This work presents the development of both weakly compressible and incompressible Smoothed Particle Hydrodynamics (SPH) models for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration with industrial significance, we have chosen to model the extrudate swell of a second-order polymeric fluid. The extrudate or die swell is a phenomenon that takes place during the extrusion of polymeric fluids. When a polymeric fluid is forced through a die to give a polymer its desired shape, due to its viscoelastic non-Newtonian nature, it shows a tendency to swell or contract at the die exit depending on its rheological parameters. The die swell phenomenon is a typical example of a free surface problem where the free surface is formed at the die exit after the polymeric fluid has been extruded. The swelling process leads to an undesired increase in the dimensions of the extrudate. To be able to obtain a near-net shape product, the flow in the extrusion process should be well-understood to shed some light on the important process parameters behind the swelling phenomenon. To this end, a systematic study has been carried out to compare constitutive models proposed in literature for second-order fluids in terms of their ability to capture the physics behind the swelling phenomenon. The effect of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell problem was solved for a wide range of Deborah numbers and for two different Re numbers. The numerical model was validated through the solution of fully developed Newtonian and Non-Newtonian viscoelastic flows in a two-dimensional channel, and the results of these two benchmark problems were compared with analytic solutions, and good agreements were obtained.
Existence of solution for a resonant p-Laplacian second-order m-point boundary value problem on the half-line with two dimensional kernel
