A long gas bubble moving in a tube with flowing liquid

The steady axisymmetric behavior of a long gas bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier-Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh for wide ranges of capillary number Ca and Reynolds number Re. As illustrated with a series of computed results, the hydrodynamic stresses due to liquid flow around the bubble tend to shape the middle section of long bubbles into a cylinder of constant radius, with a uniform annular liquid flow adjacent to the tube wall. But the surface tension effect tends to cause nonuniformities in the annular liquid film thickness. The annular liquid film thickness generally increases with increasing Ca, but decreases with increasing Re. In units of the bubble velocity relative to the tube wall, the average liquid flow velocity U¯ is always less than unity, indicating that the bubble always moves faster than the average liquid flow. For convenient practical applications, a least-square fitted empirical formula is obtained in a form of U¯=1-A∼/(1+B∼Ca-C∼) with A∼,B∼, and C∼ being functions of Re. The fact that the behavior of long bubbles moving in a tube appears independent of the bubble length is consistent with the inconsequential influence of the uniform annular film flow in bubble's middle section to the bubble dynamics. Whereas all the long bubbles exhibit similar nose profile, various tail shapes can be obtained by adjusting the values of Re and Ca.