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Measurements of the amount of fluid left behind when a viscous liquid is blown from an open-ended tube are described.

The disintegration of drops in strong electric fields is believed to play an important part in the formation of thunderstorms, at least in those parts of them where no ice crystals are present. Zeleny showed experimentally that disintegration begins as a hydrodynamical instability, but his ideas about the mechanics of the situation rest on the implicit assumption that instability occurs when the internal pressure is the same as that outside the drop. It is shown that this assumption is false and that instability of an elongated drop would not occur unless a pressure difference existed. When this error is corrected it is found that a drop, elongated by an electric field, becomes unstable when its length is 1.9 times its equatorial diameter, and the calculated critical electric field agrees with laboratory experiments to within 1 %. When the drop becomes unstable the ends develop obtuse-angled conical points from which axial jets are projected but the stability calculations give no indication of the mechanics of this process. It is shown theoretically that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3°. Apparatus was constructed for producing the necessary field, and photographs show that conical oil/water interfaces and soap films can be produced at the calculated voltage and that their semi-vertical angles are very close to 49.3°. The photographs give an indication of how the axial jets are produced but no complete analytical description of the process is attempted.

Since the time of Osborne Reynolds it has been known that turbulence produces virtual mean stresses which are proportional to the coefficient of correlation between the components of turbulent velocity at a fixed point in two perpendicular directions. The significance of correlation between the velocity of a particle at one time and that of the same particle at a later time, or between simultaneous velocities at two fixed points was discussed in 1921 by the present writer in a theory of “Diffusion by Continuous Movements.” The recent improvements in the technique of measuring turbulence have made it possible actually to measure some of the quantities envisaged in the theory and thus to verify some of the relationships then put forward. The theory has also been developed in several directions which were not originally contemplated. The theory, as originally put forward, provided a method for defining the scale of turbulence when the motion is defined in the Lagrangian manner, and showed how this scale is related to diffusion. It is now shown that it can be applied either to the Lagrangian or to the Eulerian conceptions of fluid flow.

(Abstract.) Part I .-The stability for symmetrical disturbances of a viscous fluid in steady motion between concentric rotating cylinders is investigated mathe matically. It is shown that at slow speeds the motion is always stable, but that at high speeds the motion is only stable when the ratio of the speed of the outer cylinder to that of the inner one exceeds a certain value. When the ratio is less than* this or when it is negative the motion becomes unstable at high speeds. The " criterion " for stability is found, and in cases suitable for experimental verification an approximate form for the " criterion " is developed which is useful for numerical computation.The type of instability which may be expected to appear when the speed of the cylinders is slowly increased is shown to consist of symmetrical ringshaped vortices spaced at regular intervals along the length of the cylinders. These vortices rotate alternately in opposite directions. Their dimensions are calculated and it is shown that they are contained in partitions of rectangular cross-section. In the case when the instability arises while both cylinders are rotating in the same direction, these rectangles are squares, so that the vortices are spaced at distances apart equal to the thickness of the annular space between the two cylinders. In the case when the cylinders rotate in opposite directions the spacing, or distance between the centres of neigh bouring vortices, is smaller than this ; and at the same time two systems of vortices develop-an inner system which is similar to the system which appears when the two cylinders rotate in the same direction, and an outer system, which is much less vigorous and rotates in the opposite direction to the adjacent members of the inner system.Part I I .-In Part II experiments are described in which all the conclusions of Part I are subjected to numerical verification. I t is shown that the predicted vortices do in fact appear at the predicted speeds, and photographs of the vortices are shown. The accuracy with which the critical speeds can be predicted is good, the error being about 2 per cent. This compares favourably with the best previous criterion for stability, namely, that of Reynolds, on May 12, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from

The viscosity of a fluid in which small solid spheres are suspended has been studied by Einstein as a problem in theoretical hydrodynamics. Einstein’s paper gave rise to many experimental researches on the viscosity of fluids containing solid particles, and it soon became clear that though complete agreement with the theory might be expected when the particles are true sphered, some modification is necessary when the particles are flattened or elongated. The theory of such systems was developed by G. B. Jeffery, who calculated the motion of ellipsoidal particles in a viscous fluid and their effect on the mean viscosity. Some of his conclusions have been verified by observation. So far no one seems to have extended Einstein’s work to liquids containing small drops of another liquid in suspension. The difficulties in the way of a complete theory when solid particles are replaced by fluid drops are almost insuperable, partly because the correct boundary conditions are not known, and partly because a fluid drop would deform under the combined action of viscous forces and surface tension. Even if the boundary conditions were known to be those commonly used in hydrodynamical theory, the calculation of the shape of the deformed drop would be exceedingly difficult. When the radius of the suspended drops or the velocity of distortion of the fluid are small, surface tension may be expected to keep them nearly spherical, and in that case Einstein’s analysis may be extended so as to include the case of liquid drops.