This paper is concerned with the motion in water of air bubbles whose equivalent spherical radii are in the range 0.5-4.0 mm. These bubbles are not spherical but are, approximately, oblate spheroids; and they may rise steadily in a vertical straight line, or along a zig-zag path, or in a uniform spiral. The rectilinear motion occurs when the radius is less than about 0.7 mm, and the other motions occur for larger bubbles. There is disagreement in the literature as to whether it is the zig-zag or the spiral motion that occurs. It was found experimentally that, when the bubbles are produced in the manner described in this paper, only the zig-zag motion occurs when the radius of the bubble is less than about 1 mm, but bubbles of larger radius either zig-zag or spiral depending upon various factors.

The spiralling bubble is treated theoretically by assuming that the flow near the front of the bubble is inviscid (the Reynolds number of the motion is several hundred) and considering the distribution of pressure over the front surface. Equations are obtained relating the geometrical parameters of the spiral, the shape of the bubble and the velocity of rise. The analysis is simplified by assuming that the pitch of the spiral is large compared with its radius, and the velocity of rise and shape of the bubble are determined as functions of the radius. The experimental and theoretical values are compared, and fair agreement found. Reasons to account for the disagreement are proposed.

A modification of the theory is proposed to take account of the presence of impurities or surface-active substances in the water, and the velocities of rise thus predicted are in agreement with the experimental observations.

The zig-zag motion is treated in a similar way, and the analysis leads to an equation which determines the stability of the rectilinear motion. The value of the Weber number at which the rectilinear motion. The value of the Weber number at which the rectilinear motion becomes unstable is deduced, and is found to be in fair agreement with experiment. The experimental evidence on the wake behind solid bodies is described briefly, and reasons are given for suggesting that the zig-zag motion is due to an interaction between the instability of the rectilinear motion and a periodic oscillation of the wake.

Results of an experimental investigation of the heat loss of fine heated wires immersed in a supersonic stream at right angles to the flow direction are presented. The measurements show that the heat loss of the wire is independent of the free-stream Mach number for values of the latter between 1.3 and 4.5. Since the wire is always in the wake of a detached shock wave, the streamlines in the neighbourhood of the wire pass through a normal shock wave. The Reynolds number Re2 based on conditions behind a normal shock, thus becomes the characteristic parameter for the heat transfer. The measurements covering a Reynolds number range of 3 to 220 show the existence of two flow regimes. For Re2 > 20 the Nusselt number is a linear function of the square root of the Reynolds number, and the equilibrium temperature is nearly independent of Re2 < 20 the Nusselt number decreases more slowly, and the equilibrium temperature rises sharply with decreasing Reynolds number.

A method is presented for treating problems of the propagation and ultimate decay of the shocks produced by explosions and by bodies in supersonic flight. The theory is restricted to weak shocks, but is of quite general application within that limitation. In the author's earlier work on this subject (Whitham 1952), only problems having directional symmetry were considered; thus, steady supersonic flow past an axisymmetrical body was a typical example. The present paper extends the method to problems lacking such symmetry. The main step required in the extension is described in the introduction and the general theory is completed in §2; the remainder of the paper is devoted to applications of the theory in specific cases.

First, in §3, the problem of the outward propagation of spherical shocks is reconsidered since it provides the simplest illustration of the ideas developed in §2. Then, in §4, the theory is applied to a model of an unsymmetrical explosion. In §5, a brief outline is given of the theory developed by Rao (1956) for the application to a supersonic projectile moving with varying speed and direction. Examples of steady supersonic flow past unsymmetrical bodies are discussed in §6 and 7. The first is the flow past a flat plate delta wing at small incidence to the stream, with leading edges swept inside the Mach cone; the results agree with those previously found by Lighthill (1949) in his work on shocks in cone field problems, and this provides a valuable check on the theory. The second application in steady supersonic flow is to the problem of a thin wing having a finite curved leading edge. It is found that in any given direction the shock from the leading edge ultimately decays exactly as for the bow shock on a body of revolution; the equivalent body of revolution for any direction is determined in terms of the thickness distribution of the wing and varies with the direction chosen. Finally in §8, the wave drag on the wing is calculated from the rate of dissipation of energy by the shocks. The drag is found to be the mean of the drags on the equivalent bodies of revolution for the different directions.

The dynamical theory of the motion of a body through an inviscid and incompressible fluid has yielded three relations: a first, due to Kirchhoff, which expresses the force and moment acting on the body in terms of added masses; a second, initiated by Taylor, which expresses added masses in terms of singularities within the bòdy; and a third, initiated by Lagally, which expresses the forces and moments in terms of these singularities. The present investigation is concerned with generalizations of the Taylor and Lagally theorems to include unsteady flow and arbitrary translational and rotational motion of the body, to present new and simple derivations of these theorems, and to compare the Kirchhoff and Lagally methods for obtaining forces and moments. In contrast with previous generalizations, the Taylor theorem is derived when other boundaries are present; for the added-mass coefficients due to rotation alone, for which no relations were known, it is shown that these relations do not exist in general, although approximate ones are found for elongated bodies. The derivation of the Lagally theorem leads to new terms, compact expressions for the force and moment, and the complete expressions of the forces and moments in terms of singularities for elongated bodies.

Ward's slender-body theory of supersonic flow is applied to bodies terminating in either (i) a single trailing edge at right angles to the oncoming supersonic stream, or (ii) two trailing edges at right angles to one another as well as to the oncoming stream, or (iii) a cylindrical section with two or four identical fins equally spaced round it. The wave drag at zero lift, D, is given by the expression $\frac {D}{\frac {1}{2}\rho U^2} &=& \frac {1}{2\pi}\int^l_0 \int^l_0 log\frac{1}{|s-z|}S^{\prime \prime}(s)S^{\prime \prime}(z)dsdz - \\ &-& \frac{S^\prime (l)}{\pi}\int^l_0 log \frac {l}{l-z}S^{\prime \prime}(z)dz + \frac{S^{\prime 2}(l)}{2\pi} \{ log \frac{l}{(M^2-1)^{1|2}b}+k \} $ where l is the length of the body, b the semi-span of the trailing edge (or length of trailing edge of a single fin), and S(z) is the cross-sectional area of the body at a distance z behind the apex. The constant k depends on the distribution of trailing-edge angle along the span for each trailing-edge configuration. In case (i) it is 1·5 for a uniform distribution of trailing-edge angle and 1·64 for an elliptic distribution. In case (ii) it is 1·28 for a uniform distribution and 1·44 for an elliptic distribution. Study of case (iii) indicates that interference effects due to the presence of the body reduce the drag of the fins. For example, with a uniform distribution of trailing-edge angle, k for two fins falls from 1·5 in the absence of a body to 1·06 when the body radius equals the trailing-edge semi-span, while k for four fins falls from 1·28 to 0·45 under the same conditions.

Where ordinary finite-wing theory is applicable, the present method must agree with it for small $(M^2-1)^{1|2}b|l$, and this is confirmed by two examples (§3), but within the limit imposed by slenderness the present method is of course more widely applicable, as well as simpler, than finite-wing theory.

It is not known experimentally whether slender-body theory gives accurate predictions of drag at zero lift, for the shapes here discussed, under the conditions for which on theoretical grounds it might be expected to do so. It should be noted that, although tests have not yet been made on ideally suitable bodies, no clear the drag is therefore twice that of a wing made up of two of them. The final stages of the process cannot be represented by slender-body theory, but the initial trend may well be indicated fairly accurately.