady, uniform states would be reached. An integral condition is obtained which, together with the relevant boundary conditions; determines the asymptotic shock profiles of the transverse quantities whenever the disturbance is such that a steady state is reached. This removes the puzzling arbitrariness of these profiles.

It is also shown that the ‘1–4’ trans-Alfvénic shock is unstable to magnetoacoustic waves and contact fronts. A qualitative description of how it may be broken up is given. If the disturbance is of finite extent, a steady state is reached. An integral equation is obtained which, together with the relevant boundary conditions, determines the asymptotic steady-state shock-profiles for this case. This removes the apparent arbitrariness of these profiles.

The behaviour of ‘2–3’ trans-Alfvénic shocks and of switch-on and switch-off shocks is discussed.

A symposium on the subject ‘Concentrated vortex motions in fluids’ was held at the University of Michigan, Ann Arbor, Michigan, from 6 to 11 July 1964. The symposium was organized by the International Union for Theoretical and Applied Mechanics, and participation was restricted to about 150 people invited on the basis of their active interest in the subject. The author was the secretary of the international committee which planned the meeting, and has prepared the following account of the scientific developments of the symposium in order to make them widely available.[dagger] There will be no other publication of the proceedings of the symposium.Editors’ note. This novel method of reporting the proceedings of a symposium to the scientific community at large deserves consideration by the organizers of other meetings. There are obvious objections to the common plan of publishing in one volume the papers presented at a meeting: being an isolated volume it is not certain to be acquired by libraries; the papers are published at a time and under conditions not known in advance to authors; and papers are not usually subjected to the scrutiny of referees before publication. However, some kind of widely available report on an important meeting seems to be desirable, particularly when participation is by invitation only. Dr Küchemann's detailed account of the papers presented and the discussion that followed them should enable readers to see whether there were any developments at the Michigan Symposium of direct interest to them, and any such developments can be followed up with the aid of the very complete list of references at the end of the report.The Editors will be glad to consider other similar reports of conferences concerned with some aspect of fluid mechanics for publication in the Journal. Further details of the work described at the symposium, and of related previous work, are given in the references quoted.

The structure of plane steady shock waves in a gas with several internal energy modes which relax in parallel is investigated. Transport effects are neglected. Conditions for continuity and monotonicity of the velocity profile are discussed; when all modes have constant specific heats and relaxation times it is established that velocity must decrease monotonically. Internal mode energy contents may overshoot their local equilibrium values.

Numerical results for waves in a hypothetical gas with two relaxing modes are presented for purposes of illustration.

The three-dimensional equations for a general inertial ocean current are transformed so that the temperature rather than the vertical co-ordinate appears as an independent variable. A downstream power series expansion is made of the equations and boundary conditions, which involves an expansion about the mean potential vorticity. A general first-order solution is obtained for a boundary current between two level surfaces, one of no motion and one of uniform temperature. The case of constant potential vorticity is treated for arbitrary inviscid boundary conditions; it is found that the current can exist as a boundary layer only if the open ocean geostrophic drift is westward everywhere in the depth interval. This result is extended to arbitrary potential vorticity distributions by an asymptotic analysis in physical space.

The present paper deals with the practical and rigorous solution of the potential problem associated with the harmonic oscillation of a rigid body on a free surface. The body is assumed to have the form of either an elliptical cylinder or an ellipsoid. The use of Green's function reduces the determination of the potential to the solution of an integral equation. The integral equation is solved numerically and the dependency of the hydrodynamic quantities such as added mass, added moment of inertia and damping coefficients of the rigid body on the frequency of the oscillation is established.

The local and total mass flow through a circular orifice and a two-dimensional slit with large upstream to downstream pressure ratio is calculated for large Knudsen number. The solution is obtained by performing one iteration on an integral form of the kinetic equation, starting with the free molecular solution for the distribution function. A relaxation model is used to describe the inter-molecular collisions. There is found to be a variation of approximately 40% in the local mass flow perturbation (from the free molecular value) across the orifice. For the slit the variations are smaller when the Reynolds number is less than 0·3, but are comparable for greater values. This Reynolds number, defined by $Re = 2 R \rho (kT|m)\frac {{1}}{{2}}/\m$ with R the radius of the orifice or half-width of the slit and with the flow field quantities evaluated far upstream, appears to be a less ambiguous quantitative parameter than the inverse Knudsen number. The ratio of total mass flow to the free molecular value is given by [1+0·083 Re+o(Re)] for the orifice and [1−0·057 Re ln Re+0·055 Re+o(Re)] for the slit. There is agreement within experimental error between the theoretical results and available experimental data for the circular orifice when Re < 1.

In a backward boundary layer the fluid has, in the mathematical model, been flowing along a solid wall through an infinite distance. The co-ordinate distance x along the boundary is measured upstream, and the velocity U of the flow outside the boundary layer is taken as negative. The main application is to flow in converging passages.

The existence of similar solutions is considered, with emphasis on the correct asymptotic behaviour for large values of the stretched co-ordinate normal to the wall. This emphasis is shown to be necessary in considering backward boundary layers.

For two-dimensional flow in converging passages the requirement that a boundary layer should be possible for vanishingly small viscosity with a potential core flow is shown to lead directly to Hamel's spirals as the shape of the boundary streamlines.

Flow in axisymmetric converging passages is considered. For flow in a cone there is no limit as the viscosity tends to zero, and no potential core flow with a boundary layer is possible. The nature of a solution of the Navier-Stokes equations for laminar flow is considered.

A digital computer method for solving certain problems involving two-dimensional incompressible viscous flow is described. The time-dependent case is treated; the mathematical problem is thus that of solving a non-linear fourth-order partial differential equation in three variables. The choice of difference equations, of relaxation procedure, the kind of approximation to boundary conditions, and the resulting computational stability, speed, and accuracy are considered. Most experience so far has been for a rectangular region for which boundary velocities are prescribed as certain functions of time; an example of one such problem showing vortex formation and break-up is given.

This paper considers the barotropic stability of an asymmetric zonal current on a rotating earth. The current is of hyperbolic tangent form in latitude. For this stability problem the neutral wave solutions are found and estimates of the growth rates for the unstable waves are obtained from the neutral solutions as in earlier investigations by the writer. It is again found that the beta effect which is due to the earth's curvature and rotation tends to stabilize the basic flow.

The stability of the basic flow is examined for a special case. For this case the current is centered at 35 degrees latitude, has a total shear of 30 m sec−1 and a half-width of 550 km. The most unstable waves are found to be wave-numbers 6 and 7 which amplify by a factor of e in 2·9 days. In addition, wave-numbers 5, 8 and 9 are also unstable. The stability of the symmetric jet is also examined for a comparable case. It is found that a wider band of wave-numbers is unstable. The most unstable wave is wave-number 8 which amplifies by a factor of e in 2·7 days.

In conclusion it is noted that these growth rates are slower than the amplification rates for the unstable waves associated with the baroclinic stability problem.

This paper presents an investigation into the influence of rotation on the laminar asymptotic velocity and temperature profiles obtained when fluid flows through a vertical tube which rotates about a parallel axis with uniform angular velocity, and which is subjected to a uniform temperature gradient. Rotation induces secondary free convection flow in the plane perpendicular to the axis resulting in non-symmetrical axial-velocity and temperature profiles which modify the solved using a series expansion in ascending powers of the rotational Rayleigh number, the resulting solution being approximate and valid only for low rates of heating.

The nature of the steady-state viscous flow between two large rotating disks has often been discussed, usually qualitatively, in the literature. Using a version of the numerical method described in the preceding paper (Pearson 1965), digital computer solutions for the time-dependent case are obtained (steady-state solutions are then obtainable as limiting cases for large times). Solutions are given for impulsively started disks, and for counter-rotating disks. Of interest is the fact that, at high Reynolds numbers, the solution for the latter problem is unsymmetrical; moreover, the main body of the fluid rotates at a higher angular velocity than that of either disk.

The flow in the wake of a two-dimensional model with a blunt trailing edge was examined at Reynolds numbers (based on model chord) between 1·4 × 105 and 2·56 × 105. The ratio of total boundary-layer thickness at the trailing edge to model base height was approximately 0·5. Measurements were taken of base pressure and vortex shedding frequency together with traverses of the wake using a hot-wire anemometer. Traverses carried out along the wake showed a peak in the root-mean-square velocity-fluctuation at a distance equal to one base height from the model rear face. The position of the peak is referred to as the position of the fully formed vortex. The investigation was extended to a model fitted with splitter plates up to four base heights long. For each plate tested, a position of the fully formed vortex was found, and its distance from the model base was discovered to be inversely proportional to the base pressure coefficient. The flow about the model with splitter plates is described as being separated into five régimes of flow.

When the upward flow of a fluid through a bed of particles of appropriate and almost uniform size is rapid enough so that the drag on each particle is as great as the particles buoyant weight, the particles do not remain close packed and the bed is said to be fluidized. Industrial uses of fluidized beds in the chemical and petroleum industries in particular are already extensive. Uses in the atomicenergy industry are being developed.

In this paper a mathematical model which describes the phenomena on a continuum basis is deduced. With this model we find that the system is unstable to small internal disturbances. Alternatively, we find that surface waves can be propagated (with attenuation) in the composite fluid and these waves for fluidized beds with a high ratio of solids density to fluid density are stable. These results are in agreement with experiment. Hot beds, where strongly exothermic reactions may be taking place, centrifugal beds (beds fluidized within a rotating system), and electromagnetic beds (those in which the particulate phase is electrically conducting) are all shown to be unstable to small internal disturbances.

The equations derived here may also be used as approximate equations for dispersed particle two-phase flow.

The steady, axisymmetric, converging motion of a viscous incompressible fluid inside an infinite right circular cone is considered. It is shown that the exact solution of the Navier-Stokes equation for the stream function Ψ is of the form Ψ(r,θ) = AF(rv/A,θ), where (r, θ) are spherical polar co-ordinates chosen so r = 0 is the apex and θ = 0 is the axis of the cone, 2πA is the volumetric flow rate, and v the kinematic viscosity of the fluid. Asymptotic expansions of the stream function are found for large and small rv/A.

For large rv/A, Stokes's method for slow motions is generalized to obtain a complete asymptotic expansion. Except for cones of special angles, all terms in this expansion may theoretically be found.

For small rv/A a solution is constructed in two parts, namely, an inner expansion which starts from boundary-layer type equations as well as the no-slip condition at the wall, and an outer expansion in unstretched variables rv/A and cosθ which satisfies the boundary conditions at the axis of the cone. The condition that the inner solution merge with the outer solution with an exponentially small error requires an outer solution near the apex which is not potential sink flow, as might perhaps have been expected from the solution for two-dimensional flow in a wedge. The simplest outer flow satisfying the requirement is a vortex motion. Complete inner and outer expansions are developed and it is shown that they contain only six undetermined constants which must be determined by joining this solution numerically to the Stokes solution upstream. The inclusion of logarithmic terms in these expansions has not been found necessary.

The stability of a viscous liquid between two concentric rotating cylinders with an axial flow has been investigated. Attention has been confined to the case when the cylinders are rotating in the same direction, the gap between the cylinders is small and the axial flow is small. A perturbation theory valid in the limit when the axial Reynolds number R → 0 has been developed and corrections have been obtained for Chandrasekhar's earlier results.

In addition to the random processes described as horizontal turbulence, there are certain more regular processes which may contribute to horizontal mixing. One of these occurs in a shearing current, where the vertical gradient of velocity combined with vertical turbulent mixing leads to an effective diffusion in the horizontal direction. It is shown that this effect occurs in an alternating flow, such as a tidal current, as well as in a steady flow. In estuaries and coastal waters horizontal mixing by the shear effect may be associated with tidal currents, density currents or wind-driven currents. In each case the effective coefficient of horizontal diffusion, Kx or Ky, is inversely proportional to the coefficient of vertical eddy diffusion Kz. The occurrence of a stable gradient of density therefore increases the effective horizontal mixing very considerably. Results obtained from observations in the Mersey estuary and Irish Sea are compared with theoretical estimates of these effects.

An analysis is made of the effect of a longitudinal gravity field on two-dimensional supercavitating flow past wedges. Under the assumption that the flow is both irrotational and incompressible, a non-linear theory is developed for steady flow. By utilizing conformal mapping in combination with the Schwarz reflexion principle, the mathematical problem is reduced to a three-parameter, non-linear integral equation with one constraint. The equation is derived by reflecting the flow about the rigid boundaries; the constraint is obtained by requiring the net singularity strength inside the cavity-wedge system to be zero. A successive-approximation procedure is used to obtain a numerical solution of the integral equation. Typical results are presented in graphs and tables, and the results of the present work are compared to those of Acosta's linear theory.

The Newtonian theory of inviscid hypersonic flow is extended to obtain a solution uniformly valid in the subsonic region, and that is used to determine the position and shape of the sonic line. The main modification to the theory has to be made near the body surface and an expansion, essentially in terms of the stream function, is employed.

For simplicity the solution is limited to the cases of axially- and plane-symmetric flows. As an illustration of the theory the flows past a sphere and a circular cylinder are treated in some detail. Comparison with the numerical results of Garabedian and Lieberstein gives favourable agreement.