Numerical modelling of viscoelastic flows based on a log-conformation formulation

Resumen

Viscoelastic fluids are a type of non-Newtonian fluids which are formed by complex internal structures and high-molecular-weight, whose typical examples are the polymer solutions and molten polymers. Also, the viscoleastic fluid flow presents a combination of two fluid properties: viscosity and elasticity. The main characteristic regarding the behavior of these flows is the strong dependence of the stresses on the flow history. Due to this complexity, computing the viscoelastic fluid flow involves a wide range of difficulties, in particular when elasticity becomes dominant, i.e., when the dimensionless Weissenberg number is high. These difficulties are considered one of the biggest challenges in computational rheology; this is known as the High Weissenberg Number Problem (HWNP). This study presents different strategies to deal with the numerical shortcomings that appear when the viscoelastic fluid is particularly elastic. These strategies are carried out in the Finite Element (FE) framework and by using the Variational Multiscale (VMS) formulation as stabilization approach. A term-by-term is also design. The cornerstone of this work is the application of a reformulation of the equations associated to the standard formulation, namely, the logarithmic reformulation, which permits simulating more elastic flows due to the fact that it eliminates the exponential stress profiles in the vicinity of stress singularities. Another topic addressed in this work is the study of the effect of temperature in viscoelastic fluid flow, where a two-way strategy is considered: the viscoelastic properties have now a dependence with the temperature, and the energy equation takes into account has to consider the viscous dissipation. That study is particularly interesting due to the fact that non-isothermal flow in many industrial applications. On the other hand, the incorporation of time-dependent subscales for solving the viscoelastic fluid flow problem is crucial to address two issues: the first one related with the instability produced when solving anisotropic space-time discretizations, and the second, the already mentioned exponential growth typical in viscoelastic flows with high Weissenberg number. In this work, time-dependent subgrid-scales are presented for both formulations: standard and logarithmic. Finally, as the logarithmic formulation is particularly expensive, above all when the scheme considered is monolithic, a fractional step for this formulation is designed, in which the system of equations is defined in a fully decoupled manner. This algorithm is especially useful when purely elastic instabilities need to be captured. These instabilities lead in some cases to elastic turbulence: a physical phenomenon in which the fluid flow becomes chaotic even for low Reynolds numbers.