Phase-field modeling of an immiscible liquid-liquid displacement in a capillary

We develop a numerical model for a two-phase flow of a pair of immiscible liquids within a capillary tube. We assume that a capillary is initially saturated with one liquid and the other liquid is injected via one of the capillary's ends. The governing equations are solved in the velocity-pressure formulation, so the pressure levels are imposed at the capillary inlet and outlet ends. We model the structure of the flow and the shape of the interface. We are able to reproduce the flow for a wide range of capillary numbers, when the meniscus preserves its shape moving together with the flow, and when the meniscus constantly stretches resembling the transport of a passive impurity. We demonstrate that the phase-field approach is capable of reproducing all features of the liquid-liquid displacement, including the motion of a contact line, the dynamic changes of the capillary pressure, and the dynamic changes of the apparent contact angle.

Figure 1

(a) The velocity of the tip of the liquid-liquid meniscus and (b) the length of the meniscus vs time. The data are obtained for $\text{Pe}={10}^4$, $\text{Cn}=4\times {10}^{-4}$, $\text{Re}=1$, $M={10}^{-4}$ using the numerical grids with the grid size of $1/250$ (solid lines), $1/300$ (dotted lines), $1/350$ (dashed lines), and $1/400$ (dash-dotted lines).

Figure 2

(a) The velocity of the tip of the liquid-liquid meniscus and (b) the length of the meniscus vs time. The data are obtained for $\text{Pe}={10}^4$, $\text{Re}=1$, $M={10}^{-2}$ for different Cahn numbers (different interface thicknesses, as $\delta =\sqrt{2\text{Cn}}$) $\text{Cn}=5\times {10}^{-5}$ (long-dashed lines), $\text{Cn}=2\times {10}^{-4}$ (solid lines), $\text{Cn}=4\times {10}^{-4}$ (dotted lines), and $\text{Cn}=8\times {10}^{-4}$ (dashed lines).

Figure 3

The typical fields of concentration for different time moments. The data are obtained for $\text{Pe}={10}^4$, $\text{Cn}=4\times {10}^{-4}$, $\text{Re}=1$, and two different Mach numbers, $M={10}^{-2}$ (a)–(c) and $M={10}^{-4}$ (d)–(f).

Figure 4

The typical fields of pressure for different time moments. All parameters as in Fig. 3.

Figure 5

(a) The fields of pressure (isolines with the density fields); (b) concentration (isolines) and velocity (vectors); (c) chemical potential; and (d) concentration and deviation of the velocity from the Poiseuille profile. The fields are shown for the time moment $t=3$, and parameters $\text{Pe}={10}^4$, $\text{Re}=1$, $\text{Cn}=4\times {10}^{-4}$.

Figure 6

The velocity of the meniscus's tip for the numerical runs with $M={10}^{-1}$ (a), $M={10}^{-2}$ (b), $M={10}^{-3}$ (c), and $M={10}^{-4}$ (d). The other parameters are $\text{Pe}={10}^4$, $\text{Cn}=4\times {10}^{-4}$, and different Reynolds numbers, $\text{Re}=0.25$ (long-dashed lines), $\text{Re}=0.5$ (solid lines), $\text{Re}=1$ (dotted lines), and $\text{Re}=2$ (dashed lines).

Figure 7

The time changes of the meniscus's length. The parameters are the same as in Fig. 6.

Figure 8

The profiles of the $x$ component of the velocity across the tube at the points of the meniscus's tip (solid lines) and near the inlet or outlet (dashed lines). The data are obtained for for $t=3$, $\text{Pe}={10}^4$, $\text{Re}=1$, $\text{Cn}=4\times {10}^{-4}$, and two different Mach numbers, $M={10}^{-2}$ (a) and $M={10}^{-4}$ (b).

Figure 9

The profiles of the pressure along the centerline of the tube. The parameters are the same as in Fig. 8.

Figure 10

The pressure profiles along the centerline and along the wall for $M=0.01$, $\text{Re}=1$, $\text{Pe}={10}^4$, and three different Cahn numbers, $\text{Cn}=8\times {10}^4$ (dashed line), $\text{Cn}=2\times {10}^4$ (solid line), and $\text{Cn}=5\times {10}^5$ (dotted line). The profiles are shown for $t=3$.

Figure 11

The time evolution of the surface tension coefficient. The parameters are the same as in Fig. 6.

Figure 12

The time evolution of the dynamic contact angle. The parameters are the same as in Fig. 6. In (d), only two curves are shown for $\text{Re}=0.25$ and $\text{Re}=0.5$, as the adopted algorithm for calculations of the contact angle generates too noisy results for higher Reynolds numbers.

Figure 13

The mass fraction (20) vs capillary number. The results are shown for $\text{Pe}={10}^4$, $\text{Cn}=4\times {10}^{-4}$, and different Reynolds numbers. Circles correspond to the fingering displacement, and crosses depict the pistonlike displacements.

Figure 14

The capillary pressure as calculated from Eq. (23) vs time. The curves are plotted for $M={10}^{-2}$ and other parameters as in Fig. 6.

Figure 15

The position of a meniscus on a wall vs time. The curves are plotted for $M={10}^{-2}$, $\text{Re}=1$, $\text{Cn}=4\times {10}^{-4}$ and three different Peclet numbers, $\text{Pe}={10}^3$ (dashed line), $\text{Pe}={10}^4$ (solid line), and $\text{Pe}={10}^5$ (dotted line).