On the motion of bubbles in vertical tubes of arbitrary cross-sections: some complements to the Dumitrescu–Taylor problem

Abstract

We first study the rising velocity $U_b$ of long bubbles in vertical tubes of different cross-sections, under the acceleration due to gravity $g$. The vessel being initially filled with a liquid of kinematic viscosity $\nu$, it is known that for cylindrical tubes of radius $R$, high-Reynolds-number bubbles ($\hbox{\it Re}\,{\equiv}\,U_{b}R/\nu \,{\gg}\, 1$) are characterized by {Newton's law} $U_{b}\,{\propto}\,\sqrt{gR}$ and low-Reynolds-number bubbles by {Stokes' law} $U_{b}\,{\propto}\, gR^2/\nu$. We show experimentally that these results can be generalized for vessels of ‘arbitrary’ cross-section (rectangles, regular polygons, toroidal tubes). The high-Reynolds-number domain is shown to be characterized by $U_{b}\,{=}\,(8\pi)^{-1/2} \sqrt{gP}$, and the low-Reynolds-number range by $U_{b}\,{\approx}\, 0.012 gS/\nu$, where $P$ and $S$ respectively stand for the wetted perimeter and the area of the normal cross-section of the tube. We derive an analytical justification of these results, using the rectangular geometry. Finally, the problem of long bubble propagation in an unsteady acceleration field is analysed. The theory is compared to existing data.