A nearly uniform shear flow, was obtained in the working section of a wind tunnel by inserting a grid of parallel rods with varying spacing.

The function of such a grid is to impose a resistance to the flow, so graded across the working section as to produce a linear variation in the total pressure at large distances downstream without introducing an appreciable gradient in static pressure near the grid. A method of calculating a suitable arrangement of the rods is described. Although this method is strictly applicable only to weakly sheared flows, an experiment made with a grid designed for a shear parameter as large as 0·45 gave results in close agreement with the theory. There was no evidence from the experiment of any large-scale secondary flow accompanying the shear–a danger inherent in an empirical attempt to grade the resistance of the grid–nor was any tendency observed for the shear to decay with increasing distance from the grid.

An exact solution is found in a fairly simple form for two-dimensional progressive waves of arbitrary amplitude on a fluid of unlimited depth, when only surface tension and not gravity is taken into account as the restoring force. The calculated wave forms are exhibited graphically for various amplitudes, and the relation between wave velocity and amplitude is plotted. The wave of greatest height occurs when the vertical distance between trough and crest is 0·730 wavelengths (compared with 0·142 for gravity waves). Higher waves are prevented from appearing by the enclosing of air bubbles in the troughs.

A circular cylinder is rotating in a uniform stream of viscous liquid, the circumferential velocity of the cylinder being much greater than that of the uniform stream. It is shown that the effect of the uniform stream can be regarded as a small perturbation of the rotary flow due to the cylinder, and that a uniformly valid first approximation to the flow field can be obtained in this way. This approximate solution has the interesting property that the circulation about any circular contour concentric with the cylinder is the same. The lift on the cylinder is shown to be that given by the classical formula, and the drag force is zero to the order of the approximation considered.

The supply of water to the outside of a long vertical tube was adjusted until the water film was seen to be only just disturbed by a train of travelling waves. Under these conditions the Reynolds number, defined as the discharge per unit width divided by the kinematic viscosity, was 4·4. The wave train was slightly irregular, and average values of the length and velocity of the waves were about 0·45 in. and 5½ in./sec, the temperature being 19° C.

This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R. The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering.

Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R, and thereby avoids the severe mathematical difficulties usual in stability problems at high R. The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R. In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R. However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable.

The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.

A proof is given that, in spite of certain singularities in the oblique area distributions, the supersonic area rule can be used to determine correctly the wave drag, according to linearized theory, of thin symmetrical rectangular wings at zero incidence. The proof could be extended to cover the case of swept wings with supersonic edges, for which similar singularities also exist.

Isolated masses of buoyant fluid were released in a water tank. Their width, 2r, and the distance travelled, z, were measured as functions of time and were found to follow roughly the laws $r = nz,\;\;\;\;\; w = C(g \overline {B}r)^ \frac{1}{2},$ where w is the vertical velocity, $\overline {B}$ the mean buoyancy, and n and C are constants. These equations are predicted by dimensional analysis, assuming viscosity to be negligible, and the constants appear to be independent of the Reynolds number. It is found that C [eDot ] 1·2 and n is in the neighbourhood of 4.

Since the Froude number relating the buoyancy and inertia forces is the same as for isolated masses of buoyant air in the atmosphere, it is concluded that the constants will have the same value in this latter case. This is confirmed roughly by observation of cumulus cloud towers.

Some of the characteristics of the motion observed in the experiments are described and comparison is made with vortex rings.

An expression is obtained for the jump in the vorticity across a gasdynamic discontinuity in an inviscid flow. This result generalized results of Truesdell (1952) and Lighthill (1957) for the vorticity behind a steady curved shock in a uniform flow and that of Emmons (1957) for the vorticity jump across a steady flame. The derivation is a dynamical one, and no assumptions on the composition or thermodynamic properties of the fluid are made. The jump in vorticity in the steady flow case is found to depend upon the jump in density and upon gradients along the discontinuity surface of the tangential velocity component and of the normal mass flow. An analogous result is found with unsteady flow.

It is shown that the interpenetration of two ionized streams is arrested, as a rule, not because of individual collisions between particles belonging to opposite streams, but because the whole system of charged particles is unstable. The smallest wavelength of an unstable oscillation is λmin where $\lambda_{min} = \surd \left(\frac {\pi m_0 U^2}{2N \epsilon^2}\right) \left(1 - \frac {U^2}{c^2} \right) ^{-\frac {3}{4}}$ Here ± U are the velocities of the undisturbed streams, and N is the density of electrons in each.

A further calculation for the non-relativistic case deals with the amplification of the plasma oscillations present in two colliding streams. It is shown that these grow rapidly and that $\tau_{crit} \surd (m_0 U^2|\pi N \epsilon^2$ is the distance of interpenetration achieved before the counterstreaming of the electrons is brought to a halt. The value of τcrit depends only insensitively on the ratio of the internal plasma energy densities Tpl to the kinetic energy densities Tkin in the streams. For example, τcrit = 9·0 when Tpl:Tkin = 1:10, and τcrit = 19·0 when Tpl : Tkin = 1:105.