Synergetic Fluid Mixing from Viscous Fingering and Alternating Injection

We study mixing of two fluids of different viscosity in a microfluidic channel or porous medium. In recent work, we suggested that miscible viscous fingering--a hydrodynamic instability that takes place when a less viscous fluid displaces a more viscous fluid--can enhance mixing in Darcy flows, such as flows in Hele-Shaw cells or porous media [1]. Enhanced mixing due to viscous fingering emerges from the velocity disorder and the additional interfacial area created between the two fluids as a result of the hydrodynamic instability. Here, we show that the synergetic action of alternating injection and viscous fingering leads to a dramatic increase in mixing efficiency at high Péclet numbers. Based on observations from high-resolution simulations, we develop a theoretical model of mixing efficiency that combines a hyperbolic mixing model of the channelized region ahead, and a mixing-dissipation model of the pseudo-steady region behind. Our macroscopic model quantitatively reproduces the evolution of the average degree of mixing along the flow direction, and can be used as a design tool to optimize mixing from viscous fingering in a microfluidic channel. [1] B. Jha, L. Cueto-Felgueroso and R. Juanes, Fluid mixing from viscous fingering, Physical Review Letters, 106, 194502 (2011).