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This work builds on previous studies demonstrating the computational efficiency of a 2D gap-averaging model developed for the displacement of power-law fluids in Hele-Shaw cells [3], with a particular focus on the instability of the viscous fingering and the growth rates of the perturbation. In this study, the theoretical growth rate for power-law fluids is derived and compared with results from 2D computational fluid dynamics (CFD) simulations. To ensure a quantitative comparison, simulations are conducted under controlled initial conditions with well-defined wave numbers, which allows tracking of the growth of the perturbation over time.
The theoretical framework established in this work is a tailored version of growth rate formulas, inspired by the approach of Mora and Manna [2] for generalized Newtonian fluids. This adaptation modifies their methodology to suit power-law fluids, simplifying the theoretical model to closely align with the classical Saffman-Taylor problem.
Simulations using the developed 2D model begin with Newtonian fluids, setting a baseline by comparing results with the existing literature, including 3D simulations and linear stability analysis by Lu et al. [1]. Moving to power-law fluid simulations, this study explores a variety of fluid pairs, including oil with xanthan gum and polyacrylamide solutions. Key variables such as the consistency index (k), flow behavior index (n), and surface tension are meticulously varied to examine their effects on the dynamics of viscous fingering.
For each simulation, the dimensionless growth rate of the perturbation is extracted and compared with theoretical results, offering a detailed analysis of how different parameters influence the growth rate and stability of viscous fingers in two-phase fluid systems. By systematically exploring critical parameters such as rheological properties, the ratio of friction gradients, surface tension, and wall boundary conditions, this research not only confirms the applicability of the developed 2D model but also brings new insights into the understanding of interface stability and the dynamics of viscous fingering.
References
- Daihui Lu, Federico Municchi, and Ivan C Christov. Computational analysis of interfacial dynamics in angled Hele-Shaw cells: Instability regimes. Transport in Porous Media, 131(3):907–934, 2020.
- S. Mora and M. Manna. Saffman-Taylor instability for generalized Newtonian fluids. Phys. Rev. E, 80: 016308, Jul 2009. doi: 10.1103/PhysRevE.80.016308.
- Yao Zhang, Hans Joakim Skadsem, Knut Erik Teigen Giljarhus, and Benjamin Barrouillet. Numerical modeling of fluid displacement in Hele-Shaw cells: A gap-averaged approach for power-law and Newtonian fluids. Rheologica Acta, page Under Review, 2025.
