Dilute polymer solutions Kinetic theory, constitutive modelling and simulations

Abstract

Rheological models for dilute polymer solutions are typically derived from simplified micro mechanical representations of the individual macro-molecules. For example, the elastic dumbbell model, featuring two beads connected by an entropic spring, serves as a basic representation of a polymer molecule. The simplistic Hookean spring force gives rise to Oldroyd-B type constitutive models, which suffers largely in strong flows. While Warner’s finitely extensible non-linear elastic spring (FENE) provides realistic rheological predictions, it leads to closure problems. The FENE-P approximation provides a closed-form constitutive equation for the conformation tensor. However, in strong flows and transient flows it leads to erroneous predictions. We revisit approximate constitutive modelling for the FENE model. We recognize that the non-linearity of the spring, departure from Gaussian distribution and correct tensor structure, are crucial for predicting accurate rheological behaviour and one needs an additional nonlinear scalar variable to represent highly non-linear situations seen in strong flows. We show that this new model (termed as FENE-NP), drastically improves the steady state and time dynamics results over the FENE-P model, especially in the ability to capture a transient second normal stress difference in shear flows and the ability to capture hysteresis in strong uni-axial extensional flows. Once the polymer solver is bench-marked for viscometric flows, we formulate the fluid solver. We develop a novel lattice Boltzmann method (LBM) framework that uses crystallographic discrete velocities (termed RD3Q35) for subsonic flows. This model is then coupled with the FENE-NP constitutive model for exploring the low to moderate Reynolds regime for two sets of problem: internal flow (flow past non-circular duct) and external flow (flow past cylinder). We also discuss the development of a flow force calculation routine (discrete Reynolds transport theorem : DRTT), that aims at more accurate calculations of drag and lift coefficients for visco-elastic flow past solid objects