The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier–Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space.

The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviaboricThe deviation from the zonal mean. wave possesses some of the characteristics of the theoretical Eady wave and it is possible to define the complete deviaboric wave structure by means of two-dimensional quasi-phase, amplitude diagrams. These diagrams may also typify the nature of certain solutions to the non-separable baroclinic instability problem.

The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviakoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure.

Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers.

The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer.

The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall.

The large differences between the axisymmetric and zonal mean states indicate that linear baroclinic instability analysis of the axisymmetric state is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation.

This paper discusses two infinite length models (planar and cylindrical) for ciliary propulsion of microscopic organisms. Through the concept of an extensible envelope (instantaneous surface covering the numerous cilia) over the organism, we consider a small amplitude analysis for the velocity of propulsion. A comparison of the velocities of propulsion for the infinite models reveals that they are just over twice that obtained for the finite spherical model (Blake 1971). This indicates that planeness is more important than finiteness, as the solution for typical micro-organisms (e.g. disk-shaped) should occur somewhere between these two models. The maximum velocity of propulsion obtained is one quarter of the wave velocity; that inferred for Opalina, the organism modelled, is nearly one fifth of the wave velocity. Associated shapes of the surface and paths of movement of the tips of the cilia are illustrated.

Non-linear interactions among wind-generated gravity waves transfer energy to low frequency waves in a coastal zone. A transfer function is derived for a straight coastline of constant bottom slope. This model is applied to three actual cases, and numerical evaluation of the energy transfer produces low frequency spectra which are compared with observations.

An integral equation for unsteady inviscid surface waves has been obtained. Existing known approximations are all derived from the one equation. The Boussinesq equation is obtained and criticized.

This paper is intended to evaluate the wall effects in the pure-drag case of plane cavity flow past an arbitrary body held in a closed tunnel, and to establish an accurate correction rule. The three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant-jet models, are employed to provide solutions in the form of some functional equations. From these theoretical solutions several different rules for the correction of wall effects are derived for symmetric wedges. These simple correction rules are found to be accurate, as compared with their corresponding exact numerical solutions, for all wedge angles and for small to moderate ‘tunnel-spacing ratio’ (the ratio of body frontal width to tunnel spacing). According to these correction rules, conversion of a drag coefficient, measured experimentally in a closed tunnel, to the corresponding unbounded flow case requires only the data of the conventional cavitation number and the tunnel-spacing ratio if based on the open-wake model, though using the Riabouchinsky model it requires an additional measurement of the minimum pressure along the tunnel wall.

The numerical results for symmetric wedges show that the wall effects in-variably result in a lower drag coefficient than in an unbounded flow at the same cavitation number, and that this percentage drag reduction increases with decreasing wedge angle and/or with decreasing tunnel spacing relative to the body frontal width. This indicates that the wall effects axe generally more significant for thinner bodies in cavity flows, and they become exceedingly small for sufficiently blunt bodies. Physical explanations for these remarkable features of cavity-flow wall effects are sought; they are supported by the present experimental investigation of the pressure distribution on the wetted body surface as the flow parameters are varied. It is also found that the theoretical drag coefficient based on the Riabouchinsky model is smaller than that predicted by the open-wake model, all the flow parameters being equal, except when the flow approaches the choked state (with the cavity becoming infinitely long in a closed tunnel), which is the limiting case common to all theoretical models. This difference between the two flow models becomes especially pronounced for smaller wedge angles, shorter cavities, and with tunnel walls farther apart.

In order to gauge the degree of accuracy of these theoretical models in approximating the real flows, and t o ascertain the validity of the correction rules, a series of definitive experiments was carefully designed to complement the theory, and then carried out in a high-speed water tunnel. The measurements on a series of fully cavitating wedges at zero incidence suggest that, of the theoretical models, that due to Riabouchinsky is superior throughout the range tested. The accuracy of the correction rule based on that model has also been firmly established. Although the experimental investigation has been limited to symmetric wedges only, this correction rule (equations (85), (86) of the text) is expected to possess a general validity, at least for symmetric bodies without too large curvatures, since the geometry of the body profile is only implicitly involved in the correction formula. This experimental study is perhaps one of a very few with the particular objective of scrutinizing various theoretical cavity-flow models.

An asymptotic series solution is obtained for the low Reynolds number flow through an axisymmetric tube whose radius varies slowly in the axial direction. Expressions for the pressure drop along the tube and the shear stress at the wall are derived. The analysis is applicable to such problems as the flow through viscometric capillary tubes and the flow through blood vessels.

Some experiments are described in which steady-state shearing flows are developed in stratified brine solutions contained in a cyclically continuous tank of rectangular cross-section. Over the range of overall Richardson numbers studied, the results suggest that whenever turbulent layers are present on either side of a region of fluid with a gravitationally stable density gradient, they cause erosion of this region to occur. The erosion leads to the formation of two homogeneous layers separated by a thin layer of strong density and velocity gradients. The gradient Richardson number, computed by using the velocity and density gradients in this transition layer, tends to have a value of order one.

If we define an overall Richardson number Ri* by averaging the velocity and density gradients over the entire depth of fluid in the tank, we find that the non- dimensional buoyancy flux, Q, is functionally related to Ri* by an equation of the form Q = C1(Ri*)−1 where C1 is a constant, approximately, and Ri* ranges in value between one and thirty.

To check the effect of a large variation of the molecular diffusivity coefficient on flow conditions, we ran a limited number of experiments with thermally stratified fluid. Over a restricted range, 1·0 < Ri* < 5·0, velocity profiles very similar to those measured in the brine-stratified experiments at like values of Ri* were obtained. This suggests that the coefficient of molecular diffusion is not an important parameter in either type of experiment.

Other experiments, made in the same apparatus, describe the entrainment by a turbulent, homogeneous layer of an initially quiescent layer of fluid with a linear density gradient. The depth of the turbulent layer, D, increases with time, t, according to the relation. \[ D^3\propto t. \] This result is consistent with that found by Kato & Phillips (1969), although the turbulent layer in the present experiment is generated in a different manner.

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

The velocity fields in and around a deformed drop suspended in an arbitrary (albeit Stokesian) unbounded flow field are solved. The usefulness of the solution is demonstrated by solving the drag force and lateral migration of a drop suspended in an unbounded Poiseuillian field.

It is demonstrated that, due to the deformation of the drop, there exists a radial component of the settling velocity. The direction of the radial migration depends primarily on the product UHR (the Hadamard-Rybczynski terminal settling velocity) by U0 (the maximum Poiseuillian velocity). A positive product results in a lateral migration away from the location of maximum velocity; the converse also holds.

The velocity measuring technique described in this paper consists of measuring the time of flight of a tracer of heated air from an electrically pulsed wire to one of two sensor wires which are operated as resistance thermometers. These sensor wires are at right angles to the pulsed wire and are placed one on either side of the pulsed wire. The instrument may be used in highly turbulent flows including regions in which flow reversals occur. The paper discusses the theoretical behaviour of the probe and the results of some calibration experiments.

An apparatus that approximates a two-dimensional, infinite train of peristaltic waves yields measurements of mean flow, of mean pressure rise, and of pressure-time pulses at fixed locations. In addition, visual observations of ‘reflux’ and ‘trapping’, using dyed fluid, are shown. The inertia-free range extends up to a Reynolds number of about 1. In this range, the theory of Shapiro, Jaffrin & Weinberg (1969) is confirmed with respect to mean pressure vs. mean flow, pressure vs. time, reflux, and trapping. The controversy regarding the criterion of material reflux is settled in favour of the Lagrangian time-mean velocity rather than the Eulerian time-mean velocity. Experiments at higher Reynolds numbers show that the second-order expansion theory of Jaffrin (1971) is valid up to a Reynolds number of about 10.

The effects of sinusoidal protrusions on steady laminar free convection between vertical walls is investigated in this paper. Numerical results are presented for various values of the size and spacing of the protrusions. In particular optimum values are found which yield maximum wall heat-transfer coefficients.

We consider the stability of the motion generated in a differentially heated vertical slot filled with a linearly stratified salt solution. The theoretical mean motion field between infinite plates is a function of the Rayleigh number Rs, = gβ(∂S0/∂z) D4/k8ν. If Rs is zero the salinity does not enter the problem and one finds instability in the form of stationary rolls which obtain most of their energy from the basic velocity field. Even at very small Rs of order -1000 these shear instabilities are replaced by diffusively destabilized convective rolls which appear at a thermal Rayleigh number Ra = gαΔTD3/kTν which is two orders of magnitude less than that required for the shear generated modes. The present calculations, which take proper account of both the mean fields and the boundary conditions, give results which compare somewhat more favourably with the experimental results of Thorpe, Hutt & Soulsby (1969) than the theory put forward by these authors. It is shown why their theory, which deals with different boundary conditions from those in the experiment, gives adequate results as Rs tends to negative infinity.

This paper gives a brief account and interpretation of bed forms produced during mass transfer from beds of hardened Plaster of Paris (calcium sulphate) exposed to turbulent streams of water. Defects within or upon the initially plane plaster beds perturbed the profile of mass-transfer (solution) rate as the result of the setting up of separated flows. Those defects that exceeded a certain critical dimension set by flow conditions grew in length, breadth and depth into heel-shaped hollows of parabolic plan called flutes. The continued growth and eventual interference of coextensive flutes gave rise to a bed with scallops, an assemblage of intersecting, saucer-shaped depressions. Defects which fell below the critical dimension set by flow conditions grew initially in length and breadth to give long, narrow grooves lying parallel with flow, but after a sufficient time were erased so that the bed reverted to plane. Many plane beds became fashioned into continuous systems of very long furrows and ridges aligned parallel with flow and coupled with paired longitudinal vortices resembling Taylor–Görtler vortices klinematically. The dimensions and orientation of this structure suggested that Klineian streaks, which on rigid beds occur randomly in space and time, had become stabilized in space in the new context of the deformable beds. The furrows commonly experienced a higher-order instability which resulted in the production of secondary flutes and scallops similar to those resulting from pre-existing defects. All of the bed forms described are associated with separated flows, and it is suggested that separation of flow may be a phenomenon necessary for their maintenance and, in the case of flutes, perhaps also for their origin.

A thermal boundary layer is established by heating a vertical plate in a dielectric liquid. An alternating voltage is applied between the heated plate and another plate which is not parallel to the heated plate. This voltage produces a non-uni- form electric field which in turn produces electrical forces acting on the gradients in dielectric permittivity which result from the temperature gradients. These electrical forces alter the boundary layer. In this paper approximate equations are developed which allow one to calculate the boundary-layer,thickness, velocity, and Nusselt numbers for the boundary layer in the presence of the non-uniform electric field. Numerical calculations show that the heat-transfer coefficient can be either increased or decreased by the non-uniform field, depending on whether the field is strongest at the top or bottom of the plates and also on the field strength. Experiments were performed which demonstrate the change in heat transfer caused by the non-uniform field.

The mathematical representation of acoustical sources in motion relative to the surrounding fluid is discussed. It is observed that several types of moving sources exist, and that it is sometimes necessary to choose the proper type. One of these sources currently appears to be more physically realistic than the others.

A method of solution is proposed for flows through small apertures in otherwise impermeable barriers. This method, which is an application of the method of matched asymptotic expansions, is used to solve a specific water-wave problem, yielding an approximate formula for the transmission coefficient.

The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers. For larger Reynolds numbers an approximate Legendre polynomial series representation is used to reduce the governing system of equations to a non-linear ordinary differential equation boundary-value problem which is solved numerically. The resulting flow pattern and the torque required to rotate the spheres are presented for various cases considered.

On the basis of a statistical treatment of the local structure of a flowing monodisperse system of particles suspended in a viscous fluid, a complete set of equations is proposed which governs the mechanical behaviour of both phases looked upon as interpenetrating co-existent continua. The set includes the dynamic equations of mass and momentum conservation of both phases in their mean flow and the kinetic equation for suspended particles. A technique is developed for the calculation of various quantities describing random local motion of the particles and fluid superimposed on their mean flow.

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.