Relevance of dynamic wetting in viscous fingering patterns

We demonstrate that wetting effects at moving contact lines have a strong impact in viscous fingering patterns. Experiments in a rotating Hele-Shaw (HS) cell, dry or prewetted, show consistent morphological differences. When the wetting fluid invades a dry region, contact angle dynamics yield a kinetic contribution to the interface pressure drop that scales with capillary number as ${\mathrm{Ca}}^{2∕3}$ but is significantly larger than the Park-Homsy kinetic correction. Numerical results are in very good agreement with experiments and show that standard HS equations work best for prewetted cells.

Figure 1

Time sequence of two experiments performed in a dry cell (left) and in a prewetted cell (right) for the same set of parameters. The time lapse between consecutive snapshots is $9\mathrm{s}$. The parameters are $b=0.5\mathrm{mm}$ and $\Omega =60\mathrm{rev}∕\min$. The initial radii, $R_0=7.6\mathrm{cm}$ (left) and $7.9\mathrm{cm}$ (right), are the same within experimental error because the initial drop is not perfectly circular, so that $S\simeq 7\times {10}^4$. Maximal values of Ca (for the fastest fingertips in the last snapshot) are $3.3\times {10}^{-3}$ (left) and $4.2\times {10}^{-3}$ (right).

Figure 2

Time sequence of fingering patterns formed in a wet cell ($t=9$, 13.5, 18, and $22.5\mathrm{s}$). The oil is Rhodorsil 47V 50, and $b=0.5\mathrm{mm}$, $\Omega =120\mathrm{rev}∕\min$, $R_0=5\mathrm{cm}$ $(S=2.3\times {10}^5)$. The maximal Ca is $1.6\times {10}^{-2}$.

Figure 3

Time sequence of fingering patterns formed in a dry cell ($t=8.4$, 10.4, 14.4, and $18.4\mathrm{s}$). The oil is Rhodorsil 47V 50, and $b=0.5\mathrm{mm}$, $\Omega =150\mathrm{rev}∕\min$, $R_0=6\mathrm{cm}$ $(S=6.6\times {10}^5)$. The maximal Ca is $1.7\times {10}^{-2}$.

Figure 4

Dimensionless maximum radial coordinate of outgoing fingers $r_{+}∕R_0$ vs dimensionless time $t∕\tau$ in log-linear scale, for wet (circles and diamonds) and dry (squares and triangles) conditions. The two solid lines are the numerical results for the corresponding cases. See text for details. The range of maximal Ca is $(1.0\times {10}^{-3})\text{–}(2.5\times {10}^{-2})$.

Figure 5

Patterns from numerical integration, using the liquid properties of Rhodorsil 47V 50, and (a) $b=0.4\mathrm{mm}$, $R_0=8.0\mathrm{cm}$, $\Omega =40\mathrm{rev}∕\min$ $(S=2\times {10}^4)$, $t=210\mathrm{s}$; (b) $b=0.4\mathrm{mm}$, $R_0=10.0\mathrm{cm}$, $\Omega =60\mathrm{rev}∕\min$ $(S=1.2\times {10}^5)$, $t=140\mathrm{s}$. ${\rm Y}$ given by Eq. (6) with $l=100$ has been used on the left panels, and the standard boundary condition on the right ones. The maximal Ca are $1.1\times {10}^{-3}$ [wet (a)], $8\times {10}^{-4}$ [dry (a)], $2.0\times {10}^{-3}$ [wet (b)], $1.2\times {10}^{-3}$ [dry (b)].