Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution

The simulation of transient flows is relevant in several applications involving viscoelastic fluids. In the last decades, much effort has been spent on deriving time-marching schemes able to efficiently solve the governing equations at low computational cost. In this direction, decoupling schemes, where the global system is split into smaller subsystems, have been particularly successful. However, most of these techniques only work if inertia and/or a large Newtonian solvent contribution is included in the modeling. This is not the case for polymer melts or concentrated polymer solutions. In this work, we propose two second-order time-integration schemes for discretizing the momentum balance as well as the constitutive equation, based on a Gear and a Crank-Nicolson scheme. The solution of the momentum and continuity equations is decoupled from the constitutive one. The stress tensor term in the momentum balance is replaced by its space-continuous but time-discretized form of the constitutive equation through an Euler scheme implicit in the velocity. This adds velocity unknowns in the momentum equation thus an updating of the velocity field is possible even if inertia and solvent viscosity are not included in the model. To further reduce computational costs, the non-linear relaxation term in the constitutive equation is taken explicitly leading to a linear system of equations for each stress component. Four benchmark problems are considered to test the numerical schemes. The results show that a Crank-Nicolson based discretization for the momentum equation produces oscillations when combined with a Crank-Nicolson based scheme for the constitutive equation whereas, if a Gear based scheme is implemented for the constitutive equation, the stability is found to be dependent on the specific problem. However, the Gear based scheme applied to the momentum balance combined with both second-order methods used for the constitutive equation is stable and accurate and performs much better than a first-order Euler scheme. Finally, a numerical proof of the second-order convergence is also carried out.