Buoyant miscible displacement flows in a nonuniform Hele-Shaw cell

Miscible displacement flows within the gap of a nonuniform Hele-Shaw cell are considered, theoretically and experimentally. The cell is vertical and it can be diverging or converging. A light fluid displaces a heavy fluid downwards. The displacement imposed velocity is sufficiently large so that diffusive effects are negligible within our time scale of interest. For certain flow parameters, the displacement flow is characterized by a symmetric, two-dimensional penetration of the light fluid into the heavy one, for which a lubrication approximation approach is developed to simplify the governing equations and find a semianalytical solution for the flux functions. The solutions reveal how the cell nonuniformity may affect the propagation of the interface between the two fluids, versus the other flow parameters, i.e., the viscosity ratio $\left(m\right)$ and a buoyancy number $\left(\chi \right)$, for which a detailed flow regime classification is presented. Our results demonstrate that the presence of nonuniformity adds a unique spatiotemporal nature to these displacements which is not the case for uniform cell flows. The combination of the model and experiments reveals the existence of self-spreading, spike, and unstable (viscous fingering) flow regimes, which may occur at various spatial positions within the cell. A converging cell may allow a transition from spike to self-spreading or unstable regime, whereas a diverging cell may offer a transition from self-spreading or unstable to spike regime. Our work demonstrates that the novel spatiotemporal nature of nonuniform cell flows must be considered through the numerical solution of the interface propagation equation, to yield accurate predictions about the flow behaviors at various spatial positions.

Figure 1

Schematic of the displacement flow geometry considered. Here $\alpha$ is negative so that the light fluid displaces the heavy fluid in a slightly converging vertical channel. For a positive value of $\alpha$, the channel would be slightly diverging. Note that $\left|\alpha \right|\ll 1$.

Figure 2

Examples of displacements showing interface evolution in time at $t=0,2.5,...,22.5,25$, for (a) $m=0.1,\chi =1$; (b) $m=1,\chi =10$; (c) $m=10,\chi =1$. In (a), the convergence of the results is shown using two typical mesh sizes: $dx=0.1$ (line) and $dx=0.2$ (dashed line). The inset for (b) is zoomed at the displacing tip in the range $25\le x\le 40$. Also, note that $h_f,h_{sp}$, and ${\ell }_{sp}$ denote the front height, the spike height, and the spike length.

Figure 3

Comparison between analytical regime classification for uniform Hele-Shaw cell displacement flows and simulation results of the spike $\left(\square \right)$, self-spreading $\left(\circ \right)$, and frontal shock $\left(▵\right)$ regimes. The boundaries between the regimes, marked by different colors, can be obtained by analyzing the analytical form of the flux function, for which the details can be found in [14]. Throughout the paper, we adopt the convention of denoting simulation and experimental datapoints by hollow and filled symbols, respectively.

Figure 4

Results for a uniform channel. (a) Contours of $h_s$ versus $\chi$ and $m$. (b) Contours of $V_f$ versus $\chi$ and $m$.

Figure 5

Interface evolution in time, $t=0,3,...,27,30$ at $\chi =10$. In each row $m$ is fixed $(m=0.1,1,10$ from top to bottom). In each column, $\alpha$ is fixed $(\alpha =-0.01,0,+0.01$ from left to right). The insets show interface heights close to the tip and cover the same range of $x$ as in the main figures.

Figure 6

Simulation datapoints representing ${\chi }_{\text{cr}}^{*}\equiv {\chi }_{\text{cr}}{\left(1+\alpha x\right)}^3$, denoting the critical buoyancy number for the transition from spike to self-spreading regime (diamonds) and the transition from spike to frontal shock regime (stars). (a) Critical values in the plane of ${\chi }^{*}$ and $\alpha x$. The symbol size represents the viscosity ratio magnitude, varying as $m=0.1,0.2,...,1,2,...,9,10$, except for $m=0.6$ which is undetermined due to it being the zone in between self-spreading and shock flows. (b) Critical values in the plane of ${\chi }^{*}$ and $m$. The symbol size represents the growth in $\alpha x$, varying as $\alpha x=-0.8,-0.6,...,0.6,0.8$. The shaded areas mark the regimes predicted by the analytical results for uniform cells, i.e., the same as presented in Fig. 3. Note that nonuniform cell results cannot necessarily be extended from uniform ones since the critical values for nonuniform cells vary with $\alpha x$ in (a) and they do not collapse on uniform cell boundaries in (b).

Figure 7

Experimental results in uniform channels: spike $(\blacksquare )$, self-spreading $(•)$, and unstable $(▴)$ flows. The shaded areas mark the regimes predicted by the analytical results, as discussed in the analytical section. The field of view for all the inset images is $6\times 7.5{\mathrm{cm}}^2$. Note that flows in the range of frontal shock (predicted by the model) are unstable flows in experiments.

Figure 8

Simulated versus experimental results for the front height as well as the spike length and height, in a uniform channel for $m=1$ and measured at $x=650$. The color bar represents the buoyancy number $\left(\chi \right)$, here and elsewhere. The axes of some plots are multiplied by ${log}_{10}\left(\chi \right)$ to offer a better distribution of data.

Figure 9

Experimental results (lines) compared to simulation outputs (dashed lines) for $m=1$: (a) $\chi =11,\alpha L=-0.8,t\approx 0,50,100,...,500$; note that due to the progressive decrease in buoyancy, the front height disappears after a certain distance and the self-spreading flow now takes on a shape analogous to a wedge; (b) $\chi =8,\alpha L=2.5,t\approx 0,130,230,...,1300$.

Figure 10

Experimental results of spike $(\blacksquare )$, self-spreading $(•)$, and unstable $(▴)$ flows compared to the transition from spike to frontal shock regime (stars and lines) and the transition from spike to self-spreading regime (diamonds and lines), predicted by simulations for $m=0.1,1,10$ (from left to right). In each graph, the intersecting line represents the boundary of the uniform geometry. This figure presents the new domain classification offered for nonuniform flows, analogous to the classification provided in Fig. 7.

Figure 11

Experimental results at $m=0.1$. Top row: (a) a stable interface in a uniform cell compared to (b) a destabilized interface in a converging cell $\left(\alpha L=-0.8\right)$, both at $\chi =7.5$. Bottom row: (c) an unstable interface in a uniform cell compared to (d) a stabilized interface in a diverging cell $\left(\alpha L=2.5\right)$, both at $\chi =1$. The field of view is $8.5\times 12.4{\mathrm{cm}}^2$ in each image.

Figure 12

Snapshots of displacements in a frame that moves with the front. Initial destabilization of the interface followed by stabilization of the flow in a diverging channel of $\alpha L=2.5$ with $\chi =0.6$ and $m=0.1$ for $\widehat{t}=0,2,7,12,22,32,52,200$ s. The field of view is $3\times 4{\mathrm{cm}}^2$.

Figure 13

Analyzing the stabilization effect for the experiment of Fig. 12. (a) Finger characteristics as time grows for $t=25,90,155,285,415,545,650,675$: number of fingers $\left(N_F,+\right)$; average finger width $(W_F,*)$; average finger length $(L_F,\times )$. We remind one that all lengths have been rendered dimensionless using ${\widehat{D}}_0$ throughout the paper. (b) Interface profiles from simulation for $t=0,200,400,...,1800,2000$, with the same parameters as in the experiment. Profiles for which the experimental flow is unstable are marked by thicker lines.

Figure 14

Simulated versus experimental results for the spike length and height, measured at $x=650$ in a converging cell with $\alpha L=-0.8$. Datapoints for $m=1$ are marked by superposed circles and for $m=10$ without. Note that, due to the wedgelike nature of the interface profiles [e.g., evident in Fig. 9], comparisons of front heights were not performed for the converging geometry.

Figure 15

Simulated versus experimental results for the front height as well as the spike length and height, measured at $x=1150$ in a diverging cell with $\alpha L=0.8$. Datapoints for $m=1$ are marked by superposed circles and for $m=10$ without.