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.

我们基于Stokes近似分析地研究了围绕长圆形管轴上的球形气泡周围的轴对称慢粘性液体流动。气泡在远离气泡的哈根–泊瓦流中以恒定的速度沿管的轴线平移。可以任意给出气泡的平移速度和哈根-泊瓦伊流的平均速度。为了分析斯托克斯方程，我们使用了复特征函数展开法和最小平方误差法。结果，针对气泡的某些半径绘制了流场中的流线样式和压力轮廓线。根据气泡的半径确定施加在气泡上的阻力和由气泡引起的压力变化。如果管中的气泡半径较小，我们将结果与以前的渐近结果进行了比较。根据气泡半径，计算圆管中由于哈根-泊瓦流动引起的气泡漂移速度。我们表明，由于气泡的半径，哈根-泊瓦流中的漂移气泡导致的额外压降变为正值或负值。当气泡在没有哈根-泊肃叶流的情况下沿圆形管运动时，在气泡的前后方向会出现一系列粘性环形涡流。此外，我们讨论了气泡表面上的压力和法向应力分布，这取决于气泡的半径。我们表明，由于气泡的半径，哈根-泊瓦流中的漂移气泡导致的额外压降变为正值或负值。当气泡在没有哈根-泊肃叶流的情况下沿圆形管运动时，在气泡的前后方向会出现一系列粘性环形涡流。此外，我们讨论了气泡表面上的压力和法向应力分布，这取决于气泡的半径。我们表明，由于气泡的半径，哈根-泊瓦流中的漂移气泡导致的额外压降变为正值或负值。当气泡在没有哈根-泊肃叶流的情况下沿圆形管运动时，在气泡的前后方向会出现一系列粘性环形涡流。此外，我们讨论了气泡表面上的压力和法向应力分布，这取决于气泡的半径。