The axisymmetric creeping motion of a neutrally buoyant deformable drop flowing through a circular tube is analysed with a boundary integral equation method. The fluids are immiscible, incompressible, and the bulk flow rate is constant. The drop to suspending fluid viscosity ratio is arbitrary and the drop radius varies from 0.5 to 1.15 tube radii. The effects of the capillary number, viscosity ratio, and drop size on the deformation, the drop speed, and the additional pressure loss are examined.

Drops with radius ratios less than 0.7 are insensitive to substantial variation in capillary number and viscosity ratio, and computed values of drop speed and extra pressure loss are in excellent agreement with small deformation theories (Hestroni et al. 1970; Hyman & Skalak 1972a). For this drop size range, significant deformation will result only for Ca > 0.25. The onset of a re-entrant cavity is predicted at the trailing end of the drop for Ca ≈ 0.75. Drop speed and meniscus shape become independent of drop size for radius ratios as small as 1.10. The extra pressure loss can be positive or negative depending mainly on the viscosity ratio, however a relatively inviscid drop can cause a positive extra pressure loss when capillary forces are significant. Computed values for extra pressure loss and drop speed are in good agreement with the experimental data of Ho & Leal (1975) for drops of sizes comparable with the tube radius.

Although unsteady, high-Reynolds-number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary-layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady three-dimensional separating flows follow and depend on the symmetry properties of the flow (e.g. line symmetry, axial symmetry). In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi two-dimensional with a displacement thickness in the form of a crescent-shaped ridge. Physically the singularities can be understood in terms of the behaviour of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

A theory to explain the initial stages of unsteady separation has been proposed by Van Dommelen & Cowiey (1990). In the present paper, this theory is verified for the separation process that occurs at the equatorial plane of a sphere or a spheroid which is impulsively spun around an axis of symmetry. A Lagrangian numerical scheme is developed which gives results in good agreement with Eulerian computations, but which is significantly more accurate. This increased accuracy, and a simpler structure to the solution, also allows verification of the Eulerian structure, including the presence of logarithmic terms. Further, while the Eulerian computations broke down at the first occurrence of separation, it is found that the Lagrangian computation can be continued. It is argued that this separated solution does provide useful insight into the further evolution of the separated flow. A remarkable conclusion is that an unseparated vorticity layer at the wall, a familiar feature in unsteady separation processes, disappears in finite time.

Linear water-wave theor is used in conjuctin with a wide-spacing approximation to develop closed-form expressions for the reflection and transmission coeffcients appropriate to a plane wave incident upon any number of identical equally spaced obstacles in two dimensins, and also to derive a real expressin from which the sloshing requencies, which occur when the bodies are bounded by rigid walls, can be determined. In each case the solutin is in terms of known properties of radiation problems associated with any one of the bodies in isolation.