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

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 (Re equivalent to U-b, R/v much greater than 1) are characterized by Newton's law U-b proportional torootg-R and low-Reynolds-number bubbles by Stokes' law U-b proportional to gR(2)/nu. We show experimentally that these results can be generalized for vessels of 'arbitrary' crosssection (rectangles, regular polygons, toroidal tubes). The high-Reynolds-number domain is shown to be characterized by U-b = (8pi)(-1/2)rootgP and the low- Reynolds-number range by U(b)approximate to0.012gS/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.