The terminal velocity of rise of small, distorted gas bubbles in a liquid of small viscosity is calculated. Small viscosity means that the dimensionless group gμ4/ρT3 where g is the acceleration of gravity, μ the viscosity, ρ the density and T the surface tension, is less than 10−8. It is assumed—and the numerical accuracy of the assumption is discussed—that the distorted bubbles are oblate ellipsoids of revolution. The drag coefficient is found by extending the theory given recently (Moore 1963) for the boundary layer on a spherical gas bubble. The results are in reasonable quantitative agreement with the experimental data.

The equations governing the boundary layer on a spherical gas bubble rising steadily through liquid of small viscosity are derived. These equations are linear are linear and are solved in closed form. The boundary layer separates at the rear stagnation point of the bubble to form a thin wake, whose structure is determined. Thus the drag force can be calculated from the momentum defect. The value obtained is 12πaaUμ, where a is the bubble radius and U the terminal velocity, and this agrees with the result of Levich (1949) who argued from the viscous dissipation in the potential flow round the bubble. The next term in an expansion of the drag in descending fractional powers of R is found and the results compared with experiment.

We have carried out a comparison study for a set of benchmark problems which are relevant for convection in the Earth's mantle. The cases comprise steady isoviscous convection, variable viscosity convection and time-dependent convection with internal heating. We compare Nusselt numbers, velocity, temperature, heat-flow , topography and geoid data. Among the applied codes are finite-difference, finite-element and spectral methods. In a synthesis we give best estimates of the 'true' solutions and ranges of uncertainty. We recommend these data for the validation of convection codes in the future.

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