Cost-effective multi-mode FENE bead-spring models for dilute polymer solutions

Abstract

New multi-mode finitely extensible nonlinear elastic (FENE) bead-spring models, FENE-M chain and FENE-MR chain, are presented as cost-effective alternatives to the FENE chain. The cornerstone of the new models is an approximate form of the spring force law which remains invariant under the transformation to normal coordinates. This allows the governing equations to be solved in normal coordinates, where different modes are decoupled. Utilizing this decoupling, the FENE-M chain closely reproduces the dynamics of a FENE chain in general flow kinematics at a computational cost which is O(1/ N 2) times the most efficient semi-implicit numerical solution of a FENE chain or ∼( N+1)/3 times the cost of a FENE dumbbell, where N denotes the number of beads in the chain. A reduced form of this model, referred to as the FENE-MR chain, is also proposed in which the cumulative behaviour of modes 2 through ( N−1) is captured in a single representative mode. The FENE-MR chain replicates the stresses predicted by a FENE chain in general flow kinematics at a cost only 35% higher than that for a FENE dumbbell. The approach used to derive the FENE-M and FENE-MR models can also be applied to closure approximations. Here, we apply these concepts to the FENE-LS dumbbell closure [J. Non-Newtonian Fluid Mech. 87 (1999) 179] to develop the FENE-LSM and FENE-LSMR chain closures. The FENE-LSMR chain closure closely approximates the dynamics of a FENE chain at a cost only twice that of a FENE-P dumbbell. Tests of the proposed models have been performed in standard steady-state and time-dependent rheometrical flows and in the flow kinematics obtained from a DNS database of a turbulent channel flow over a wide range of polymer and flow parameters (10 4≤ b( N−1)≤10 6; 1≤ We≤100) to demonstrate the robustness of the proposed models in general flow kinematics.