The additional pressure drop arising from the presence of a neutrally-buoyant, eccentrically positioned, spherical particle in a Poiseuille flow is calculated to O(a/Ro)5 (a = sphere radius; Ro = tube radius). Similar calculations (of a lesser order of accuracy) are given for non-circular conduits and for ellipsoidal particles. Due to changes in particle orientation resulting from rotation, the instantaneous pressure drop for an ellipsoid of revolution varies periodically with time. This pressure diminution is averaged over one period to obtain the time-average pressure drop.

Does the harbour of an artificial island need a bottom? The excitation of waves inside a partially immersed open circular cylinder is considered. An incident plane wave is expanded in Bessel functions and for each mode the problem is formulated in terms of the radial displacement on the cylindrical interface below the cylinder. The solution is obtainable either from an infinite set of simultaneous equations or from an integral equation. It is shown that the phase of the solution is independent of depth and resonances are found at wave-numbers close to those of free oscillations in a cylinder extending to the bottom. If the resonances of the cylinder are made sharper (by increasing the depth of immersion) the peak response of the harbour increases, but the response to a continuous spectrum remains approximately constant. Numerical results are obtained by minimizing the least squares error of a finite number N of simultaneous equations. Convergence is slow, but the error is roughly proportional to 1/N and this is exploited. The solution obtained from a variational formulation using the incoming wave as a trial function is found to give a very good approximation for small wave-numbers, but is increasingly inaccurate for large wave-numbers. Away from resonance the amplitude of the harbour oscillation is less than 10% of the amplitude of the incoming wave provided the depth of the cylinder is greater than about ¼ wavelength, and it is argued that in practice at the resonant wave-number oscillations excited through the bottom of the harbour will leak out through the entrance before they can reach large amplitudes. In an appendix the excitation of harbour oscillations through the harbour entrance is discussed, and some results of Miles & Munk (1961) on an alleged harbour paradox are re-interpreted.

A two-dimensional, semi-infinite tank, closed by a vertical wall at one end, is initially filled to a certain level with an in viscid liquid at rest. A gate in the lower part of the wall is suddenly opened to a region where the gas pressure is greater or less than the gas pressure over the free liquid surface in the tank. The liquid escapes (or moves inwardly) through the gate under the combined action of gravity and the driving pressure differential. The fluid motion, the shapes of the two free surfaces, and the discharge through the gate are calculated for small times after the opening.

The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion variable θΦ, this equation is equivalent to Whitham's (1958) approximate characteristic rule for the propagation of shock waves in non-uniform flow. While some further insight into the accuracy of Whitham's rule does emerge, the theory is not a selfcontained rational approach, since some knowledge of the wave interaction parameter Φ must be assumed. Analytical and numerical solutions to the basic shock refraction relation are presented for a broad range of flows in which the principal interaction occurs with disturbances generated upstream of the shock. These solutions include the passage of a weak oblique shock wave through: a supersonic shear layer, a converging or diverging flow, a pure pressure disturbance, Prandtl–-Meyer expansions of the same and opposite family, an isentropic non-simple wave region, and a constant pressure rotational flow. The comparison between analytic and numerical results is very satisfactory.

The function of a healthy ureter is analyzed in terms of a fluid-mechanical model. To the extent that the Reynolds number is of the order of one, the fundamental equations are shown to reduce to those of the theory of lubrication. It is found that from the point of view of the pressure variation with time (the urometrogram) the important part of the peristaltic wave is the constricting part. For this reason this part of the wave is represented with an algebraic expression of the form h ∼ xn making it possible to find closed form solutions. Using Fourier analysis in defining the complete wave shape of the ureter it was also possible to obtain numerical solutions. For both cases it is shown that there is good agreement between the theoretical and experimental pressure distributions, this not being the case for sinusoidal wave shapes. An approximate equation for the flux is developed and a universal relation is presented connecting the maximum pressure, flux and kinematic behaviour of the ureter.

The exact solution of the Stokes equations for the creeping motion of two spheres of arbitrary size and arbitrarily oriented with respect to a shear field is obtained by use of spherical bipolar co-ordinates. Numerical results are given for two special cases: (1) the free motion of two equal-sized spheres in simple shear flow and (2) the free motion of a sphere near a wall in the rotational shear field between two parallel disks rotating at different rates. The sphere trajectories calculated for the first of these problems are found to agree fairly well with those observed experimentally.

An experimental investigation of a large long air bubble moving into stationary water in a horizontal channel of rectangular cross-section is presented and three well-defined flow régimes for the water discharged beneath the bubble are described. The influence of surface tension on the bubble velocity is explained using the hypothesis that the radius of curvature of the two-phase interface close to the upper wall does not vary greatly with channel depth and is close to the theoretical value for a channel of such depth that the bubble is just motionless.

Aircraft downwash consists initially of a vortex pair descending with its accompanying fluid through the atmosphere. Condensation trails are formed in exhaust emitted into the accompanying fluid and the shapes of them and their evolution depend on the positions of the engines in relation to the wing tip vortices.

The atmosphere is stably stratified and so the descending accompanying fluid acquires upward buoyancy. Consequently vorticity is generated at the outside of the accompanying fluid and the flow pattern in the vortex pair is altered so as to produce detrainment of its exterior part. So long as any air which is a mixture of accompanying fluid and exterior air is detrained, the vortices remain stable, but the width of the pair decreases and its downward velocity increases with time as a result of the buoyancy. Eventually the upper stagnation point in the motion relative to the vortices begins to move upwards relative to the vortices so that some mixed fluid is entrained into the circulation and the vortices immediately become unstable, mixing occurs, the pressure in the core rises, and any vortex core trails that may exist appear to burst.

The motion produces downward-thrust blobs in trails from centrally placed engines, which correspond to the holes sometimes seen in cloud when distrails are formed.

We give further consideration to flow situations which are steady in the sense that ∂/∂t ≡ 0 but for which individual fluid elements are subjected to a small sinusoidal deformation. The particular situation studied involves the flow between eccentric circular cylinders which rotate about their axes with the same angular velocity Ω. The eccentricity is assumed to be small. It is shown that measurements of the force on the inner cylinder can be used to determine the complex dynamic viscosity of an elastico-viscous liquid.

The theory provides the necessary mathematical background for the operation of a new commercial rheometer. Consideration is given to the possibility of ‘on-line’ use of such an instrument for control purposes.

The paper examines the scattering properties of a cavitated line vortex when excited by line and point sources of sound. It is found that the vortex resonances discovered by Kelvin for incompressible flow are essentially unmodified by fluid compressibility, and that many of the resonant modes radiate energy to infinity. Those resonant modes dominate the vortex response, their amplitude growing algebraically with time in a driven instability of the model flow.

The off resonance response is dependent on the value of a normalized frequency parameter (ω/Ω)2|ln ka|. Ω denotes the angular velocity of the steady vortex flow, k the acoustic wave-number, and a the cavity radius. Even off resonance the cavity is an extremely efficient wave scatterer, the scattering efficiency increasing with source order. For example, the scattered energy of a point quadrupole is shown to exceed that of the direct field by a factor of 108 for the typical underwater flow Mach number of 10−2.

The equations of motion and energy governing a three-dimensional fluctuating flow of an incompressible fluid in the vicinity of a stagnation point on a regular surface have been integrated analytically. The velocity of the oncoming flow relative to the body oscillates in magnitude but not in direction.

It has also been shown that the analysis of Lighthill for the two-dimensional fluctuating flow may be extended to the three-dimensional flow (both chordwise and spanwise), namely for each point on the body there is a critical frequency ω0 such that for frequencies ω > ω0 the oscillations are to a close approximation ordinary ‘shear waves’, unaffected by the mean flow; the phase advance in the skin friction is then 45°. For frequencies ω < ω0 the oscillations may be closely approximated by the sum of two parts: one quasi-steady part and the other proportional to the acceleration of the oncoming stream. The phase advance in the skin friction is then tan−1 (ω/ω0).

In non-equilibrium inviscid blunt-body flows, the state of the gas at the stagnation point is known to be in thermodynamic equilibrium for all finite relaxation times. The dependence of this state on the non-equilibrium processes and body geometry is investigated for the most general conditions. The stagnation-point state is always found to be in a narrow range bounded on one side by the state obtained in an equilibrium flow. The other bound, called the frozen limit, is far removed from the state obtained in an identically frozen flow (infinite relaxation times). For certain state variables, the frozen-limit value lies outside the range determined by frozen and equilibrium flow. Significant errors are found in several published predictions of the stagnation-point state, resulting from the non-analytic approach to equilibrium in nearly frozen flow.

The two bounds on the pressure are expressible in terms of the normal shock density ratios for equilibrium and frozen flow. The actual pressure for an arbitrary flow situation is found to depend only on the shock nose radius and the relation between density and time in the relaxation zone behind a normal shock wave. If the density law is given by a single relaxation model, a closed form expression for the pressure is obtained. The analysis is carried out for both plane and axisymmetric flows, and is also valid for non-equilibrium free-stream conditions.

The time-mean characteristics of the laminar boundary layer near a two-dimensional stagnation point, when the velocity of the oncoming flow relative to the body oscillates are investigated analytically. First, when the amplitude of the oscillating velocity is small compared with the oncoming flow velocity, a series expansion is made and the obtained equations are solved numerically. The equations are also solved approximately in the extreme cases when the frequency is low and high. The obtained approximate solutions are compared with the numerical solutions in terms of skin friction. Next, when the frequency is high, the finite-velocity-amplitude case is treated. Time-mean velocity profiles and skin friction are obtained and compared with the small-amplitude case.

The error minimization technique is introduced as a method of obtaining the simultaneous solution to a set of partial differential equations. Because this numerical technique is at worst neutrally stable, it is believed to have fundamental advantages over existing techniques. The method is formulated in general for the steady fluid dynamic conservation equations and then applied to a specific case of irrotational isentropic flow of a perfect gas through a nozzle. Flow-field solutions were found in nozzle contours having a throat radius of curvature as low as 0·5. Comparison between experimental Mach number contours and the theoretical solution for a finite inlet nozzle with a conical exit is excellent.

Experimental studies of drag reduction and polymer degradation in turbulent pipe flow with dilute water solutions of unfractionated polyethylene oxide are described. Drag reduction results indicate that the magnitude of the reduction cannot be correlated on the basis of weight average molecular weight, rather the phenomenon depends strongly on the concentration of the highest molecular weight species present in the molecular weight distribution. Polymer degradation in turbulent flow is found to be severe for high molecular weight polymers causing appreciable changes in drag reduction and molecular weight with the duration of flow. Data indicates that drag reduction exists in the limit of infinite dilution suggesting that the phenomenon is due to the interaction of individual polymer molecules with the surrounding solvent and that the extent of reduction is relatively independent of pipe diameter when a comparison is carried out at equal solvent wall shear stresses. Consideration of the high viscosity obtained with solutions in an irrotational laminar flow field suggests this is due to polymer molecule deformation and that this phenomenon is central to the mechanism of turbulent flow drag reduction.

The three-dimensional mixing layer between two grazing perpendicular streams is investigated both theoretically and experimentally. Similar shear velocity profiles are present about 40 initial momentum thicknesses downstream of the splitter edges. Thereafter the velocity component in the resultant direction of the two streams is shown to be constant through the layer; the profile perpendicular to this (the shear direction) has approximately an error function form. The ratio of the maximum shear stress to the square of the shear velocity difference is about 50% greater than in a two-dimensional layer.

The non-linear stability of hydromagnetic flows is investigated by applying energy methods. A universal stability estimate, namely a stability limit for motions subject to arbitrary non-linear disturbances, is obtained for bounded or periodic domains. Our analysis is restricted to fluids possessing constant density and electrical conductivity and we do not take into account temperature or Hall effects. This result establishes the existence of an open region of certain stability near the origin of the $(\tilde{R}_e, \tilde{R}_m)$ Cartesian plane for every fixed Pm (where $(\tilde{R}_e, \tilde{R}_m)$ and Pm are the Reynolds number, magnetic Reynolds number and magnetic Prandtl number, respectively). The universal stability limit can then be improved by suitably defining a maximum problem using variational techniques, and obtaining the relevant Euler–Lagrange equations. The tentative solution to this problem gives a stability limit which enlarges the universal stability region. Our results are then compared with linear and experimental ones, with special emphasis given to the role played by the magnetic field.

An analytic relation is derived for the shock front velocity as a function of the initial parameters (pressure, density, and particle velocity) in a continuous, in-homogeneous medium. This relation was verified experimentally by using it to predict the propagation of a shock wave through a known rarefaction wave.

The Green's functions are found for a line source of internal waves in a wedge of stratified fluid of constant Brunt–Väisälä frequency, and are used to discuss the diffraction of internal waves by a wedge in all cases when the vertex angle of the wedge of fluid exceeds the acute angle between a characteristic and the horizontal. Robinson's (1970) results are confirmed and extended.

It is found that the diffracted waves are as important as the incident and reflected ones at all points that lie within a quarter-wavelength or so of either characteristic that passes through the apex. Also, in cases when all the reflected waves are inclined forwards, the diffracted waves lead to a positive backscatter of energy. When the vertex angle of the fluid wedge is less than the characteristic angle, the diffraction problem appears to be ill-posed, and, instead, the motion due to a vibrating body in the wedge of fluid is considered.

A general conclusion is that the so-called ray theory for internal waves, in which the incident and reflected waves alone are considered, has similar limitations to the geometrical theory of optics. Both theories involve the assumption that the typical dimensions in the problem are large compared to the wavelength.

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.