The present paper deals with the temperature field past an isothermal sphere. Acrivos & Taylor (1962) have obtained results for the case R [les ] 1 with no restriction on the Prandtl number. The results of the present paper are for the case R < 1 in which the Prandtl number is of order unity. An expansion for the Nusselt number is given up to and including the term of order R2.

The heat transfer appropriate to low Reynolds number hot-wire anemometry is calculated from the full non-linear equations of motion and of heat transfer by an iterative method starting with the Oseen solution and its heat flux analogue. The second and third iterates yield close agreement with measured data.

Theoretical results are obtained for forced heat convection from a circular cylinder at low Reynolds numbers. Consideration is given to the cases of a moderate and a large Prandtl number, the analysis in each case being based upon the method of matched asymptotic expansions. Comparison between the moderate Prandtl number theory and known experimental results indicates excellent agreement; no relevant experimental work has been found for comparison with the large Prandtl number theory.

This paper presents some recent theoretical results on the energy exchange between a swimming flexible two-dimensional hydrofoil of finite profile thickness and the inviscid incompressible fluid in which the body swims. The rate at which kinetic energy is transferred to the fluid by the undulating hydrofoil, the power required to maintain the prescribed motion, and the resulting power available for propulsion are calculated in terms of the thickness to chord ratio and the displacement and rate of displacement of the hydrofoil. With a small unsteady perturbation theory, the analysis is decomposed to show separately the effects of the circulatory and non-circulatory flows, both depending on the first-order terms of the unsteady perturbation velocity components. In addition, an analysis is presented showing the effect of the non-linear unsteady pressure distribution on the surface of the hydrofoil. Contrary to what might be expected, this latter effect is of the same order of magnitude for a thick rounded-nose profile as for the flat plate where the effect is concentrated at the sharp leading edge and is related to the so-called suction force. However, except for small values of the reduced frequency, the non-linear contribution is negligible in comparison with the linear contribution.

New functions associated with the retarded flow in the wake are introduced and special techniques for their solution are presented, these being related to Theodorsen's function of unsteady airfoil theory for the special case of the undulating flat plate.

The numerical results reveal that the kinetic energy imparted to the fluid, the power required to maintain the motion, and the resulting propulsive power, follow closely those of an infinitesimal model for small values of the reduced frequency of oscillation, but diverge somewhat from the classical thin plate analysis for large reduced frequencies. Of particular interest is the fact that a very large percentage of the power required to maintain the motion is used in the generation of the wake, whereas a very small percentage of the power available for propulsion comes from the wake. This indicates that, if some mechanism could be devised to control the wake, very high swimming efficiencies could be attained. Fish, in all probability, have been succeeding in doing this for millions of years.

The equations governing high-frequency oscillatory viscous flows are investigated through the separation of the steady and the unsteady parts. All Reynolds number ranges are studied and the orders of magnitude of the steady streaming produced by the Reynolds stresses are established.

The oscillating circular cylinder at low Reynolds numbers is studied through the method of inner and outer expansions. Steady recirculating cells exist near the cylinder. The results compare very well with experiments. Analytic expressions for the streamfunction and the drag coefficient are obtained.

The oscillating flow towards an infinite plate is investigated in detail. The steady streaming is caused by the steady component of the Reynolds stress. The pressure gradient always causes reverse flow near the solid boundary.

The motion which develops in a deep layer of a viscous, thermally conducting fluid initially hot below and cold above some horizontal plane, so that the system is gravitationally unstable, is studied by laboratory and numerical experiments. Three cases are considered: (i) the flow which occurs in a porous medium when the interface is the lower boundary of the system; (ii) a similar study in a viscous fluid; (iii) an interface distant from the confining horizontal boundaries, in a viscous fluid. In all cases the initial development of the flow—assuming an initial source of noise, for example as temperature fluctuations—occurs within the thermal interface between the hot and cold fluid. The scale of the motion is set by the thickness of the interface.

The development of the disturbances in the interface involves: a period of local thickening and induced, damped motions in which the diffusion of heat and vorticity dominate; a period of gestation, involving rapid amplification, with the disturbance imbedded in the interface and diminishing importance of the role of diffusion of heat; a period of emergence of the disturbances from the interface, during which the accelerations are sufficiently rapid for molecular processes to be unimportant, entrainment being the dominant process, and the gravitational energy accumulated locally in the interface is largely removed; and finally a period of adjustment of the large eddies. The amplification process is adequately described by the linearized equations of motion.

It has been observed by Walters & Davidson (1963) that release of a mass of gas in water sometimes produces a rising toroidal bubble. This paper is concerned with the history of such a bubble, given that at the initial instant the motion is irrotational everywhere in the water. The variation of its overall radius a with time may be predicted from the vertical impulse equation, and it should be possible to make the same prediction by equating the rate of loss of combined kinetic and potential energy to the rate of viscous dissipation. This is indeed seen to be the case, but not before it is recognized that in a viscous fluid vorticity will continually diffuse out from the bubble surface, destroying the irrotationality of the motion, and necessitating an examination of the distribution of vorticity. The impulse equation takes the same form as in an inviscid fluid, but the energy equation is severely modified. Other results include an evaluation of the effect of a hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension.

Townsend has derived a relation between mean vorticity and Reynolds stress valid in the wall layer of a turbulent flow. The vorticity Ω appears as a function of the local stress τ and its gradient. Such a relation is better suited for use in the vorticity equation than in the momentum equation. The Rayleigh problem, whose vorticity equation is simply ∂Ω/∂t = ∂2τ/∂y2, is introduced as a setting for Townsend's theory. Certain wall speed programmes are shown to generate Rayleigh layers that are exactly self-similar in the fully turbulent part of the flow. Those layers correspond to Clauser's equilibrium boundary layers. A formal analogy between the two families is found; the analogy becomes quantitatively exact in the limit of infinite Reynolds number. The Rayleigh problem is posed in similarity form. A composite non-linear, ordinary differential equation for the stress profile is deduced from a two-layer model incorporating Townsend's relation for the wall layer and Clauser's constant eddy-viscosity assumption for the outer layer. The profile depends on the wall-speed programme selected and on two empirical constants: the combination λ = √(k)/k of Clauser's k and Kármán's k, and Townsend's constant B. Closed-form solutions for arbitrary λ and B are obtained in two important cases: constant wall stress, analogous to constant pressure above a boundary layer, and zero wall stress, corresponding to continuous separation. The velocity profile in the wall region of a continuously separating Rayleigh layer is found to depend sensitively on B.

The stability of viscous flow between eccentric cylinders is analysed for the case in which the inner cylinder rotates while the outer cylinder remains stationary, and where the difference in radii of the cylinders is small in comparison with their mean radius. The linearised equations governing the marginal stability of axially periodic disturbances are derived in general for the case where the cylinders are infinitely long, and are solved approximately to give estimates of the critical Taylor number at which vortex flow occurs for a range of relative eccentricity of the cylinders.

The results give an upper bound to the stability boundary, and certain results of DiPrima are used to establish a lower bound, and consequently the stability boundary is well established for eccentricity ratios less than about 0·6. One important conclusion is that for a considerable range of eccentricity ratio the flow is less stable than when the cylinders are concentric.

The rate of generation of internal waves by a thin turbulent boundary layer was calculated in a previous paper for a stably-stratified atmosphere with no significant wind-shear outside the boundary layer by considering the excitation of normal modes of wave propagation. By using the concept of wave-packets propagating upwards from the boundary layer, the effects of wind-shear can be included. Conditions for the validity of the approximation are given. In general, the spectral distribution of wave-energy at a particular height takes large values in two bands of horizontal wave-number, one band deriving from wave-packets undergoing internal reflexion near that height and the other from wave-packets of very small local frequency that accumulate there. The ‘reflexion’ wave-numbers are dominant if the wind increases with height and the ‘accumulation’ wave-numbers if the wind initially decreases with height. The spectral energy distributions and intensities of the wave-motion are discussed in more detail for an atmosphere of uniform stability and unidirectional wind-shear. The accumulation process may lead to instability or overturning of the waves, and estimates are made of the probable scale and intensity of the ‘clear-air’ turbulence produced. An interesting point is that the rate of energy loss from the boundary layer by radiation of internal waves turns out to be comparable with the rate of production in the outer nine-tenths of the layer, both for atmospheric boundary layers and for the surface layer of the ocean. It seems likely that radiation limits the layer thickness to some extent.

The formation and growth of three-dimensional wave packets in a laminar boundary layer is treated as a linear problem. The asymptotic form of the disturbed region developing from a point source is obtained in terms of parameters describing two-dimensional instabilities of the flow. It is shown that a wave caustic forms and limits the lateral spread of growing disturbances whenever the Reynolds number is √2 times the critical value. The analysis is applied to the boundary layer on a flat plate and shapes of the wave-envelope are calculated for various Reynolds numbers. These show that all growing disturbances are contained within a wedge-shaped region of approximately 10° semi-angle.

The paper considers the refraction of a plane shock wave at an interface between two streams of different Mach number. Particular attention is paid to the irregular wave systems. It is found that when the interface is slow-fast, that is when the speed of sound a0 in the first or incident wave medium is less than the speed of sound aB in the second or transmitted wave medium, then there are two irregular systems, one being a double Mach reflexion type and the other being a four-wave confluence type. There are also two irregular systems when the refraction is fast-slow; these are a single Mach reflexion type and an expansion wave type. This last system has a central expansion wave when the flow is steady and a continuous band expansion wave when the flow is self-similar. Only two of the irregular wave systems have been observed experimentally in the fully developed state. Possible degeneracies are discussed.

The spatial distribution of uniformly sized spheres falling in a viscous liquid is investigated experimentally for a solid volume fraction of 0·025 and a single sphere Reynolds number of 0·6. The observed spatial distribution agrees closely with a random distribution based on allocation of spheres to space cells according to a binomial probability mechanism.