On the Film Thickness Between a Bubble and the Wall in Liquids in Vertical Tubes

We study numerically the film thickness that is formed between the free surface of a bubble and the inner wall in a vertical tube. The bubble is formed by gas injection in a tube filled with a viscous fluid. The computations were performed through the use of the Boundary element method (BEM) to solve the Stokes equations and a fourth order Runge–Kutta scheme to build the bubble shape. After the computation of the bubble shape, the thickness of the annular film was calculated for low Bond numbers, Bo, and a wide range of Capillary numbers, Ca. For the case Ca ≪ 1 (inviscid approximation) it is found that the film actually touches the wall, meanwhile for the viscous case we found that the film thickness, scaled by radius of the tube, is a function of Ca and Bo. We also discuss experiments that validate the numerical results.