Journal of Fluid Mechanics



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The motion of a viscous drop through a cylindrical tube


S. R. HODGES a1, O. E. JENSEN a2 and J. M. RALLISON a1
a1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
a2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Article author query
hodges s   [Google Scholar] 
jensen o   [Google Scholar] 
rallison j   [Google Scholar] 
 

Abstract

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.

(Received January 27 2003)
(Revised September 22 2003)



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