In this paper the two dimensional flow of a power-law fluid is studied analytically using a singular perturbation method in order to determine the residual liquid film thickness of power-law fluids on the wall of a circular tube or a rectangular channel when displaced by another immiscible fluid. Inner and enter expansions are developed in terms of a small parameter C-A(1/3) (modified capillary number). A differential equation for the shape of gas bubble is solved numerically in order to determine the inner solution. The method of matched asymptotic expansions is used to match the inner and outer solutions. This approach indicated that the residual liquid film thickness of non-Newtonian fluids increases with decreasing power-law index.