The motion of a viscous drop through a cylindrical tube

Liquid of viscosity mu moves slowly through a cylindrical tube of radius R under the action of a pressure gradient. An immiscible force-free drop having viscosity lambdamu almost fills the tube; surface tension between the liquids is gamma. The drop moves relative to the tube walls with steady velocity U, so that both the capillary number Ca = muU/gamma and the Reynolds number are small. A thin film of uniform thickness epsilonR is formed between the drop and the wall. It is shown that Bretherton's (1961) scaling epsilon proportional to Ca-2/3 is appropriate for all values of lambda, but with a coefficient of order unity that depends weakly on both lambda and Ca. The coefficient is determined using lubrication theory for the thin film coupled to a novel two-dimensional boundary-integral representation of the internal flow. It is found that as lambda increases from zero, the film thickness increases by a factor 4(2/3) to a plateau value when Ca-1/3 much less than lambda much less than Ca-2/3 and then falls by a factor 2(2/3) as lambda --> infinity. The multi-region asymptotic structure of the flow is also discussed.