Saffman-Taylor Instability and the Inner Splitting Mechanism

For viscously unstable, miscible Hele-Shaw displacements, we investigate the origin of the streamwise vorticity shown to be responsible for the inner splitting mechanism by Oliveira and Meiburg [J. Fluid Mech. 687 431 (2011)]. Towards this end, we compare 3D Navier-Stokes simulation results for neutrally buoyant, viscously unstable displacements and gravitationally unstable, constant viscosity ones. Only the former exhibit the generation of streamwise vorticity. The simulation results show that it is caused by the lateral displacement of the more viscous fluid by the less viscous one, with the variable viscosity terms playing a dominant role.

Figure 1

(a) Constant viscosity, upward vertical displacement of denser fluid by a lighter fluid, for $(M,\mathrm{Re},\mathrm{Pe},F_x)=(0,1,1000,20)$. Gravity points in the $-x$ direction, and the $c=0.5$ concentration isosurface is plotted at time $t=20$. (b),(c) Neutrally buoyant displacement of a more viscous fluid by a less viscous one, for $(M,\mathrm{Re},\mathrm{Pe},F_x)=(3,1,1000,0)$. The $c=0.5$ and the $c=0.26$ isosurfaces are displayed at $t=20$. Spanwise interfacial waviness and inner splitting is observed for the viscous finger, but absent for the density finger.

Figure 2

Cross sections of the density finger of Fig. 1 as it advances through $x=30$. Darker gray contours indicate the lighter, injected fluid. Red (blue) contours correspond to positive (negative) streamwise vorticity. The times shown are (a) $t=18.2$, (b) $t=20$, and (c) $t=22$. In this constant viscosity displacement, streamwise vorticity appears only along the solid walls. Panel (b) also shows $(v,w)$-velocity vectors, which indicate that the denser resident fluid is displaced laterally by the lighter injected fluid.

Figure 3

Cross section of the neutrally buoyant viscous finger of Figs. 1 and 1 passing through $x=37.3$. Darker gray contours correspond to less viscous fluid. The same vorticity contour levels as in Fig. 2 are shown. The displayed times are (a) $t=19.6$, (b) $t=20.8$, and (c) $t=38.4$. Streamwise vorticity is seen to form at the solid walls, and also along the interface where the less viscous injected fluid displaces the more viscous resident fluid laterally within the $y$, $z$ plane.

Figure 4

The same cross sections as in Fig. 3. The contours represent the sum of all the terms involving viscosity derivatives in Eq. (5). They indicate that these terms are primarily responsible for the generation of the streamwise vorticity.

Figure 5

Same cross section as in Figs. 3 and 4, with the contours representing the term $-{\mu }_{zz}{v}_z$ in Eq. (5). The contour levels are the same as in Fig. 4, which demonstrates the dominance of this term in the generation of the streamwise vorticity.

Figure 6

(a) Perspective view of the $c=0.5$ isosurface showing the interaction of many fingers after small random initial perturbations are introduced at $t=2$ in a spanwise domain of size 20. The governing parameters are the same as before, i.e., $(M,\mathrm{Re},\mathrm{Pe},F_x)=(3,1,1000,0)$ and time is $t=16.6$. (b),(c) Cross sections at $x=29$ show the concentration field in gray levels and positive (negative) streamwise voriticy in red (blue). Nonsymmetrical streamwise vorticity quadrupoles are responsible for the interfacial waviness in the spanwise direction. During finger merging, the regular quadrupole structure is disturbed, as demonstrated near $y=2$.