Multiphase CFD modeling of front propagation in a Hele-Shaw cell featuring a localized constriction

We study a liquid-gas front propagation in a modulated Hele-Shaw cell by means of multiphase computational fluid mechanics based on the three-dimensional Navier-Stokes equations. In the simulations an obstacle that partially fills the gap is placed at the center of the cell, and the liquid-gas interface is driven at a constant velocity. We study the morphological differences between imbibition and drainage for a wide range of capillary numbers, and explore how the wetting properties of the constriction affect the amount of liquid that remains trapped in the draining process. We observe increasing remaining volumes with increasing capillary number and decreasing contact angle. The present CFD implementation for a single mesa defect provides insight into a wide number of practical applications.

Figure 1

Sketch of the problem: a modulated Hele-Shaw cell with a localized obstacle in the middle (constriction, black square). Two stages are shown: (a) first, the cell is filled with a viscous fluid (blue) at constant displacement velocity $V_{\infty }$. (b) Once the obstacle is completely covered with the imbibed fluid, the flow direction is reversed at the same magnitude $-V_{\infty }$. (c) Side view: cell of gap thickness $L_z$ modulated by an obstacle of height $l_z$. The imbibed liquid forms a contact angle ${\theta }_t$ and ${\theta }_b$ with the top and bottom plates, respectively.

Figure 2

Snapshots of the simulation for oil 1 (red), ${\theta }_h={\theta }_o=0^{\circ }$, $\beta =0^{\circ }$ and (a) $V_{\infty }=0.2$ mm/s (Ca $=4.83\times {10}^{-4}$) and (b) $V_{\infty }=0.6$ mm/s (Ca $=1.449\times {10}^{-3}$). In this complete-wetting case, the instabilities at the front appear more easily as the front velocity increases.

Figure 3

Volume of liquid that remains on top of the obstacle $\mathcal{V}$ as function of $\beta$ for oil 1 and $V_{\infty }=0.2$ mm/s (Ca $=4.83\times {10}^{-4}$). As the tilt angle increases, the volume decreases. The interface remains flat after passing the obstacle when $\beta$ is above the critical angle ${\beta }_c$. Thumbnail pictures show snapshots of the simulation at two times during the drainage stage for $\beta =6^{\circ }(<{\beta }_c)$ and $\beta ={60}^{\circ }(>{\beta }_c)$.

Figure 4

Snapshots during the drainage stage for oil 1-air and $V_{\infty }=0.2$ mm/s (Ca $=4.83\times {10}^{-4}$). Here a Hele-Shaw cell under complete wetting conditions on the walls ${\theta }_h=0^{\circ }$ is used, but the contact angle imposed at the obstacle wall is (a) ${\theta }_o={10}^{\circ }$ and (b) ${\theta }_o={30}^{\circ }$.

Figure 5

Snapshots of the simulation for water-air (Ca $=2.8\times {10}^{-6}$), and ${\theta }_o={45}^{\circ }$ (a) and ${\theta }_o={60}^{\circ }$ (b). The obstacle is represented by the gray square. In the drainage stage a small volume of water remains on top of the obstacle for ${\theta }_o={45}^{\circ }$.

Figure 6

Snapshots of the simulation for (a) ether (Ca $=3.59\times {10}^{-6}$) and (b) ethanol (Ca $=1.08\times {10}^{-5}$) for ${\theta }_o={45}^{\circ }$. The volume of liquid that remains over the obstacle is (a) $2.84{\mathrm{mm}}^3$ and (b) $4.06{\mathrm{mm}}^3$.

Figure 7

Snapshots of the simulation for oil 1 (Ca $=4.83\times {10}^{-4}$) and (a) ${\theta }_o={45}^{\circ }$ and (b) ${\theta }_o={60}^{\circ }$. During drainage, the interface breaks up when it reaches the inlet boundary ($t=360\mathrm{s}$). The volume of liquid that remains over the obstacle is $1.78{\mathrm{mm}}^3$ in case (a).

Figure 8

Variation of the dimensionless volume $\mathcal{V}/\left(l_x{x}_n^2\right)$ with the contact angle ${\theta }_o$ using the modified capillary number defined by Eq. (12).

Figure 9

Velocity field for (a) ethanol and (b) oil during the drainage phase for several times. The velocity field at locations occupied by air is omitted from the plots for clarity. Here $V_{\infty }=0.2$ mm/s and ${\theta }_h={45}^{\circ }$. Once the interface reaches the obstacle, a recirculation zone is created.

Figure 10

Component of vorticity $\langle {\omega }_z\rangle$ for several cases. As the capillary number increases, the vorticity increases. The time $t_0$ corresponds to the initial time of the drainage.