A Numerical Study of Several Viscoelastic Fluid Models

Abstract

Viscoelastic fluids are a type of fluid with a solvent and immersed elastic filaments which create additional stresses on the fluid. The Oldroyd-B equations are a well accepted model of the flow of viscoelastic fluids but in extensional flows, a characteristic of flows where liquids approach or separate from each other, as the Wiessenberg number (Wi), a number that measures the relaxation time of the fluid, approaches infinity the stress of the polymer also goes to infinity. For small Wi, the polymer stress remains bounded but as Wi gets bigger the polymer stress approaches a cusp shape until the solution eventually becomes unbounded. Modifications to the Oldroyd-B model have been proposed that keep the solutions bounded, such as the Polymer Diffusion, Giesekus Model, and Phan-Thien and Tanner model. Here we study how well these modifications approximate the Oldroyd-B model when the stress is very large. An ideal model for numerical simulations would be close to the Oldroyd-B model outside of a small region near the cusp or singularity but still be well-resolved near the singularity. Analysis has been done to see how the proposed solutions differ in regards to stress, time and other factors. When finding such results it is desirable to use minimal computing resources when resolving these near singular solutions. Several different modifications to the Oldroyd-B system with stress diffusion are investigated using MATLAB and discussed to identify which modifications perform the best in this flow geometry.