A numerical technique based on the simplified arbitrary Lagrangian-Eulerian (simplified ALE) method for simulating the behavior of time-dependent interfacial stratified fluids is described. From the calculation of an integral-type constitutive equation, the strain history is obtained by the memory integration method in a moving Lagrangian grid. Excess distortion of mesh elements due to fluid straining is overcome by the use of an arbitrary Lagrangian-Eulerian treatment of the continuous mesh re-zoning. The scheme is applied to simulate an interfacial deformation of stratified fluids at the Y-junction of two laminar layers. From results of a benchmark calculation, the present scheme is found to show good agreement with previous numerical works of pure viscous and shear-thinning fluids. Moreover, the present results predict some important flow phenomena that can appear in the flow geometry of viscoelastic fluids. (C) 2000 Elsevier Science B.V. All rights reserved.