A numerical study to analyze the interplay of Weissenberg number and viscosity ratio in a log-strain tensorial model for viscoelastic fluids

Technical Summary: Numerical Analysis of the Log-Strain Tensorial Model for Viscoelastic Fluids

Problem Statement
This study addresses the computational challenges associated with simulating viscoelastic fluids, specifically focusing on the interplay between the Weissenberg number ($Wi$) and the viscosity ratio ($\xi$). The viscosity ratio is defined as the ratio of the polymeric contribution to the effective viscosity ($\eta_p = \kappa\tau_r$) to the solvent viscosity ($\eta$). While the role of $Wi$ in viscoelastic flows is well-documented, the independent influence of $\xi$ is often neglected. Furthermore, numerical simulations at high $Wi$ frequently encounter the High Weissenberg Number Problem (HWNP), characterized by the loss of positive definiteness in the conformation tensor and numerical breakdown in regions of high deformation rates. The authors investigate a recently proposed log-strain tensorial model, which utilizes a logarithmic relation between elastic stress and elastic strain, to explore these parameters systematically.

Methodology
The authors employ a stabilized mixed finite element formulation to solve the governing equations of the log-strain model. The methodology comprises the following key components:

• Governing Equations: The model describes the evolution of the elastic strain tensor ($B_{el}$) using an upper-convected derivative balanced by a nonlinear relaxation term involving $\log B_{el}$. The elastic stress is defined as $T_{el} = \kappa \log B_{el}$. This formulation inherently preserves the symmetry and positive definiteness of the strain tensor.
• Temporal Discretization: A semi-implicit first-order Backward Euler scheme is used. To decouple the kinematic fields from nonlinearities, the convective and constitutive relaxation terms are linearized using known quantities from the previous time step.
• Advection Treatment: The evolution of the strain tensor is handled via a generalized Lie derivative formulation. A semi-Lagrangian characteristic mapping is employed to update the tensor field, ensuring frame-invariance and enhancing stability. A projection step is applied at each increment to strictly enforce the determinant and positive definiteness constraints of $B_{el}$.
• Spatial Discretization: The spatial domain is discretized using equal-order continuous linear Lagrange elements (P1-P1) for velocity and pressure, and piecewise constant discontinuous Galerkin elements (DG0) for the strain tensor. To circumvent the inf–sup condition and stabilize the solution, the Variational Multiscale (VMS) framework is applied to the velocity-pressure system. The strain tensor field is stabilized through the temporal discretization and Lie derivative treatment, rendering additional VMS stabilization for the tensor unnecessary in the tested regimes.
• Benchmark Cases: The study analyzes three paradigmatic pressure-driven flows: flow in a straight channel, flow past a cylinder, and flow through a 4:1 planar contraction. Simulations are conducted at low Reynolds numbers ($Re \ll 1$) to isolate the effects of elasticity and dissipation.

Key Contributions and Results

1. Role of the Viscosity Ratio ($\xi$):
   The study demonstrates that $\xi$ is a critical parameter in determining the extent of non-Newtonian behavior as $Wi$ increases.

   • In straight channel flows, for low $\xi$, velocity profiles remain approximately parabolic. As $\xi$ increases, the profiles develop a "plug-like" region near the centerline, characteristic of shear-thinning fluids. This plug region shrinks as $Wi$ increases, but the phenomenon is only observable when $\xi$ is sufficiently large.
   • The results indicate that $Wi$ alone does not dictate flow properties; the magnitude of $\xi$ determines the degree of shear-thinning and the resulting deviation from Newtonian profiles.

2. Drag Reduction and Flow Past a Cylinder:
   Simulations of flow past a cylinder reveal a drag reduction effect consistent with shear-thinning behavior. The drag coefficient ($C_D$) decreases as the effective strain rate increases. The study highlights that for high $\xi$, the model exhibits a strong dependence on the local flow type. While the model displays shear-thinning in simple shear, it shows no rate-dependence in extensional flows, a feature that distinguishes it from simpler models.

3. 4:1 Planar Contraction:
   In the contraction flow, the model predicts a non-monotonic variation in the size of the corner vortex as $Wi$ increases. Notably, the nucleation of secondary lip vortices is entirely absent across the investigated parametric range. This absence aligns with the behavior of shear-thinning models (e.g., PTT, Giesekus), where high strain rates near the re-entrant corner reduce viscous resistance, preventing the formation of secondary recirculation zones often seen in constant-viscosity models like Oldroyd-B.

4. Comparison with Generalized Newtonian Fluid (GNF) Models:
   The authors compared the log-strain model solutions against a GNF model where the viscosity was fitted to match the log-strain model's shear-thinning behavior in simple shear.

   • Straight Channel: The GNF and log-strain models yielded nearly identical velocity profiles (relative error ~2.6%), confirming that in simple shear-dominated flows, a rate-dependent viscosity is sufficient.
   • Flow Past a Cylinder: Significant discrepancies arose between the two models. The velocity profiles differed not just by a scaling factor but in shape. The authors attribute this to the presence of extensional flow regions upstream and downstream of the cylinder, where the GNF model's viscosity dependence (derived from shear) fails to capture the dissipation mechanisms of the full tensorial log-strain model.

Significance and Claims
The paper claims that the log-strain model, combined with the proposed stabilized numerical scheme, successfully simulates viscoelastic flows up to relatively high Weissenberg numbers ($Wi \sim 35$) without the numerical breakdown typical of the HWNP. The study emphasizes that the viscosity ratio $\xi$ is an independent and crucial factor in generating non-Newtonian flow profiles, a nuance often overlooked when focusing solely on $Wi$.

Furthermore, the comparison with the GNF model underscores the limitations of inelastic approximations. While GNF models may suffice for simple shear-dominated flows, the log-strain model captures essential rheological features dependent on the flow type (shear vs. extension), which are necessary for accurately predicting flows in complex geometries. The authors conclude that the intrinsic mathematical structure of the log-strain model, which preserves the positive definiteness of the conformation tensor, is vital for numerical stability and physical admissibility.

Future Directions
The authors suggest that future work should focus on developing computational methods capable of handling three-dimensional flows and more complex geometries, as well as investigating the interplay between inertia and elasticity at higher Reynolds numbers. Additionally, they propose extending the constitutive model to include multiple relaxation modes or state-dependent relaxation times.