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 $\lambda\mu$ 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 ${\hbox{\it Ca}}\,{=}\,\mu U/\gamma$ and the Reynolds number are small. A thin film of uniform thickness $\epsilon R$ is formed between the drop and the wall. It is shown that Bretherton's (1961) scaling $\epsilon\propto{\hbox{\it Ca}}^{{2}/{3}}$ is appropriate for all values of $\lambda$, but with a coefficient of order unity that depends weakly on both $\lambda$ and ${\hbox{\it Ca}}$. 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 ${\hbox{\it Ca}}^{-{1}/{3}}\,{\ll}\,\lambda\,{\ll}\,{\hbox{\it Ca}}^{-{2}/{3}}$ and then falls by a factor $2^{{2}/{3}}$ as $\lambda\,{\rightarrow}\,\infty$. The multi-region asymptotic structure of the flow is also discussed.