Numerical solution of the FENE-CR model in complex flows

• Introduction of a finite difference technique for solving the FENE-CR model.
• Numerical solution of confined and free surface flows.
• Numerical simulation of cross-slot flow instability.
• Numerical predictions of a drop impacting a rigid surface.
• Analysis of the effects of Re, Wi and L2 on the flow.

A finite difference technique for solving the FENE-CR (Finite Extendable Non-linear Elastic – Chilcott and Rallison) closure constitutive model in complex flows has been developed and tested. The governing equations are solved using a Marker-and-Cell type method on a staggered grid. The momentum equation is integrated employing an implicit method while the FENE-CR constitutive equation is approximated by a second-order Runge–Kutta scheme. To demonstrate that the numerical technique can cope with complex flows governed by the FENE-CR model, three flow problems were analysed: the fully-developed channel flow, the 2D cross-slot flow and the impacting drop problem. The analytic solution for fully-developed channel flow of FENE-CR fluids with a solvent viscosity is also presented for validation purposes. This flow problem is used to verify the numerical method and to quantify its accuracy by comparing numerical results of fully-developed channel flow with the analytic solution. The second flow is employed to assess whether the numerical methodology is capable of capturing the purely-elastic instabilities predicted in the literature for 2D cross-slot confined flows. Additionally, the complex free surface flow corresponding to the filling of a 2D cross geometry has also been investigated. The last problem concerns the flow dynamics of a FENE-CR fluid drop impacting on a rigid surface, which allows the assessment of the capability of the model to deal with free surfaces. The effects of varying the Reynolds number, the Weissenberg number and the finite extensibility of the polymer molecules (L2)(L2) on the resulting flow patterns are analysed.