A hybrid lattice Boltzmann model for simulating viscoelastic instabilities

Abstract

Low-Reynolds-number viscoelastic fluids exhibit a chaotic behaviour linked to the growth of elastic instabilities when the polymer molecules are stretched excessively. Despite offering various exciting benefits, the numerical difficulties associated with simulating this complex regime have left a lot still unexplored regarding the transition and onset of these low-Reynolds number viscoelastic instabilities. Recently, methods based on the lattice Boltzmann method (LBM) have emerged as a viable numerical tool for studying the behaviour of viscoelastic fluids largely due to the simplicity, adaptability, and intrinsic parallel features of LBM. However, extensive memory requirements and the inability to preserve numerical stability have restricted previous attempts from simulating viscoelastic instabilities. In this work, a hybrid lattice Boltzmann model is applied to simulate low-Reynolds number viscoelastic instabilities. In this approach, the hydrodynamic field is solved using a lattice Boltzmann model, whereas the polymer field is solved separately using a high-resolution finite difference scheme with the logarithmic Cholesky decomposition. The model is first validated for steady viscoelastic flows using the four-roll mill case, where the results are compared against the analytical solution and previous numerical studies. The model is then applied to simulate a chaotic low-Reynolds number viscoelastic instability by initialising a small perturbation to the isotropic polymer stress. The results at different Weissenberg numbers experience complex flow dynamics at sufficiently long time durations, which transition from quasi-periodic to periodic to aperiodic states with increasing elastic effects. The quasi-periodic regime was found to experience the fastest transition into the transient regime, with the transitioning time being delayed as the Weissenberg number increased. This work proves the capability of applying a coupled lattice Boltzmann method for exploring the complex flow behaviour of elastic instabilities, allowing the potential to explore more challenging and practical viscoelastic instability cases.