Saffman-Taylor fingers with kinetic undercooling

The mathematical model of a steadily propagating Saffman-Taylor finger in a Hele-Shaw channel has applications to two-dimensional interacting streamer discharges which are aligned in a periodic array. In the streamer context, the relevant regularization on the interface is not provided by surface tension but instead has been postulated to involve a mechanism equivalent to kinetic undercooling, which acts to penalize high velocities and prevent blow-up of the unregularized solution. Previous asymptotic results for the Hele-Shaw finger problem with kinetic undercooling suggest that for a given value of the kinetic undercooling parameter, there is a discrete set of possible finger shapes, each analytic at the nose and occupying a different fraction of the channel width. In the limit in which the kinetic undercooling parameter vanishes, the fraction for each family approaches $1/2$, suggesting that this “selection” of $1/2$ by kinetic undercooling is qualitatively similar to the well-known analog with surface tension. We treat the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analog with surface tension, since kinetic undercooling permits finger shapes which are corner-free but not analytic. We provide numerical evidence for the selection mechanism by setting up a problem with both kinetic undercooling and surface tension and numerically taking the limit that the surface tension vanishes.

Figure 1

Numerical results for kinetic undercooling $\epsilon =0.1$. (a) The shape of the fingers for the surface tension values $\gamma =0.03,0.5,1$ (from the innermost to the outermost curve). (b) The dependence of the finger width $\lambda$ on surface tension $\gamma$. Numerically computed data points are indicated by the solid (blue) circles. The open (red) circle is the estimated finger width for this family at $\gamma =0$.

Figure 2

Dependence of finger width $\lambda$ on surface tension $\gamma$ for fixed values of kinetic undercooling. In each case, three solution branches are shown. The open (red) circles represent an extrapolated value for $\gamma =0$.

Figure 3

The condition number of the Jacobian for solutions computed with $N=1000$ nodes plotted against $\gamma$ for the first branch (circles), second branch (diamonds), and third branch (squares) for fixed values of $\epsilon$. The (blue) dashed line is a rough linear fit through the data on the first branch for small $\epsilon$.

Figure 4

This figure constitutes our main result. (a) The selection of two distinct branches shown as the solid (red) circles. Note the distinction between these and the continuous solution space found by Dallaston and McCue [41], bounded below by ${\lambda }_{\text{min}}\left(\epsilon \right)$, the dashed (black) curve. The primary branch seems to asymptote to the curve ${\lambda }_{\text{min}}\left(\varepsilon \right)$ as $\epsilon \to \infty$. (b) The entire primary branch from part (a). Recall that as $\epsilon \to \infty ,c\to 1^{-}$ and $\lambda \to 1^{-}$. The portion of the branch shown in part (a) is boxed in the lower left corner for reference. The inset shows a comparison with Eq. (18) provided by Chapman and King [16], shown as the smooth (blue) curve.

Figure 5

The shape of the finger for solutions on the primary branch with $\gamma =0.03$ and $\epsilon =0,1,10,3500$ (from the innermost to the outermost curve). For comparison, the dashed (red) curve shows a semicircle of unit radius.

Figure 6

The structure of a typical Jacobian for $N=100$. The plot shows ${log}_{10}\left|\mathbf{J}\right|$, for $\gamma =0.02,\epsilon =0.2$, on the first solution branch.