Fluid Mixing from Viscous Fingering: An Analytical Overview
The phenomenon of viscous fingering in fluid dynamics is a significant hydrodynamic instability occurring when a less viscous fluid displaces a more viscous fluid. This paper by Jha, Cueto-Felgueroso, and Juanes from the Massachusetts Institute of Technology delves into fluid mixing arising from this instability. The authors employ a combination of high-resolution numerical simulations and a novel two-equation dynamic model to explore and quantify the interfacial dynamics and mixing rates during such displacement processes.
The research targets the enhancement of mixing efficiency in porous media at low Reynolds numbers by leveraging the natural instabilities that promote fluid motion heterogeneity. The focus is on simulating viscous fingering within a two-dimensional porous medium—a classical setup representing flow in structures like a Hele-Shaw cell. The mathematical framework utilized includes the advection-dispersion equation, Darcy’s law, and a divergence-free velocity condition. Critical to the simulations are two dimensionless parameters: the viscosity contrast between the two fluids and the Péclet number, defined as the ratio of advection to diffusion processes.
The numerical simulations particularly spotlight a viscosity ratio of 33 and a Péclet number of 10,000. Through progressive snapshots of the concentration field, the research documents the transformation of the fluid interface. In these simulations, fluid displacement proceeds from left to right with a moving reference frame synchronized to the flow's average velocity. The boundary conditions are periodic transversely and open in the flow direction, leading to the development of viscous fingers. The emergent fingers extend quickly longitudinally but less so transversely, inducing interfacial stretching.
Notably, the paper identifies various nonlinear interactions intrinsic to the fingering process, such as shielding, merging, and channeling of the fingers. These interactions are impacted by the non-local dependence of viscosity on fluid concentration, a key aspect of the displacement dynamics that significantly changes the efficiency of mixing. While viscous fingering enhances interfacial fluid mixing by increasing surface area and velocity disorder, it simultaneously causes channeling where the low-viscosity fluid bypasses portions of the medium, resulting in unmixed regions. This dual effect constitutes a central theme of the paper, encapsulating the paradox of increased interfacial area versus the inefficiency introduced by channeling.
The authors succeed in developing a robust two-equation dynamic model that articulates the evolution of concentration variance and mean scalar dissipation rate. This model facilitates the quantitative assessment of mixing efficiency in scenarios characterized by viscous instability. Using this framework, the study predicts the optimal viscosity contrast that maximizes mixing, providing insights that may be applied in broader contexts, such as enhanced oil recovery, groundwater contamination, and other subsurface flow scenarios.
In conclusion, the paper offers a comprehensive examination of the mixing dynamics associated with viscous fingering, supported by rigorous modeling and simulation results. The analytical model presented sheds light on both theoretical and practical perspectives, paving the way for future investigations into optimizing fluid mixing processes across different applications in fluid dynamics and engineering systems.
