This paper deals with Stokes flow due to a stationary axially symmetric slender body in a uniform stream, which may be either parallel or perpendicular to the axis of the body. The effect of the body is represented by distributions of singularities along a segment of its axis of symmetry. Systems of linear integral equations for these distributions are obtained, and the first few terms of uniformly valid (in the Stokes region) asymptotic expansions in the slenderness ratio are discussed. The leading terms yield the expected result that the drag on the body in a transverse stream is double that in an axial stream. The second approximation to the ratio of these two drags is also independent of the body shape.

A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUε, where U is a measure of the undisturbed velocity and ε = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of ε2 are presented, together with iterative relations which allow their determination to the order of any power of ε. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of ε, and, for a cylindrical body, to the order of ε3.

Thermal instabilities of a contained fluid are investigated for a fairly general class of problems in which the dynamics are dominated by the effects of rotation. In systems of constant depth in the direction of the axis of rotation the instability develops when the buoyancy forces suffice to overcome the stabilizing effects of thermal conduction and of viscous dissipation in the Ekman boundary layers. Owing to the Taylor–Proudman theorem, a slight gradient in depth exerts a strongly stabilizing influence. The theory is applied to describe the instability of the ‘lower symmetric régime’ in the rotating annulus experiments at high rotation rates. An example of geophysical relevance is the instability of a self-gravitating, internally heated, rotating fluid sphere. The results of the perturbation theory for this problem agree reasonably well with the results of an extension of the analysis by Roberts (1968).

The paper examines the stability of the uniform flow which approaches a two-dimensional stagnation region formed when a cylinder or a two-dimensional blunt body of finite curvature is immersed in a crossflow. It is shown that such a flow is unstable with respect to three-dimensional disturbances. This conclusion is reached on the basis of a mathematical analysis of a simplified form of the disturbance equation for the stream-wise component of the vorticity vector. The ultimate, or stable, flow pattern is governed by a singular Sturm–Liouville problem whose solution possesses a single eigenvalue. The resulting flow is one in which a regularly distributed system of counter-rotating vortices is super-imposed on the basic, Hiemenz-like pattern of streamlines. The spacing of the vortices is a unique function of the characteristics of the flow, and a theoretical estimate for it agrees well with experimental results. The analysis is extended heuristically to include the effect of free-stream turbulence on the spacing.

The problem is similar to the classical Görtler–Hämmerlin study of the stability of stagnation flow against an infinite flat plate, which revealed the existence of a spectrum of eigenvalues for the disturbance equation. The present analysis yields the same result when an infinite radius of curvature is assumed for the blunt body.

Some exact solutions of the steady M.H.D. equations for an inviscid infinitely conducting fluid are discussed, and a generalization of the Prendergast model for an idealized magnetic star is obtained, in which there is a finite azimuthal swirl velocity of the fluid. The flow in all examples is bounded by a sphere and expressions are found for the distribution of energy between the fluid motion and magnetic field.

Numerical methods have been used to investigate the steady incompressible flow past oblate and prolate spheroids for Reynolds numbers up to 100. The ratio of minor to major axis of the spheroids investigated were 0·9, 0·5 and 0·2, together with 1·0, which represents the limiting case of a sphere. The pressure distribution and the skin and form drag coefficients were numerically evaluated for the various Reynolds numbers. Streamlines, equi-vorticity lines and equivelocity lines are presented and show in detail the flow characteristics.

The idea of distributing static probe cross-sectional areas so as to render the probe insensitive to Mach number is combined here with that of using non-circular cross-sections to render probes insensitive to yaw and angle of attack. Appropriate non-circular cross-sections are described in detail, and a general means of designing blunt or slender probes to have zero sensitivity to yaw and angle of attack in potential flow is described. Four experimental probes have been tested, and test results are presented. These results show that the probes are quite insensitive to yaw and angle of attack within certain limiting angles, which are assumed to correspond to the onset of flow separation.

The boundary-layer flow over a semi-infinite horizontal circular cylinder heated to a constant temperature and immersed in a uniform axial free stream is discussed in five situations corresponding to successively greater displacements from the leading edge. In the first three cases the drift velocity due to buoyancy is assumed small compared to the axial velocity component. Close to the leading edge of the cylinder the techniques of Seban & Bond are extended to include the drift velocity; far from the leading edge the asymptotic series methods of Stewartson, of Glauert & Lighthill, and of Eshghy & Hornbeck are employed to obtain a solution for the drift velocity. In the intermediate zone where the series solutions do not apply the appropriate partial differential equations are solved numerically. Still further downstream than the region where the ‘asymptotic’ solutions hold it is assumed that the boundary-layer flow is primarily convective and that the boundary layer is thin compared with the radius of the cylinder. A series solution is obtained which is valid near the lowest generator of the cylinder. Numerical methods are used to advance this solution upwards around the cylinder by solving the full boundary-layer equations step-by-step.

This work deals with an experimental investigation of natural convection inside a horizontal cylinder. The fluid, geometry, and thermal boundary condition were chosen so as to have a high Prandtl number and unit-order Grashof number.

The thermal boundary condition was established by imposing temperatures at two points, 180° apart, on the circumference of the cylinder. The resulting boundary condition for the full 360° was found experimentally and is presented. The apparatus was constructed so that the entire cylinder could be rotated in order to introduce an arbitrary heating angle into the boundary condition.

Temperature profiles and streamline patterns were observed at steady state for various values of this heating angle and for various initial conditions. For some cases, velocity profiles were also plotted. It was found that the interior ‘core’ of fluid was thermally stratified when the diameter containing the two imposed temperatures was horizontal. The flow occurred primarily in the region close to the cylinder wall. The cylinder was rotated so that the diameter containing the two imposed temperatures made an angle with the horizontal. The hotter of the imposed temperatures was always below the horizontal diameter of the cylinder. As this angle of rotation increased, it was found that the velocities encountered in the fluid increased and the degree of thermal stratification in the core region decreased. It was also found that the steady-state results were identical for the different initial conditions imposed. The results of this study are compared with previous work, both analytic and experimental.

The paper is concerned with a class of non-Newtonian fluids, ν-fluids, all of whose properties are determined by a single dimensional constant of the same dimensions as a viscosity. A regular nth-order ν-fluid is then defined to be one whose nth order time derivative of stress is a regular function of the local stress and flow fields and any of their space and time derivatives. The regularity condition determines the constitutive relation of such a fluid completely in terms of a finite set of non-dimensional constants which define the fluid.

An obvious property of these fluids is that their motions obey the same principles of Reynolds number similarity as those of a Newtonian fluid, and the primary aim of the paper is to examine the extent to which their flow properties are the same as those of the mean turbulent flow of a Newtonian fluid.

It is shown that a third-order fluid is the simplest ν-fluid which shares enough properties with turbulent motion to be worth further consideration in this context. At infinite Reynolds number, the constitutive relation for such a fluid reduces to the form \[ AS\ddot{S} + B\dot{S}^2 + CS^2 S^{\prime\prime} + DSS^{\prime 2} + Eu^{\prime 2}S^2 = 0, \] where A,B, …,E, are isotropic tensor constants of the fluid, S is the stress tensor, u’ is the total rate of strain tensor, dots denote total time derivatives, and primes denote space derivatives. A number of illustrative examples of the properties of such a constitutive relation are then considered, representing the decay of a homogenous stress field, the effect of rigid-body rotation on such a decay, the structure of the equilibrium stress field in the presence of homogeneous rate of strain, both with and without vorticity, and the nature of flow near a plane boundary. In all cases, the results appear to be consistent with known properties of turbulent motion, to the extent that the analysis is taken.

Finally, the effect of finite Reynolds number on the decay of an isotropic and homogeneous stress field is shown to be consistent with observations on the decay of isotropic turbulence.

Turbulent mass transfer to a wall at high Schmidt numbers is controlled by the velocity field within the viscous sublayer. Measurements have been obtained of the root-mean-square fluctuating mass transfer coefficient and the frequency spectrum of the fluctuating mass transfer coefficient for a Schmidt number of about 2300. From an order-of-magnitude analysis it is concluded that flow fluctuations in the direction of mean flow have little effect on the mass transfer fluctuations. A comparison of the mass transfer spectrum with the spectrum of the component of the velocity gradient in the transverse direction sz reveals that the high-frequency portion of the sz spectrum is not effective in transferring mass. Approximate relations between the mass transfer spectrum and the sz spectrum are developed for high frequencies and for low frequencies.

Electrochemical techniques are used to measure the circumferential component of the velocity gradient sz at the wall of a pipe through which a turbulent fluid is flowing. The ratio of the root-mean-squared value of sz to the time-averaged velocity gradient is found to be 0·09 or 0·1, depending on whether corrections are made for frequency response. The frequency spectrum is similar to that for the component of the wall velocity gradient in the direction of mean flow. The amplitude distribution function for sz is very roughly approximated by a Gaussian distribution.

Quarter-wavelength resonators for harbour entrances and similar applications are usually designed and built without regard to the end effect. This paper describes tests on rectangular resonators using water waves of unique frequency in a long narrow wave channel fitted with a wave generator at one end, a wave absorber at the other end and suitable wave filters. In these tests the wavelength was kept constant while the geometry of the rectangular resonant branch canal was systematically varied. Wave transmission across the resonator, as well as wave reflexion upstream of the resonator was measured. The results clearly indicate a considerable effect of main channel width on optimum resonator width and resonator length, invalidating the usual resonance theory (which predicts complete reflexion of the incident wave train when the length of the branch canal is one-quarter of a wavelength). For example, it is found that, under certain conditions, quarter-wavelength resonators may not have any effect on the incident wave. The work described is limited to the first resonant mode and to semi-infinite domains on each side of the resonator.