Enhanced mixing via alternating injection in radial Hele-Shaw flows

Mixing at low Reynolds numbers, especially in the framework of confined flows occurring in Hele-Shaw cells, porous media, and microfluidic devices, has attracted considerable attention lately. Under such circumstances, enhanced mixing is limited due to the lack of turbulence, and absence of sizable inertial effects. Recent studies, performed in rectangular Hele-Shaw cells, have demonstrated that the combined action of viscous fluid fingering and alternating injection can dramatically improve mixing efficiency. In this work, we revisit this important fluid mechanical problem, and analyze it in the context of radial Hele-Shaw flows. The development of radial fingering instabilities under alternating injection conditions is investigated by intensive numerical simulations. We focus on the impact of the relevant physical parameters of the problem (Péclet number $\mathrm{Pe}$, viscosity contrast $A$, and injection time interval $\mathrm{\Delta }t)$ on fluid mixing performance.

Figure 1

Evolution of concentration images for Péclet number $\mathrm{Pe}=3000$, viscosity contrast $A=0.922$, and different alternating injection intervals $\mathrm{\Delta }t=0.5$ (top row), 0.25 (middle row) and 0,125 (bottom row), taken at times $t=0.5$ (left column), 0.75 (middle column), and 1 (right column).

Figure 2

Interfacial profiles, represented by concentration contours with $c=0.1$ at $t=1$, for $A=0.922$, and $\mathrm{Pe}=3000$, whose correspondent images are shown in the right column of Fig. 1. Here the dimensionless radial position of the outermost interfacial profile $r_f$ is plotted as a function of the polar angle $\theta$. Despite the active interactions amongst the fingering structures, changes in the alternating injection interval $\mathrm{\Delta }t$ do not significantly alter the ultimate shape of the peripheral fingering patterns.

Figure 3

Evolution of the rotational component of the streamlines, corresponding to the concentration images depicted in Fig. 1.

Figure 4

Concentration images obtained for $\mathrm{\Delta }t=0.1$ at time $t=1,A=0$ (top row), $A=0.848$ (bottom row), and for three values of the Péclet number: 1000 (left column), 3000 (middle column), and 6000 (right column).

Figure 5

Time evolution of the concentration variance ${\sigma }^2$ for several values of the control parameters $\mathrm{Pe},A$, and $\mathrm{\Delta }t$. Lower values of the variance imply enhanced mixing efficiency.

Figure 6

Behavior of the normalized variance ${\sigma }_n^2$ as the Péclet number $\mathrm{Pe}$ is changed, for $t=1$, and several values of the control parameters $A$ and $\mathrm{\Delta }t$. Note: the cases for $A=0.922$ and $\mathrm{Pe}\ge 6000$ are numerically unstable, so they are not presented.

Figure 7

Fingering patterns for $A=0.922$ at $t=1,\mathrm{\Delta }t=0.05$ (top row), $\mathrm{\Delta }t=0.25$ (bottom row), $\mathrm{Pe}=1000$ (left column), and $\mathrm{Pe}=3000$ (left column).