Ascending non-Newtonian long drops in vertical tubes

We report on theoretical and experimental studies describing the buoyancy-driven ascent of a Taylor long drop in a circular vertical pipe where the descending uid is Newtonian, and the ascending uid is non-Newtonian yield-shear-thinning and described by the threeparameter Herschel-Bulkley model, including the Ostwald-deWaele (OdW) model as a special case for zero yield. Results for the Ellis model are included to provide a more realistic description of purely shear-thinning behaviour. In all cases, lubrication theory allows obtaining the velocity pro les and the corresponding integral variables in closed form, for lock-exchange ow with a zero net ow rate. The energy balance allows deriving the asymptotic radius of the inner current, corresponding to a stable node of the di erential equation describing the time evolution of the core radius. We carried out a series of experiments measuring the rheological properties of the uids, the speed and the radius of the ascending long drop. For some tests, we measured the velocity pro le with Ultrasound velocimetry technique. The measured radius of the ascending current compares fairly well with the asymptotic radius as derived through the energy balance, and the measured ascent speed shows a good agreement with the theoretical model. The measured velocity pro les also agree with their theoretical counterparts. We have also developed dynamic similarity conditions to establish whether laboratory physical models, limited by availability of real uids with de ned rheological characteristics, can be representative of real phenomena on a large scale, such as exchanges in volcanic conduits. The Appendix contains scaling rules for the approximated dynamic similarity of the physical process analysed; these rules serve as a guide for the design of experiments reproducing real phenomena.