Motion of Spherical Gas Bubbles in a Viscous Liquid at Large Reynolds Numbers

An analysis is made for the motion of a gas bubble rising steadily in a quiescent liquid of infinite extent. The disturbed layer of the fluid, due to viscosity, on either side of the interface is thin when the Reynolds number is sufficiently large and thus makes possible a considerable simplification of the governing equations of motion. Simultaneous solutions of the boundary-layer equations for the flow outside and inside of the bubble are obtained by considering that the tangential velocity components and the shear stresses on both sides of the interface are equal. From the calculated external stress field, the drag of a spherical bubble with negligible flow separation is evaluated. Good agreement is obtained with published data for spherical air bubbles in four different organic liquids. The experimental drag curve due to Haberman and Morton for air bubbles rising in filtered water deviates from that predicted from the present theory. The theoretical results are applicable to any fluid sphere moving steadily in a substantially immiscible, viscous liquid provided that the internal circulation is complete, the flow separation is negligible, and the Reynolds number is sufficiently large.