Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analog of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterized by a blowup of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent $\alpha =1/3$ just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.

Figure 1

Illustration of the four geometries considered in this article. The blue represents the region filled with a viscous fluid and the white region denotes an inviscid bubble. (a) Radial geometry, with the bubble contracting due to either the inviscid fluid extracted at a point or the viscous fluid injected at $r=\infty$. (b) Cylindrical tube, with the interface propagating from left to right due to the inviscid fluid being injected at $z=-\infty$. (c) Cylindrical tube, with a finite bubble translating in the $+z$ direction due to the viscous fluid being extracted at $z=\infty$. (d) Radial geometry, with the bubble expanding due to either the inviscid fluid injected at a point or the viscous fluid extracted at $r=\infty$.

Figure 2

Time evolution of a dripping ${}^3\mathrm{He}$ crystal surrounded by liquid ${}^4\mathrm{He}$ (from [46], reproduced with permission from the American Physical Society). The ${}^3\mathrm{He}$ crystal is grown in the ${}^4\mathrm{He}$ bath, and when it becomes sufficiently large, it is dragged down due to gravity and undergoes pinch-off. The width of each frame is 3.5 mm. Analysis of the video footage both just before and after pinch-off occurs indicates a scaling exponent of 1/3.

Figure 3

Numerical solution of (8a)–(8d) found using the numerical scheme described in Sec. 3a with (a) a symmetric and (b) an asymmetric dumbbell as the initial shape of the interface. For the symmetric dumbbell, the bubble contracts and pinches off into two satellite bubbles of equal volume that contract to a point at the same time. For the asymmetric dumbbell, when the neck of the bubble pinches off, two satellite bubbles of difference size form. The smaller of these two bubbles contracts to a point first, followed by the second. Simulations are performed on the domain $0\le \theta \le \pi$ and $0\le r\le 1.5$ with $628\times 300$ equally spaced nodes.

Figure 4

Rate of change of volume $\dot{V}\left(t\right)$ of the numerical solution to (8a)–(8d), shown in Fig. 3 with (a) a symmetric and (b) an asymmetric dumbbell. The solid lines denote the rate of change of volume of the numerical solution and the dotted (black) line denotes the exact rate of change $\dot{V}=-4\pi$. The dots denote the points in time when pinch-off occurs.

Figure 5

Comparison of numerical solution to (22) and (23) (solid blue line) with the numerical solution to (25) (dashed red line).

Figure 6

(a) Minimum neck radius computed from the numerical solution to (8a)–(8d) as a function of time to pinch off with, from left to right, $\sigma =1$ (blue line), 0.5 (red line), and 0.1 (yellow line). (b) Corresponding log-log plot, where dashed curves are lines of best fit, demonstrating that our numerical scheme supports the assertion that the similarity exponent $\alpha =1/3$.

Figure 7

Time evolution of the numerical solution to (5a)–(5d) up to when pinch-off occurs for (a) a symmetric and (b) an asymmetric initial condition. The dots (in red) denote the locations of the minimum neck radius. The insets compare the interfacial profile rescaled according to (13) and (14) with $\alpha =\beta =1/3$ (solid blue line) with the numerical solution to (22) and (23) (dashed red line).

Figure 8

(a) Distance between the point at which pinch-off occurs and the tip of the right (blue line) and left (red line) satellite bubbles. (b) Corresponding log-log plot, where the dashed black lines are lines of best fit, providing evidence that our numerical solutions post pinch-off scale like (26).

Figure 9

Numerical solution to (8a)–(8d) scaled according to (26) for (a) a symmetric and (b) an asymmetric dumbbell initial condition. These solutions illustrate how the full numerical solution post pinch-off approaches a similarity solution for $\left|f\right|\ll 1$.

Figure 10

Evolution of a bubble in a cylindrical tube. The bubble initially has a flat interface with a small sinusoidal perturbation. As the interface grows, it becomes unstable and develops a long finger analogous to the two-dimensional Saffman-Taylor finger which forms in Hele-Shaw cells. As time increases, both the speed and shape of this tip tend towards a constant.

Figure 11

Finger width $\lambda$ as a function of the surface tension parameter $\sigma$ of an axially symmetric analog of the Saffman-Taylor finger. The finger width is determined by evolving a near-flat interface with a small sinusoidal perturbation until both the shape and speed of the finger are constant.

Figure 12

Numerical simulation of an expanding bubble in a cylindrical tube computed using the numerical scheme described in Sec. 3a. As the bubble expands, the neck of the interface is small enough that the curvature in the cylindrical direction is sufficiently large to pull the interface towards a pinch-off singularity.

Figure 13

(a) A log-log plot of the minimum neck radius of the bubble in cylindrical tube geometry, shown in Fig. 12. The dashed curve is a line of best fit with gradient $\alpha$. The inset shows a comparison of the solution to (22) and (23) (dashed red line) with full numerical solution scaled according to (13) (solid blue line). (b) A log-log plot of the distance between the point where pinch-off occurs and the receding tip of the bubble, compared with a line of slope $\beta =1/3$. The inset shows the numerical solution scaled according to (13) for times when $t-t_0\ll 1$.

Figure 14

Speed of an axially symmetric Taylor-Saffman-like bubble $U$ propagating through a cylindrical tube as a function of the surface tension parameter $\sigma$. The initial condition is a spheroid with semiaxis given by $\rho =0.4$ and $z=0.2$ ($▴$ blue), 0.4 ($▾$ red), 0.8 ($◂$ yellow), and 1 ($▸$ purple). The bubble is evolved until both its shape and speed appear constant, at which time $U$ is estimated.

Figure 15

Numerical simulation of bubble traveling through cylindrical tube with (a) $\sigma =5\times {10}^{-4}$, (b) $\sigma =3\times {10}^{-3}$, (c) $\sigma =7\times {10}^{-3}$, and (d) $\sigma =1.5\times {10}^{-2}$. The initial condition of these simulations is a prolate spheroid of major and minor axes 1 and 0.4. The solutions are shown at a time sufficiently large such that both the speed and shape of the bubble are constant. The $\widehat{z}$ coordinate is $z$ scaled such that the bubble is centered at the origin.

Figure 16

Numerical solution to (5a)–(5d) for an expanding bubble with $\sigma =2.5\times {10}^{-3}$. The interface of the bubble is unstable as it expands and a type of axially symmetric viscous fingering develops. When a single finger is aligned with the $z$ axis, it can pinch off due to the curvature acting in the cylindrical direction.