The one-dimensional approximate equations governing flow in a free coating and a gas-assisted displacement of liquid in a vertical or inclined tube is derived using asymptotic techniques. These techniques are employed to place the free coating flow and mass conservation equations, utilized in previous studies, on a firmer theoretical ground. The derived one-dimensional flow equation for the motion of along bubble rising steadily in a vertical or inclined tube filled with a power-law fluid is found to be identical to that of Gutfinger and Tallmadge [AIChE 11 (1965) 403] for a free coating on a flat plate withdrawn vertically from the liquid reservoir. The theoretical analysis indicated that the residual liquid film thickness increases with decreasing power-law index. This trend is similar to that observed experimentally by Huzyak and Koelling [J. Non-Newt. Fluid Mech. 175 (1997) 73] for Boger fluids. The prediction of the liquid film thickness from this analysis qualitatively agrees with the experimental results of Huzyak and Koelling [J. Non-Newt. Fluid Mech. 175 (1997) 73].