A characteristic feature of a steady trailing line vortex from one side of a wing, and of other types of line vortex, is the existence of strong axial currents near the axis of symmetry. The purpose of this paper is to account in general terms for this axial flow in trailing line vortices. the link between the azimuthal and axial components of motion in a steady line vortex is provided by the pressure; the radial pressure gradient balances the centrifugal force, and any change in the azimuthal motion with distance x downstream produces an axial pressure gradient and consequently axial acceleration.

It is suggested, in a discussion of the evolution of an axisymmetric line vortex out of the vortex sheet shed from one side of a wing, that the two processes of rolling-up of the sheet and of concentration of the vorticity into a smaller cross-section should be distinguished; the former always occurs, whereas the latter seems not to be inevitable.

In § 4 there is given a similarity solution for the flow in a trailing vortex far downstream where the departure of the axial velocity from the free stream speed is small. The continual slowing-down of the azimuthal motion by viscosity leads to a positive axial pressure gradient and consequently to continual loss of axial momentum, the asymptotic variation of the axial velocity defect at the centre being as x−1 log x.

The concept of the drag associated with the core of a trailing vortex is introduced, and the drag is expressed as an integral over a transverse plane which is independent of x. This drag is related to the arbitrary constant appearing in the above similarity solution.

The problem of defining an effective velocity of convection for turbulent fluctuations in a shear flow is considered, and the definitions adopted by various workers are discussed. An experiment in the shear layer of a circular jet has shown that the usual definitions, based on peaks of the space-time correlations of the fluctuations, yield convection velocities whose magnitudes depend on the value of space or time separation chosen. An alternative approach shows that, by considering the turbulent field as a superposition of harmonic travelling waves, a wave-number/velocity spectrum can be defined that lends itself to the definition of a wave-number-dependent convection velocity and an overall convection velocity, both of which have real physical significance. An experimental technique is described for obtaining the spectrum, and results are presented for one position in the shear layer of the jet.

The interaction between a turbulent wind and the motion of uniform saltating grains of sand or soil, so massive as to fail to enter into suspension, is examined on the basis of two complementary hypotheses. The first asserts that the effect of the moving grains on the fluid outside the region to which saltation is confined is similar to that of solid roughness of height comparable with the depth of the saltation layer. The second requires the concentration of particles engaging in the saltation to adjust itself so that the shear stress exerted by the wind on the ground—different from that acting on the fluid outside the saltation layer by an amount accountable to the change in horizontal momentum suffered by the particles in their passage through the fluid—is just sufficient to maintain the sand-strewn surface in a mobile state.

Existing experimental data on the wind profiles outside the saltation region and the horizontal flux of particles through it are shown to be consistent with these hypotheses.

The second hypothesis implies a self-balancing mechanism for controlling the concentration of saltating particles. For if the concentration is too low the shear stress at the surface rises above the value required merely to secure mobility and more particles are encouraged to leave the surface; conversely, too large a concentration depresses the surface stress, and the consequent loss of surface mobility inhibits saltation and reduces th concentration of particles until equilibrium is restored.

The instability to small two-dimensional disturbances of an electrically conducting fluid of variable density is investigated. The viscous fluid is bounded between two vertical parallel planes normal to which a magnetic field of constant intensity is applied. Significant parameters upon which the behaviour of the Rayleigh number at neutral stability depends are the Hartmann number M and the wave-number α which is associated with a periodic disturbance with periodicity in the unbounded horizontal direction.

The solution may be sought by considering basic disturbances which are either symmetric or antisymmetric about the median plane parallel to the boundary planes. It is found that for a given magnetic field strength the critical Rayleigh number governing stability is associated with an antisymmetric disturbance of zero wave-number. The least stable symmetric disturbance which arises when the wave-number is zero is less easily excited. This trend is seen again in the purely hydrodynamic case (M = 0) where, corresponding to a finite wave-numbe value, the more unstable mode at neutral stability is found to be an antisymmetric one.

The most unstable situation occurs when both the Hartmann number and the wave number zero. In this case the result of Wooding (1960) that the minimum critical Rayleigh number is zero and is associated with a symmetric disturbance is reobtained.

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

It is shown that the solution of the problem of scattering of long waves by a finite barrier in a rotating system may be obtained directly from the solution of the electromagnetic scattering problem for a finite strip. It is shown that the effect of rotation is to produce on both sides of the barrier, but away from the ends, a wave which is propagated without attenuation parallel to the barrier.

The paper is concerned with formulation of the gas dynamic conservation equations for the individual species in a non-equilibrium partially ionized gas mixture. As an example, the conservation equations for the electrons and the overall conservation equations are developed for a three component plasma consisting of electrons, singly-ionized positive ions and neutral atoms. Non-elastic collisions are represented by the collisional-radiative decay mechanism of Bates, Kingston & McWhirter (1962a, b). Maxwellian velocity distributions are assumed, but the electrons are allowed to have a temperature different from the heavier particles and to drift relative to them. Particular attention is given to the electron energy balance equation which differs from that used by other investigators.

The effect of compressibility is included here in a study of wakes created in two-dimensional, steady, aligned-fields, magnetogasdynamic flow past obstacles. The gas is assumed to be viscous, resistive, and thermally conducting. With the Oseen type of approximation as well as the magnetogasdynamic boundary-layer approximation, a great simplification in the formulation of wakes results. The boundary-layer equations, although linearized, still retain the coupling between the velocity, the magnetic, and the temperature fields. The solution of the magnetogasdynamic wake, in general, is a superposition of three individual-wake components, each satisfying a diffusion type of equation. Only one of them is capable of extending upstream. Hence the wake picture is generally characterized by a conventional downstream wake with also the possibility of the existence of an upstream one.

To be sure, the general features of the magnetogasdynamic wakes in aligned-fields flows are similar to those of an incompressible fluid; however, the flow now, instead of just being subalfvénic, must be subcritical for the upstream wake to occur. That is, the flow condition corresponding to the occurrence of the upstream wake is such that the sum of the square of the Mach number M∞ and the square of the Alfvén number A∞ is less than unity. Evidently it is the mechanism of magneto-sonic-wave propagation that modifies the transition of wakes from the Alfvénic point A∞ = 1 (for an incompressible fluid) to the subcritical are $A^2_\infty + M^2_\infty = 1$ in the Taniuti-Resler diagram.

The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re < 0·03. When equal-sized spheres of Re > 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 < Re < 0·16. But, if 0·16 < Re < 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 < Re < 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re > 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.

The classical problem of the motion of a one-dimensional unsteady shock generated by a piston moving with velocity vp = ctn is extended to take into account thermal radiation effects by the similarity method of Taylor and Sedov. Gray gas and local thermodynamic equilibrium are assumed and a modification of the Schuster-Schwarzschild differential equation for the heat flux is adopted. The optical thickness is not restricted to be thin or thick, and the absorption coefficient is assumed to vary with the density and temperature. Numerical results indicate that the pressure and velocity are not affected much by the radiation, but the density, temperature and radiant heat flux are changed considerably.

The previous works on the motions of visco-plastic fluids in porous media are generalized in order to include the phenomenon of hysteresis. Only slowly varying motions are studied to any extent, when all inertial terms can be neglected without appreciable error. A general equation is deduced and some features of the motion are discussed. Simple particular cases without free or seepage surfaces are considered, when the fluid moves in the whole porous medium. A plane example points out the existence of different regions of motion during the variation of the heads in upper and lower water reservoirs. The necessity of further research on the flow of visco-plastic fluids through porous media is emphasized.

This paper considers the transient motion of a viscous fluid in a container rotating with constant angular velocity. The principal objective is to study the manner in which an arbitrary initial state of motion becomes a rigid rotation. In order to concentrate on the effects of viscosity, only the spherical container is studied here in great detail. A general theory will be presented in a subsequent publication.

Several sources of non-uniform behaviour make the analysis difficult and complex. In particular, there are three important time scales, viscous boundary layers, boundary-layer resonances at critical latitudes and intricate side-wall effects. The basic aproach consists of an expansion procedure by means of which the general inviscid solution is corrected for viscous effects and is made uniformly valid in time through the critical spin-up phase. Uniform validity is effected through the elimination of secular terms with unacceptable growth rates arising from the asymptotic perturbation series.

The interior (inviscid) motion leads to a non-self-adjoint partial differential equation eigenvalue problem with many intriguing properties. The general expansion theorem, orthogonality relationships, and viscous decay factors are deduced and used to solve the arbitrary intial-value problem. It is shown that the depth averaged circulation about circular contours, x2 + y2 = r2, is extracted from the fluid in the spin-up time scale T = L(Ωv)½. This is accomplished by a secondary non-oscillatory convective motion produced by suction into the Ekman layer. The excess circulation not eliminated in this way excites inviscid inertial oscillations which are also caused to decay by the boundary layers in the same time scale. Some very small residual effects decay in the ordinary viscous diffusion time, but all the essential processes are concluded in the much shorter interval. All modal oscillations in a sphere are determined and several specific calculations of frequency and decay rate are made and compared with experimental data. Perhaps the most important of these concerns the mode corresponding to rigid internal motion about another axis which can be produced by impulsively changing the rotation axis of the container. Agreement between theory and experiment is very good in all cases compared thus far.

The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.

A study of the stability of flow between concentric cylinders, with the inner one rotating, distinguished three kinds of instabilities: the familiar axisymmetric mode, an azimuthal mode with the predicted exp i(θ—ωt) angular dependence, and a completely non-symmetric instability which apparently arises from the interaction of the other two. The effect of small axial flow upon all of these modes was to give an approximately parabolic dependence of the critical Taylor number on the axial flow rate. In the case of the axisymmetric mode, agreement with the theory of Krueger & Di Prima (1964) was found to be excellent.

A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n < 7, which explains why this configuration is not observed for more than six spheres.

When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated.

In this paper the turbulent flow characteristics of viscoelastic fluids are investigated quantitatively. The outstanding property of these flow fields is seen to be a very pronounced suppression of turbulence, accompanied by major reductions in the turbulent drag coefficients. Careful measurements of the rheological properties of the several fluids used suggest that the observed turbulence suppression is a function of the ratio of the elastic to the viscous forces developed in the fluid. An empirical correlation of the results, based upon this observation, is proposed; the present data, while indicative, are not sufficiently extensive to verify conclusively the existence of a quantitative correlation.

In a number of respects, the observed reduction in drag is similar to that which may be obtained through promotion of ‘slip’ at the tube wall or by addition of particulate matter. It is shown that slip phenomena are clearly distinct from those studied in the present work but that particulate effects (albeit of much greater magnitude than observed heretofore) cannot be ruled out as contributory mechanisms. Further studies are thus required to determine the relative importance of continuum (viscoelastic) and particulate effects.

The Hartmann air jet generator (Hartmann 1939) is sound wave generator in which the sound is generated by oscillations of a shock developed in an over-expanded air jet by means of a blunt body: a resonator. A theory for the instability mechanism is advanced, and it is found that the result calculated from the theory are in good agreement with the experimental observations.