The effect of fluid viscoelasticity on the fraction of liquid deposited on the walls of capillary geometry and the pressure drop at the capillary static region are theoretically investigated using the Criminale-Ericksen-Filbey (CEF) constitutive equation to describe a non-Newtonian fluid displaced by the pressurized gas in a capillary. The singular perturbation method is used to determine the residual liquid film thickness of a viscoelastic fluid on the walls of a circular tube or a rectangular channel when displaced by another immiscible fluid. Inner and outer expansions are developed in terms of both a small parameter Ca-1/3 and a small parameter De/Ca-1/3. The method of matched asymptotic expansion is used to match the inner and outer solutions by means of a transition region between the advancing meniscus and the entrained film where the fluid rheology has its greatest effect. A detailed analysis indicates that the residual liquid film thickness of the viscoelastic fluid tends to decrease and the pressure drop across the bubble front tends to increase as the fluid becomes more viscoelastic. The theoretical results presented in this paper are in agreement with some of the experimental data and theoretical analyses available in the literature. (c) 2005 Elsevier Ltd. All rights reserved.