The steady axisymmetric flow of viscous liquid relative to a gas bubble due to its buoyancy-driven motion in a round tube is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh. When the bubble is relatively small compared with the tube size (e.g. the volume-equivalent radius of the bubble is less than a quarter of the tube radius R), the bubble exhibits similar behaviour to one moving in an extended liquid, developing a spherical-cap shape with increasing Reynolds number (Re) if the capillary number is not too small. The long-bubble (also known as a Taylor bubble) characteristics can be observed with bubbles of volume-equivalent radius greater than the tube radius, especially when the surface tension effect is relatively weak (e.g. for Weber number We greater than unity). The computed values of Froude number Fr for most cases agree well with the correlation formulae derived from experimental data for long bubbles, and even with (short) bubbles of volume-equivalent radius three-quarters of the tube radius. All of the computed surface profiles of long bubbles exhibit a prolate-like nose shape, yet various tail shapes can be obtained by adjusting the parameter values of Re and We. At large Weber number (e.g. We=10), the bubble tail forms a concave profile with a gas ‘cup’ developed at small Re and a ‘skirt’ at large Re with sharply curved rims. For We≤1, the bubble tail profile appears rounded without large local curvatures, although a slightly concave tail may develop at large Re. non-uniform annular film adjacent to the tube wall is commonly observed when Weber number is small, especially for bubbles of volume <3πR3, suggesting that the surface tension effect can play a complicated role. Nonetheless the computed value of Fr is found to be generally independent of the bubble length for bubbles of volume-equivalent radius greater than the tube radius. If the bubble length reaches about 2.5 tube radii, the value of its frontal radius becomes basically the same as that for long bubbles of much larger volume. An examination of the distribution of the z-component of traction along the bubble surface reveals the basic mechanism for long bubbles rising at a terminal velocity that is independent of bubble volume.
The surface tension effect on viscous liquid spreading along a superhydrophobic surface
