Drag reduction caused by dilute, distilled water solutions of five polyethylene oxides, molecular weights from 80,000 to 6,000,000, in turbulent pipe flow was studied experimentally in 0·292 and 3·21 cm ID pipes. It was found that the onset of drag reduction occurs at a well-defined wall shear stress related to the random-coiling effective diameter of the polymer. Laminar to turbulent transition is not, in general, delayed. The extent of drag reduction induced by a homologous series of polymers in a given pipe is a universal function of concentration, flow rate, and molecular weight. The maximum drag reduction possible is limited by an asymptote that is independent of polymer and pipe diameter. Flow structure measurements in a single polymer solution, 1000 ppm of molecular weight 690,000, showed that the mean flow follows an ‘effective slip’ model. In this, the mean velocity profile consists of a ‘Newtonian plug’ convected along at an additional, ‘effective slip’ velocity. The turbulent flow structure follows the ‘effective slip’ model towards the pipe wall, but is significantly different from Newtonian towards the pipe axis; in particular, the inertial subrange observed in isotropic Newtonian turbulence was absent in an energy spectrum taken on the pipe axis in the polymer solution.

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the steady vertical motion of a sphere. The measurements of the phase configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

The one-dimensional problem of shock-wave reflexion with relaxation is treated numerically by combining the shock-wave, characteristic, and Rayleigh-line equations. The theoretical results are compared with pressure and density measurements in CO2, and the agreement is found to be excellent.

The paper discusses the reflexion of a shock wave off a rigid wall in the presence of a boundary layer. The basic idea is to treat the problem not as a reflexion but as a refraction process. The structure of the wave system is deduced by a simple mapping procedure. It is found that a Mach stem is always present and that the bottom of this wave is bifurcated—called a lambda foot. The reflexion is said to be regular if the Mach stem and the lambda foot are confined to the boundary layer and irregular if either extends into the main stream. Two types of regular reflexion are found, one that has reflected compression waves and the other that has both reflected compression and expansion waves. Initial conditions are given that enable one to decide which type will appear. There are also two types of irregular reflexion, one that has a Mach stem present in the main stream and the other that is characterized by a four-wave confluence. Finally there are also two processes by which regular reflexions become irregular. One is due to the formation of a downstream shock wave that subsequently sweeps upstream to establish the irregular system and the other is due to boundary-layer separation which forces the lambda foot into the main stream.

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.

When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.

Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.

The two-dimensional internal wave pattern produced by a small disturbance moving with constant velocity along a line of arbitrary inclination in an inviscid, density stratified liquid is studied. It is shown how the far field wave pattern evolves as the inclination changes from the extreme of horizontal motion to that of vertical motion. It is found that, in general, waves propagate ahead of the disturbance.

This paper deals with a detailed application of the Whitham theory of finite amplitude wave propagation to open-channel steady flow of water of infinite depth past a slowly modulated wavy wall. A numerical procedure first used by Garabedian & Lieberstein (1958) for the solution of ‘stably-posed’ elliptic Cauchy problems is employed to obtain a map of the wave pattern on the free surface of the water. The appearance of a ‘shock’ in this solution is discussed in terms of previous analytical and experimental results.

The stability of a periodic internal wave has been investigated experimentally and theoretically. From the analysis it is found that if a primary wave, with wave-number k0 and frequency ω0, is perturbed by two infinitesimal wave-like disturbances with wave-numbers k1 and k1 + k0 and frequencies ω1 and ω1 + ω0, exponential growth of these disturbances will take place under certain conditions. The analysis also indicates which resonantly interacting disturbances can induce an instability and, when viscous dissipation is accounted for, predicts the minimum amplitude for which a wave is unstable. Experimental results demonstrate that this type of instability can cause the breakdown of a first mode internal wave propagating in a fluid composed of two layers of uniform density separated by a thin region in which the density varies continuously.

An analysis is described for convection from a circular cylinder subjected to transverse oscillations relative to the fluid in which it is immersed. The analysis is based upon use of the acoustic streaming flow field. It is assumed that the frequency involved is sufficiently small that the acoustic wavelength in the fluid is much larger than the cylinder diameter, and that there is no externally imposed mean flow across or along the cylinder. Solutions are presented which are appropriate for a wide range of Prandtl number, and the cases of small and of large streaming Reynolds number are distinguished. The analysis compares favourably with experiments when the influence of natural convection is small.

The magnetohydrodynamic lubrication flow in a step-type thrust bearing has been investigated both theoretically and experimentally. The experimental configuration consisted of nine identical segments arranged on a circular plate. Flow was induced by rotating the upper plate. Pressures, voltages and torques were recorded. Provision was made for passing current radially through the segments, and it was shown that the torque could be reduced to zero and even reversed by this means. Good agreement has been obtained between the analysis and the experimental results.

Nonlinear resonant wave triads composed of one finite and two infinitesimal components are unstable for sum interactions and neutrally stable for difference interactions. A similar criterion holds for tertiary interactions.

The velocities and the associated pressure gradients of infinitely long liquid-borne cylinders flowing freely in pipes are related analytically to their radial positions. These velocities and pressure gradients are compared with those of liquids in cylinder-free pipes and expressed as ratios. A digital computer was used to evaluate the resultant equations for values of the cylinder/pipe diameter ratio between 0·25 and 0·97, with radial positions varying from the fully eccentric to the fully concentric position. As the clearance between the pipe and the bottom of the cylinder increases, the pressure ratio (RP) decreases and the velocity ratio (RV) increases. The relationship between RP and RV is independent of liquid viscosity and density, capsule density and pipe diameter, and is shown to be nearly linear for the larger diameter ratios. The R, R relationships are compared with data from three experimental capsule pipelines with pipe diameters from ½ to 4 in., involving a variety of diameter ratios, cylinder lengths and densities, and oil viscosities. The experimental results for single capsules of finite length are shown to be in close agreement with the predictions for infinitely long cylinders.

The relevance of the analysis to capsule pipelining is indicated by relating experimental values of capsule velocity over a wide range of densities to the theoretical clearance of the capsules. A general relationship of this type would permit the optimization of the power requirements of any particular throughput of a given commodity.

The fluid is assumed to be inviscid and to be confined within two parallel planes, each perpendicular to the axis of rotation. A sphere is set moving, relative to the rotating fluid, in a straight line with uniform velocity and the temporal development of the flow structure examined. It is found that ultimately the flow has different properties inside and outside the cylinder [Cscr ], circumscribing the sphere and having its generators parallel to the axis of rotation. Inside [Cscr ] the fluid moves with the sphere as if solid; in early experiments of Taylor (1923) this phenomenon was observed. Outside [Cscr ] the motion is a two-dimensional potential flow past [Cscr ] as if it were solid. Then the asymmetry observed by Taylor and predicted in an earlier theory of the author for an unbounded fluid (1953) is not borne out. A partial explanation is offered.

The initial value problem related to axisymmetric forced oscillations of a rigidly rotating inviscid fluid enclosed in a finite circular cylinder is examined in linear approximation with the aid of the Laplace transform technique. An impulsive starting motion is considered. The solution consists of a ‘periodic’ motion which oscillates with the forcing frequency, together with a doubly infinite set of inertial modes whose presence is determined by the initial conditions and whose frequencies form a dense set in the range (−2ω, 2ω), where ω is the angular velocity. The nature of the periodic or ‘steady-state’ part of the solution is strongly dependent on the precise value of the forcing frequency α (α > 0) when α ≤ 2ω. In particular the system will resonate if α equals any one value of the dense set of resonant frequencies. It is shown that no internal sets of discontinuities in velocity or velocity gradient are present in the inviscid flow for finite times. Effects of viscosity on the inviscid solution are also discussed, and it is argued that when the inertial modes decay the steady-state flow will contain pseudo-random patterns of internal shear layers for some values of α < 3ω. It seems possible that these shear layers may be interpreted as owing their existence indirectly to viscosity.

Helmholtz stated and Korteweg proved that of all divergenceless velocity fields in a domain, with prescribed values on the boundary, the solution of the Stokes equation minimizes the rate of viscous energy dissipation. Hill & Power and also Kearsley proved the corresponding reciprocal maximum principle involving the stress tensor. We prove generalizations of both these principles to the flow of a liquid containing one or more solid bodies and drops of another liquid. The essential point in doing this is to take account of the motion of the solids or drops, which must be determined along with the flow. We illustrate the use of these principles by deducing several consequences from them. In particular we obtain upper and lower bounds on the effective coefficient of viscosity and a lower bound on the sedimentation velocity of suspensions of any concentration. The results involve the statistical properties of the distribution of suspended particles or drops. Graphs of the bounds are shown for special cases. For very low concentrations of spheres, both bounds on the effective viscosity coefficient are the same, and agree with the results of Einstein and Taylor.

Extensive visual and quantitative studies of turbulent boundary layers are described. Visual studies reveal the presence of surprisingly well-organized spatially and temporally dependent motions within the so-called ‘laminar sublayer’. These motions lead to the formation of low-speed streaks in the region very near the wall. The streaks interact with the outer portions of the flow through a process of gradual ‘lift-up’, then sudden oscillation, bursting, and ejection. It is felt that these processes play a dominant role in the production of new turbulence and the transport of turbulence within the boundary layer on smooth walls.

Quantitative data are presented providing an association of the observed structure features with the accepted ‘regions’ of the boundary layer in non-dimensional co-ordinates; these data include zero, negative and positive pressure gradients on smooth walls. Instantaneous spanwise velocity profiles for the inner layers are given, and dimensionless correlations for mean streak-spacing and break-up frequency are presented.

Tentative mechanisms for formation and break-up of the low-speed streaks are proposed, and other evidence regarding the implications and importance of the streak structure in turbulent boundary layers is reviewed.

The differential equation for the vertical velocity of a gravity wave in an inviscid shear flow is singular at a level where the mean fluid velocity is equal to the horizontal phase velocity of the waves. It has been shown that a wave travelling through such a layer has its amplitude attenuated by a constant factor dependent on the local Richardson number. In this paper the results obtained by solving numerically the full sixth order differential equation, which is derived by including viscosity and heat conduction in the problem, (and is not singular) are discussed, and the same attenuation factor is found. Some experiments which confirm certain aspects of the theory are described in an appendix.

The equations of mean motion indicate that two-dimensional turbulent wakes, when subjected to appropriately tailored adverse pressure gradients, can be self-preserving. An experimental examination of two nearly self-preserving wakes is reported here. Mean velocity, longitudinal and lateral turbulence intensity, inter-mittency and shear stress distributions have been measured and are compared with Townsend's data from the small-deficit undistorted wake. In comparison with the undistorted case, the present wakes have slightly lower turbulent intensities and significantly lower shear stresses, all quantities being non-dimensionalized by a local velocity scale taken as the maximum mean velocity deficit. A consideration of the reasons for the shear stress reduction leads to an expression from which the shear stresses in any symmetrical free equilibrium shear flow can be found. This relationship is used to calculate the rate of growth in the measured wakes, with reasonable success.

Experiments in which a horizontal layer of convecting air is probed by one beam of a Michelson interferometer are described. When the localized interference fringes are horizontal, they indicate that the beam is wide enough to provide a suitable horizontal average. When the fringes are oriented nearly vertically, quantitative temperature measurements may be made. Results are presented for ratios (λ) of Rayleigh number to critical Rayleigh number of 1·48, 3·81 and 16·0. The temperature profile becomes more distorted from linear as the Rayleigh number is increased. An isothermal central region and thermal boundary regions each occupy one-third of the layer at λ = 3·81. By λ = 16·0 each boundary region occupies only one-quarter of the layer thickness, and the central region shows a reversed gradient. No full calculation is presently available to compare to the measurements. However, the shape assumption, using the first eigenfunctions of the linear stability problem, predicts the profile rather well for λ = 1·48 and 3·81 if Nusselt number agreement is imposed.

The stability to small inviscid disturbances of a rotating flow, whose velocity components in cylindrical polars (r, 0, z) are (0, V(r), 0), is investigated when one boundary of the flow (r = b) is a free surface under the action of surface tension (γ), and the other is either at infinity, or a rigid cylinder (r = a ≠ b), or at the axis (r = 0). The free surface may be the inner or the outer boundary. A necessary and sufficient condition for stability to axisymmetric disturbances is derived, which requires that Rayleigh's criterion of increasing circulation be satisfied, and otherwise depends only on b, V(b), γ and the density of the swirling liquid. This condition may be extended to include non-axisymmetric disturbances when V ∝ 1/r and when V ∝ r although in the latter case it is no longer a necessary one. It is shown that, in the case V ∝ r, as well as V ∝ 1/r, the ‘most unstable’ disturbance on a rotating column of fluid will be non-axisymmetric if the rotation speed at the surface is sufficiently great. Several applications of the theory are suggested, and a possible experiment to test it is described.