Axisymmetric slow viscous liquid flow around a spherical bubble translating in a circular tube

Abstract

We investigate the axisymmetric slow viscous liquid flow around a spherical bubble located on the axis of a long circular tube analytically based on the Stokes approximation. The bubble translates along the axis of the tube with a constant velocity within Hagen–Poiseuille flow flowing far from the bubble. The translating velocity of the bubble and mean velocity of the Hagen–Poiseuille flow are given arbitrarily. To analyze Stokes equation, we use the method of complex eigenfunction expansions and the method of least squared error. As results, the streamline patterns and the pressure contour lines in the flow field are drawn for some radii of the bubble. The drag exerted on the bubble and the pressure change induced by the bubble are determined according to the radius of the bubble. For a small bubble radius in the tube, we compared results with previous asymptotic results. The drift velocity of the bubble due to the Hagen–Poiseuille flow in the circular tube is calculated according to the bubble radius. We show that extra pressure drop due to the drift bubble in the Hagen–Poiseuille flow becomes positive or negative depending on the radius of the bubble. When the bubble moves along a circular tube in the absence of the Hagen–Poiseuille flow, a series of viscous toroidal eddies appears in the anterior and posterior directions of the bubble. Moreover, we discuss the pressure and normal stress distributions on the bubble surface, which depend on the radius of the bubble.