Rising velocities of bubbles

We study the rising velocity U_b of long bubbles in vertical tubes of different cross sections, under the acceleration of gravity g. The vessel being initially filled with a liquid of viscosity ν, we know that for cylindrical tubes of radius R, high Reynolds number bubbles (Re≡ U_bR/ν >> 1) are characterized by a phNewton's law U_b ∼gR and low Reynolds number bubbles by a phStokes' law U_b ∼ gR^2/ν. We show experimentally that for vessels of ``arbitrary'' cross sections (rectangles, regular polygones, toroidal tubes), the high Reynolds number domain is characterized by U_b 0.2 gP, and the low-Reynolds number range by U_b 0.03 gS/ν, where P and S respectively stand for the perimeter and the area of the normal cross section of the tube. Theoretically, we apply to all the bubbles the same local approach, based on the structure of the flow at the stagnation point and show that this method does account for the different behaviors observed.