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Advances in Mathematical Physics
Volume 2016, Article ID 1234642, 16 pages
http://dx.doi.org/10.1155/2016/1234642
Research Article

Slug Self-Propulsion in a Capillary Tube Mathematical Modeling and Numerical Simulation

1Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
2Department of Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received 1 February 2016; Accepted 20 June 2016

Academic Editor: Xavier Leoncini

Copyright © 2016 M. I. Khodabocus et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A composite droplet made of two miscible fluids in a narrow tube generally moves under the action of capillarity until complete mixture is attained. This physical situation is analysed here on a combined theoretical and numerical analysis. The mathematical framework consists of the two-phase flow phase-field equation set, an advection-diffusion chemical concentration equation, and closure relationships relating the surface tensions to the chemical concentration. The numerical framework is composed of the COMSOL Laminar two-phase flow phase-field method coupled with an advection-diffusion chemical concentration equation. Through transient studies, we show that the penetrating length of the bidroplet system into the capillary tube is linear at early-time regime and exponential at late-time regime. Through parametric studies, we show that the rate of penetration of the bidroplet system into the capillary tube is proportional to a time-dependent exponential function. We also show that this speed obeys the Poiseuille law at the early-time regime. A series of position, speed-versus-property graphs are included to support the analysis. Finally, the overall results are contrasted with available experimental data, grouped together to settle a general mathematical description of the phenomenon, and explained and concluded on this basis.