Film thickness between a bubble and the inner wall of vertical tubes containing viscous liquids

We study numerically the film thickness in between the free surface of a bubble and the inner wall in a vertical tube filled with a viscous liquid. The computations were performed using the Boundary element method 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 and a wide range of Capillary numbers, Ca. For the case Ca close to zero (inviscid approximation) it is found that the film actually touches the wall, meanwhile for the viscous cases always there is a film of finite thickness.