The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.

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SumnznryWe suppose that the plates are pulled along on top of an effectively viscous asthenosphere by their cold dense sinking leading edges, and that they also tend to slide down the flanks of ocean ridge systems. Using reasonable literature values of density, viscosity and thickness, we find that a typical strong subduction zone pulls about seven times as hard as a typical mid-ocean ridge pushes. With the simplifying assumptions that other driving forces are much smaller, and the return current in the asthenosphere is everywhere (anti)parallel to the plate velocity, we perform a torque balance for each major plate in order to find its angular velocity, thus finding a set of relative angular velocities to compare with observations. The directions fit quite well but not the magnitudes. On the additional hypothesis that oceanic lithosphere thickens with age, owing to heat diffusion, the driving forces will be greater for large old plates. The fit is thereby greatly improved: predicted speeds are from 0.81 to 1.38 times the observed and the COCOS plate is not anomalous.The model predicts tension in plates near their subduction zones. The Red Sea and African rift appear to have begun spreading as a crack moved into the former African +Arabian plate from a re-entrant corner off the Gulf of Aden which would have concentrated the stress. Gondwanaland may possibly have been split in a similar way.

Conditions for two gas bubbles in a liquid to rise steadily in a vertical line are derived theoretically with these assumptions: large Reynolds number, no surface contamination, spherical shape, negligible gas density and viscosity. Drag coefficients are found, and are lower than for single bubbles. The bubbles have equilibrium distances apart, which are calculated to a first approximation. The equilibrium is shown to be stable to small vertical disturbances but unstable to horizontal ones. Similar results exist for lines of more than two bubbles, but are not calculated in detail.

Trace impurities often collect on the upstream side of an obstacle in the surface of flowing liquid. The transition from practically free surface to surface sufficiently clogged to be treated as stationary can be quite sharp. The viscous flow underneath is nonlinearly coupled to the convective mass transfer of surface-active material. For two-dimensional flow at high Reynolds number the first observations were due to Thoreau, Langton and Reynolds over 100 years ago, and the theory was given by Harper & Dixon in 1974. If the whole problem is considered from a frame of reference moving with the stream instead of fixed to the downstream surface film, the solution refers to the leading edge of a slowly spreading oil slick.The present work gives the theory corresponding to Harper & Dixon's for low Reynolds numbers (Stokes flow), for which there is a very simple leading approximation near the transition for a soluble surfactant, and a more complicated one, which can still be found exactly, for an insoluble surfactant which spreads onto clear liquid by surface diffusion. In both cases the surface remains flat: the ridge often observed is not a Stokes flow phenomenon.The results are used to clarify the circumstances in which Savic's stagnant-cap approximation is useful for a bubble rising in a viscous liquid: the rear stagnation point now plays the role of the obstacle in the surface, and the flow near the surface transition can be treated locally as if it were two-dimensional instead of axisymmetric.

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An analytical theory is given for the viscous wake behind a spherical bubble rising steadily in a pure liquid at high Reynolds number, and for that wake's effect on the motion of a second bubble rising underneath the first. Previous theoretical work on this subject consists of just two papers: a first approximation ignoring wake vorticity diffusion between the bubbles, and a full numerical solution avoiding simplifying approximations (apart from that of spherical shape of the bubbles). A second approximation is now found; it removes much of the discrepancy between the first approximation and the full solution. The leading-order calculation of wake vorticity diffusion uses a transformation of the independent variables which appears to be new. Experimental work to date has disagreed with all the theoretical work, but it addresses a somewhat different problem: a line of many bubbles.

New data on mantle viscosity variation and plate motions suggest that the author's previous theory for the forces driving and resisting plate tectonics needs revision. The flow in the mantle associated with plate movements probably pervades the whole mantle instead of being almost entirely in a thin asthenosphere at the top of it.The first task undertaken in this paper is a calculation of the viscous torque exerted on a hemispherical plate by a vertical point force in the mantle directly beneath the plate boundary. This prototype problem is simple enough, even with asthenosphere and mesosphere assigned different viscosities, to allow the computing t o be done with a large number o f terms of the relevant spherical harmonic series. It shows that forces in the lithosphere and asthenosphere are the major source of torques on plates, especially if the asthenosphere viscosity is low, and that truncating the series at a pre-assigned degree can seriously underestimate the effect of a force in the asthenosphere. These results are used as a guide t o the approximations to adopt for more realistic boundary conditions, where accurate computing by the same method as the prototype problem would be too time-consuming t o do.The second task is estimating the torques actually driving and resisting the motion of the Earth's plates. It includes a recalculation of the 'ridge push' driving mechanism which gives a zero resultant torque about the centre of the Earth, in agreement with Archimedes' principle but contrary t o a recent conclusion of Davis & Solomon. The recalculation does not affect 'slab-pull', which still has a non-zero resultant, balanced mainly by a non-zero mean angular velocity of the plate system relative t o the mean mesosphere (in which the 'hot spots' are assumed in this work to lie). This angular velocity agrees well with observational data, and gives a test of the approximations on data not used in deriving them.Misfits between driving and resisting torques are found for each plate. They are smaller for the present method than its best-fitting competitor. Conclusions common to both are that the 'slab pull' driving force must be appropriately weighted for age and speed, and that the most important resisting 156 J. F. Harper force is viscous resistance to shear flow under the whole area of the plate, not the localized friction at subduction zones which some earlier work had suggested .