Three classes of exact solutions for steady two-dimensional flows of a stratified fluid are found. The flows which correspond to these solutions have arbitrary amplitude (however defined). Two of the three classes of solutions have close bearings on the lee-wave problem in meteorology. It is also shown that the amplitudes of the lee-wave components (if there is more than one component) depend not on the details of the shape of the barrier, but only on certain simple integral properties of the function for the singularity distribution generating the barrier.

A Theory is developed, based on the very limited available experimental evidence, to predict the distortion and disintegration of a water drop when it is exposed to a stream of air with continuously increasing relative velocity. The theory is applied to water drops situated in the path of a solid sphere moving through the air.

It is supposed that a heated liquid is rising very slowly through a semi-infinite porous medium towards the permeable horizontal surface, where it mixes with a layer of cool overlying fluid. In the steady state a thermal boundary layer of exponential form exists in the medium. It is shown that the layer is stable provided that the Rayleigh number for the system does not exceed a critical positive value, and that the wave-number of the critical neutral disturbance is finite. The stability properties of the layer are explained qualitatively from physical considerations.

This paper is concerned with the non-linear interactions between pairs of intersecting gravity wave trains of arbitrary wavelength and direction on the surface of water whose depth is large compared with any of the wavelengths involved. An equation is set up to describe the time history of the Fourier components of the surface displacement in which are retained terms whose magnitude is of order (slope)2 relative to the linear (first-order) terms. The second-order terms give rise to Fourier components with wave-numbers and frequencies formed by the sums and differences of those of the primary components, and the amplitudes of these secondary components is always bounded in time and small in magnitude. The phase velocity of the secondary components is always different from the phase velocity of a free infinitesimal wave of the same wave-number. However, the third-order terms can give rise to tertiary components whose phase velocity is equal to the phase velocity of a free infinitesimal wave of the same wave-number, and when this condition is satisfied the amplitude of the tertiary component grows linearly with time in a resonant manner, and there is a continuing flux of potential energy from one wave-number to another. The time scale of the growth of the tertiary component is of order of the (−2)-power of the geometric mean of the primary wave slopes times the period of the tertiary wave. The Stokes permanent wave appears as a special case, in which the tertiary wave-number is the same as that of the primary, but its phase is advanced by ½π. The energy transfer to the tertiary component in this case is usually interpreted as an increase in the phase velocity of the wave.

The dynamical interactions in water of finite depth are considered briefly, and it is shown that the amplitude of the secondary components becomes large (though bounded in time) as the water depth becomes smaller than the wave-length of the longest primary wave.

Two mechanisms by which a viscous fluid can be deposited on a plane surface are described. Measurement of the thickness of the deposit are compared with calculated values. It is found that the two agree within rather wide limits of experimental error provided the effect of surface tension can be neglected, and the conditions under which this is legitimate are discussed.

The stability and transition Reynolds numbers for the flow due to a disk rotating in still air were increased by low rates of suction through either a woven wire-cloth or a slitted surface. Observations on the slitted disk at rotational speeds between 550 and 1250 r.p.m. showed that the critical value of the Reynolds number r2ω/ν for instability increased from about 135,000 without suction to nearly 250,000 for a value of the suction parameter a of 0·4. The corresponding values for transition increased from about 275,000 to about 400,000. A given increase in stability Reynolds number required about 75% more suction than that theoretically predicted for uniform distributed suction, a satisfactory result in view of the limitations of the apparatus.

At higher rates of suction (0·4 < a < 1·6), the reduction in secondary flow allowed transverse turbulent contamination to spread inwards from the rim. Consequently the vortices associated with secondary-flow instability were not found, though disturbances of larger wavelength appeared. Intermittent turbulent flow was spread over a much larger region of the disk and no laminar flow could be obtained above a Reynolds number of 400,000, Owing to this feature of the flow, it was not possible to extend laminar flow to values of the unit Reynolds number (ratio of stream velocity to kinematic viscosity) corresponding to flight conditions on a swept-back wing. It is concluded that the rotating disk is not a satisfactory tool for the investigation of the effects of suction on secondary-flow instabilities such as arise in the case of a swept-back wing, or for the testing of suction surfaces.

The theory of hydrodynamic stability and the impact on it of recent work with turbulent spots is discussed. Emmons's (1951) assumptions about the growth and interaction of turbulent spots are found experimentally to be substantially correct. In particular it is shown that the region of turbulent flow on a flat plate is simply the sum of the areas that would be obtained if all spots grew independently.

An investigation of the conditions required for breakdown to turbulence near a wall, that is, to initiate a turbulent spot, suggests that regardless of how disturbances are generated in a laminar boundary layer and independent of both the Reynolds number and the spatial extent of the disturbances, breakdown to turbulence occurs by the initiation of a turbulent spot at all points at which the velocity fluctuation exceeds a critical intensity. Over most of the layer this intensity is about 0·2 times the free-stream velocity. The Reynolds number is important merely in respect of the growth of disturbances prior to breakdown.

In this paper some simple theoretical considerations concerning the movement of ships in restricted waterways are discussed, and it is shown that for a ship towed from the bank, or by any external force, there are three distinct speed ranges: subcritical, critical and supercritical. In the subcritical and supercritical ranges, Bernoulli's equation and the continuity equation are satisfied everywhere by a state of steady motion relative to the ship, but in the critical range these laws require that a quantity of fluid is piled up continuously ahead of the ship in the form of a bore. Experimental confirmation is given by means of photographs of model tests.

Self-excited oscillations have been discovered experimentally in a supersonic laminar boundary layer along a flat plate. By the use of appropriate measuring techniques, the damping and amplification of the oscillations are studied and the stability limits determined at free-stream Mach numbers 1·6 and 2·2. The wave-like nature of the oscillations is demonstrated and their wave velocities are measured using a specially designed ‘disturbance generator’. It is shown empirically that the stability limits expressed in terms of the boundary-layer-thickness Reynolds number are independent of the Mach number and dependent only on the oscillation frequency. The main effect of compressibility is an increase in wave velocity with Mach number. This has the consequence that the disturbances, although possessing the same dimensionless amplification coefficient as in the incompressible case, have less time (per unit distance) to grow in amplitude. Thus, the adiabatic compressible boundary layer is shown to be more stable than the incompressible one. In general, the experiments confirm the basic assumptions and predictions of the existing stability theory and also suggest the desirability of improvement in the theory in certain phases of the problem. Finally, on the basis of these results a rough estimate of the transition Reynolds number is made in the compressible flow range.

A transformation is introduced which, for a class of outer pressure distributions, reduces the unsteady incompressible laminar boundary-layer equations in two dimensions to an equation in which the time does not appear explicitly. A formally exact solution of the resulting equation is then presented in the form of a series and it is shown that the solution can be expressed in terms of universal functions.

The paper seeks to determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency, $\frac{\hbox{(forward velocity)} \times \hbox{(thrust available to overcome frictional drag)}} {\hbox {(work done to produce both thrust and vortex wake)}}.$ The recommended procedure is for the fish to pass a wave down its body at a speed of around $\frac {5} {4}$ of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value, and the waveform including both a positive and a negative phase so that angular recoil is minimized. The Appendix gives a review of slender-body theory for deformable bodies.