Numerical simulation of viscoelastic flows using integral constitutive equations: A finite difference approach

This work presents a numerical technique for simulating incompressible, isothermal, viscoelastic flows of fluids governed by the upper-convected Maxwell (UCM) and K–BKZ (Kaye–Bernstein, Kearsley and Zapas) integral models. The numerical technique described herein is an extension of the GENSMAC method to the solution of the momentum and mass conservation equations to include integral constitutive equations. The governing equations are solved by the finite difference method on a staggered grid using a Marker-and-Cell approach. The Finger tensor Bt′(t)Bt′(t) is computed in an Eulerian framework using the ideas of the deformation fields method. However, improvements to the deformation fields   method are introduced: the Finger tensor Bt′(x,t)Bt′(x,t) is obtained by a second-order accurate method and the stress tensor τ(x,t)τ(x,t) is computed by a second-order quadrature formula. The numerical method presented in this work is validated by comparing the predictions of velocity and stress fields in two-dimensional fully-developed channel flow of a Maxwell fluid with the corresponding analytic solutions. Furthermore, the flow through a planar 4:1 contraction is investigated and the numerical results were compared with the corresponding experimental data. Finally, the UCM and the K–BKZ models were used to simulate the planar 4:1 contraction flow over a wide range of Reynolds and Weissenberg numbers and the numerical results obtained are in agreement with published data.