It is shown that for a small sphere freely suspended in a linear shear flow at small Reynolds numbers, the Nusselt number N is given by $N = \{1 - \alpha P^{\frac{1}{2}} + o(P^{\frac{3}{2}})\}^{-1}$, where P is the Péclet number. For any given type of shear flow, the numerical value of the constant α can be obtained from a general expression derived by Batchelor (1979). The corresponding result for a particle of arbitrary shape is N/N0 = {1 − αN0P½ + O(P3/2)}−1, where N0 is the Nusselt number for pure conduction.

Measurements are presented of the Nusselt numbers N and Rayleigh numbers R for shallow layers of 4He gas heated from below. By choosing different temperatures between 2·3 K and 5·1 K and different pressures between 0·07 bar and 1 bar, the extent Q of departures from the Oberbeck-Boussinesq approximation was varied. When R was evaluated at the static temperature at the midplane of the cell, both the critical Rayleigh number Rc and the initial slope N1 of the Nusselt number were found to be independent of Q within experimental scatter. This result agrees with the prediction of Busse (1967). When R was evaluated at the cold end temperature of the cell, both Rc and N1 depended strongly upon Q.

A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number RaL.

Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as $0.26 Ra^{\frac{1}{5}}_L$ for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found.

In order to determine the heat transfer inside a TIG (tungsten/inert gas) weld pool, it is necessary to have a good understanding of the flow patterns of the liquid metal. The principal force driving the fluid motion is the electromagnetic j × B force due to the current from the welding arc and its self-magnetic field. In this paper we consider the flow of a viscous incompressible conducting fluid in a hemispherical container due to various distributions of the electric current. The problem is posed as a time-dependent problem and is solved numerically using the Du Fort–Frankel leap-frog method. Results are presented for currents of 100 A flowing through the weld pool. This is a typical current for TIG welding, and corresponds to a Reynolds number in the range 200 < Re < 600. Previous solutions of the problem were restricted to low Reynolds numbers, i.e. low currents.

We study theoretically the slow viscous motion of a long slender drop placed in a simple shear flow, the drop having a low viscosity compared with that of the suspending fluid. As a simplifying approximation, the cross-section of the drop is taken to be circular. An equilibrium shape with the drop nearly aligned with the flow is found for all shear rates, although the equilibrium is only stable to small disturbances for shear rates below some critical value. The stable equilibria just below the critical shear rate are found to be accessible only if the shear rate is increased slowly.

An extension of the analytical results of Kaup & Newell (1978) concerning the effect of a perturbation on a solitary wave of the Korteweg–de Vries equation is given and numerical studies are conducted to verify the conclusions. In all cases, the numerical results agree with the results predicted by the theory. The most striking feature of the perturbed flow is the presence of a shelf in the lee of the solitary wave whose role is to absorb (provide) the extra mass which is created (depleted) by the perturbation.

The temporal evolution of small three-dimensional disturbances on viscous flows between parallel walls is studied. The initial-value problem is formally solved by using Fourier–Laplace transform techniques. The streamwise velocity component is obtained as the solution of a forced problem. As a consequence of the three-dimensionality, a resonant response is possible, leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr–Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The phase speed of the resonant waves equals the average mean velocity. The wavenumber combination that leads to the largest amplitude corresponds to structures highly elongated in the streamwise direction. The maximum amplitude, and the time to reach this maximum, scale with the Reynolds number. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.

Small rigid spherical particles are suspended in fluid, and material is being transferred from the surface of each sphere by convection and diffusion. The fluid is in statistically steady turbulent motion maintained by some stirring device. It is assumed that the Péclet number of the flow around a particle is large compared with unity, so that a concentration boundary layer exists at the particle surface, and that the Reynolds number of the flow around the particle is sufficiently small for the velocity distribution near the particle surface to be given by the Stokes equations.

The flow around a particle is a superposition of (a) a streaming flow due to a translational motion of the particle relative to the fluid with a velocity proportional to the density difference, and (b) a flow due to the velocity gradient in the ambient fluid. An expression for the mean transfer rate which is asymptotically exact for large Péclet numbers is obtained in terms of statistical parameters of these two superposed flow fields. As a consequence of the partial suppression of convective transfer by particle rotation, the only relevant parameters are the mean translational velocity of the particle in the direction of the ambient vorticity vector and the mean ambient rate of extension in the direction of the ambient vorticity. The former is shown to be zero in common turbulent flow fields, and an expression for the latter in terms of the mean dissipation rate ε is obtained from the equilibrium theory of the small-scale components of the turbulence. The final non-dimensional expression for the transfer rate is 0·55(a2ε½/κν½)1/3, where a is the particle radius. This is found to agree well with some previously published sets of data for values of $a^2\epsilon^{\frac{1}{2}}/\nu^{\frac{3}{2}}$ less than 102.

The theory of the shear-induced small deformation of viscous drops at zero Reynolds number is reviewed. The general result for arbitrary shear and surface tension is presented, and the asymptotic forms for weak flow and for high internal viscosity are derived from it. In the latter case, numerical solutions are compared with the experiments of Torza, Cox & Mason (1972).

Numerical solutions have been obtained for steady viscous flow past a circular cylinder at Reynolds numbers up to 300. A new technique is proposed for the boundary condition at large distances and an iteration scheme has been developed, based on Newton's method, which circumvents the numerical difficulties previously encountered around and beyond a Reynolds number of 100. Some new trends are observed in the solution shortly before a Reynolds number of 300. As vorticity starts to recirculate back from the end of the wake region, this region becomes wider and shorter. Other flow quantities like position of separation point, drag, pressure and vorticity distributions on the body surface appear to be quite unaffected by this reversal of trends.

A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.

We consider the onset of Rayleigh–Bénard convection from random fluctuations arising within a fluid. In the specific case in which the fluctuations are thermodynamically determined, we reduce the problem to a random initial value problem for the Fourier modes. For the case of weak nonlinear convection, it is possible to truncate the number of modes and this truncated set is solved both by a Monte Carlo technique and by moment methods for various Rayleigh numbers. We find three stages in the evolution of ordered convection from random fluctuations which correspond to time intervals in which the fluctuations and the nonlinearity have different degrees of importance. It is shown that no simple moment truncation method will succeed and that the time for onset of convection is a mean over a distribution of times for which members of an ensemble exhibit appreciable convective transport.

A fluid drop immersed in a second incompressible fluid is deformed by a motion which, far from the drop, varies linearly with distance. Deformation of the drop is then determined by both the rate of deformation and the vorticity of the continuous fluid phase far from the drop. Experiments are described in which the vorticity and rate of deformation were independently varied by means of an eccentric-disk rheometer field. It is shown that predictions from a small-parameter expansion to first order in ε = μ0Gb/σ are in good agreement with experiment to values of ε as large as 0·4, where μ0 is viscosity of the continuous phase, G a measure of the flow strength, b the drop radius, and σ the interfacial tension.

Of the several terms which arise in the O(ε) expansion, only that one which contains a Jaumann derivative of the deformation parameter provides an important contribution beyond Taylor's original analysis of the problem. This fact greatly simplifies the computation of drop shape.

In unidirectional flows, the ratios of Reynolds shear stress to total intensity (except near positions of zero stress) remain remarkably constant from one flow to another, but curvature or strong divergence of the mean flow causes very considerable changes in the stress ratios. A scheme for calculating the changes is described, based on the rapid-distortion approximation of the equations of motion. The results depend to some extent on the effective history of distortion of the turbulence and on the magnitude of an eddy viscosity that models the effect of nonlinear transfer of energy to smaller eddies of the dissipation sequence, but the correspondence with measured values in a distorted wake and in a curved mixing layer is fairly good. In particular, the curious behaviour of stress ratios in the curved mixing-layer can be reproduced qualitatively without any difficulty. Small perturbations of wall turbulence provide a simple application, and earlier calculations of the energy transfer between wind and water waves have been repeated including the changes in the stress ratios predicted by the scheme. In the latter case, very large changes in the distributions of pressure and shear stress are found, and the rates of energy transfer are much larger and in better agreement with observations.

The problem of acoustic radiation generated by instability waves of a compressible plane turbulent shear layer is solved. The solution provided is valid up to the acoustic far-field region. It represents a significant improvement over the solution obtained by classical hydrodynamic-stability theory which is essentially a local solution with the acoustic radiation suppressed. The basic instability-wave solution which is valid in the shear layer and the near-field region is constructed in terms of an asymptotic expansion using the method of multiple scales. This solution accounts for the effects of the slightly divergent mean flow. It is shown that the multiple-scales asymptotic expansion is not uniformly valid far from the shear layer. Continuation of this solution into the entire upper half-plane is described. The extended solution enables the near-and far-field pressure fluctuations associated with the instability wave to be determined. Numerical results show that the directivity pattern of acoustic radiation into the stationary medium peaks at 20 degrees to the axis of the shear layer in the downstream direction for supersonic flows. This agrees qualitatively with the observed noise-directivity patterns of supersonic jets.

The results of an experimental investigation of the entry, into a curved pipe, of the fully developed oscillatory laminar flow in a straight section are presented. Laser anemometry has been used to measure velocity profiles in the plane of the bend at various stations around a 180°-curved section. The flow development is found to depend upon both the frequency parameter of the flow and the amplitude of oscillation.

Results are presented for two values of the frequency parameter α. The first is for small α where the flow can be considered quasi-steady and the flow development is found to proceed as in previous steady-flow studies. The other case, more extensively studied, is where α has a value such that both viscous and inertial effects play important roles in establishing the basic flow at different parts of the pressure cycle. The flow development process around the curve is found to be complicated, but a general trend is found and the results are explained in terms of those already established for steady-flow development in a curved pipe.

The wave resistance of a two-dimensional pressure distribution which moves steadily over water of finite depth is computed with the aid of four approximate methods: (i) consistent small-amplitude perturbation expansion up to third order; (ii) continuous mapping by Guilloton's displacements; (iii) small-Froude-number Baba & Takekuma's approximation; and (iv) Ursell's theory of wave propagation as applied by Inui & Kajitani (1977). The results are compared, for three fixed Froude numbers, with the numerical computations of von Kerczek & Salvesen for a given smooth pressure patch. Nonlinear effects are quite large and it is found that (i) yields accurate results, that (ii) acts in the right direction, but quantitatively is not entirely satisfactory, that (iii) yields poor results and (iv) is quite accurate. The wave resistance is subsequently computed by (i)-(iv) for a broad range of Froude numbers. The perturbation theory is shown to break down at low Froude numbers for a blunter pressure profile. The Inui-Kajitani method is shown to be equivalent to a continuous mapping with a horizontal displacement roughly twice Guilloton's. The free-surface nonlinear effect results in an apparent shift of the first-order resistance curve, i.e. in a systematic change of the effective Froude number.

A general solution for Stokes’ equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids.