Fluid Mixing from Viscous Fingering

Mixing efficiency at low Reynolds numbers can be enhanced by exploiting hydrodynamic instabilities that induce heterogeneity and disorder in the flow. The unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid-fluid interfacial area and enhance mixing. Here we describe the dissipative structure of miscible viscous fingering, and propose a two-equation model for the scalar variance and its dissipation rate. Our analysis predicts the optimum range of viscosity contrasts that, for a given Péclet number, maximizes interfacial area and minimizes mixing time. In the spirit of turbulence modeling, the proposed two-equation model permits upscaling dissipation due to fingering at unresolved scales.

Figure 1

Snapshot of the concentration field during the unstable displacement of a more viscous fluid (dark) by a fully-miscible, less viscous fluid (light). The formation, splitting, and nonlinear interaction of viscous fingers induce disorder in the velocity field that affects the mixing rate between the fluids. The displacement corresponds to a viscosity ratio $M=\exp (3.5)\approx 33$ and Péclet number $\mathrm{Pe}={10}^4$. See video 1 26.

Figure 2

Decay of miscible viscous fingering. A binary mixture of initially segregated fluids is driven at constant flow rate in a periodic domain. The variance of concentration, ${\sigma }^2$ (red), a proxy for the degree of mixing, decreases monotonically with time. The mean scalar dissipation rate, $ϵ$ (blue), increases at early times due to the onset of fingering, and decays monotonically at later times. The simulation parameters are $R=2$, $\mathrm{Pe}={10}^4$. See video 2 26.

Figure 3

The dissipative structure of viscous fingering. (a),(b) The mechanical and scalar dissipation rates synchronize as mixing advances, as shown by the scatter plots of mechanical (${ϵ}_u$) against scalar ($ϵ$) dissipation rates at early (a) and late (b) times. At late times we find ${ϵ}_u\sim {ϵ}^{\alpha }$, with $\alpha \approx 1$. Inset, logarithm of the scalar dissipation rate. (c),(d) Probability density function (PDF) of the derivatives of the concentration field and scalar dissipation rate. (c) At early times, $\partial c/\partial y$ exhibits characteristic exponential tails. This behavior is similar to that of passive scalars in turbulent flows [22]. (d) At late times, the PDFs tend toward a Gaussian behavior. The skewness of $\partial c/\partial x$ reflects the inhomogeneity of the flow.

Figure 4

(a),(b) The mean dissipation rate (blue) and variance (red) predicted by the two-equation model (dashed) compare well with the results from direct numerical simulations (solid). (c) Mixing time, that is, time to reach 80% degree of mixing, plotted for different mobility contrasts and $\mathrm{Pe}={10}^4$, from both simulations and the ${\sigma }^2-ϵ$ model. Mixing time behaves nonmonotonically with $R$, increasing at high $R$ due to channeling. Inset: mixing time surface from the model as a function of $R$ and Pe: $A=0.76$, $B=0.84$.