An important step towards the understanding of many industrial coating processes is a solution of the dragout problem, which is to determine the thickness of the film of liquid which clings to a plate when it is drawn steadily out of a bath of the liquid. An approximate solution, valid for small capillary numbers, was given by Landau and Levich, and considerable effort has been exerted to extend or ref'me this work. In this paper we show that the Landau-Levich result is an asymptotic solution valid as the capillary number tends to zero, a fact not properly appreciated hitherto, and show how correction terms may be obtained by the method of matched expansions. We also show how the results may be applied to the coating of a horizontal roller.

A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions φ < 0.5 and small deformations (Ca 1) is developed. The model has its theoretical foundation in a physical analysis of dilute emulsions. The constitutive equation takes the form of a linear Jeffreys model involving observable material parameters: the viscosity of the continuous phase, gas volume fraction, the relaxation time, bubble size distribution and an empirically determined dimensionless constant. The model is validated against observations of the deformation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup) subjected to forced oscillations. The effect of φ and frequency of oscillation f on the elastic and viscous components of the deformation are investigated. At low f , increasing φ leads to an increase in viscosity, whereas, at high f , viscosity decreases as φ increases. This behaviour can be understood in terms of bubble deformation rates and we propose a dimensionless quantity, the dynamic capillary number Cd, as the parameter which controls the behaviour of the system. Previously published constitutive equations and observations of the rheology of bubble suspensions are reviewed. Hitherto apparently contradictory findings can be explained as a result of Cd regime. A method for dealing with polydisperse bubble size distributions is also presented.

A small drop of liquid 1 falls through a less dense liquid 2 and approaches the horizontal interface between liquid 2 and an underlying layer of liquid 1. After a short time the drop will be brought to rest (or nearly) in a hollow in the interface. Before the drop can coalesce with its bulk phase, the thin film of liquid 2 trapped between them must be squeezed out, and become sufficiently thin that rupture can occur. This is the film drainage problem. Early calculations, based on simple lubrication theory, fail to take proper account of two effects which are investigated here and shown to be decisive. They are the circulation induced in the drop and in the lower bulk fluid, which tends to speed up drainage, and the constriction in the film thickness at its periphery, which tends to slow it down. This constriction has been observed and some existing theories have attempted to model it in an ad hoc manner. We give here a physical explanation and calculate the minimum thickness explicitly. The effect of circulation in the adjacent fluids is also calculated.

The problem considered is the determination of the mass of the drops which break away when a viscous liquid drips slowly out of a narrow vertical tube. A simple one-dimensional theory of the unsteady extension of a viscous thread under its own weight is given, which holds when viscosity, capillarity and gravity are important but inertia is negligible. A comparison with experiment is given. There are several systematic errors, the most important of which are associated with detailed behaviour at the pipe exit where die-swell and wetting are difficult to assess. With due allowance for these errors, agreement is fairly good.

The Taylor–Saffman problem concerns the fingering instability which develops when one liquid displaces another, more viscous, liquid in a porous medium, or equivalently for Newtonian liquids, in a Hele-Shaw cell. Recent experiments with Hele-Shaw cells using non-Newtonian liquids have shown striking qualitative differences in the fingering pattern, which for these systems branches repeatedly in a manner resembling the growth of a fractal. This paper is an attempt to provide the beginnings of a hydrodynamical theory of this instability by repeating the analysis of Taylor & Saffman using a more general constitutive model. In fact two models are considered; the Oldroyd ‘Fluid B’ model which exhibits elasticity but not shear thinning, and the Ostwald–de Waele power-law model with the opposite combination. Of the two, only the Oldroyd model shows qualitatively new effects, in the form of a kind of resonance which can produce sharply increasing (in fact unbounded) growth rates as the relaxation time of the fluid increases. This may be a partial explanation of the observations on polymer solutions; the similar behaviour reported for clay pastes and slurries is not explained by shear-thinning and may involve a finite yield stress, which is not incorporated into either of the models considered here.