Edge states control droplet breakup in subcritical extensional flows

A fluid droplet suspended in an extensional flow of moderate intensity may break into pieces, depending on the amplitude of the initial droplet deformation. In subcritical uniaxial extensional flow the nonbreaking base state is linearly stable, implying that only a finite-amplitude perturbation can trigger breakup. Consequently, the stable base solution is surrounded by its finite basin of attraction. The basin boundary, which separates initial droplet shapes returning to the nonbreaking base state from those becoming unstable and breaking up, is characterized using edge tracking techniques. We numerically construct the edge state, a dynamically unstable equilibrium whose stable manifold forms the basin boundary. The edge state equilibrium controls if the droplet breaks and selects a unique path towards breakup. This path physically corresponds to the well-known end-pinching mechanism. Our results thereby rationalize the dynamics observed experimentally [H. A. Stone and L. G. Leal, J. Fluid Mech. 206, 223 (1989)].

Figure 1

Sketch of the axisymmetric droplet, its parametrization, and the coordinate system. The arrows indicate the direction of the extensional flow.

Figure 2

Edge tracking orbits for $\mathrm{Ca}=0.1$ and $\lambda =1$. (a) Droplet elongation versus time. The orbits reach the stable base state or break up through the end-pinching mechanism. Triangles denote the initial shapes and dots subsequent snapshots. (b) Corresponding normal velocity residuals measured in the $L_{\infty }$-norm versus time. The growth and decay rates are given by the most unstable and least stable eigenvalues of the edge state and the base state ${\sigma }_1^E,{\sigma }_2^E$, and ${\sigma }_1^B$, respectively.

Figure 3

Pressure field and velocity streamlines for (a) the base state and (b) the edge state for $\mathrm{Ca}=0.1$ and $\lambda =1$. The change in pressure along the $z$ axis is shown when (c) the base state is perturbed with its least stable eigenmode and (d) the edge state is perturbed with its most unstable eigenmode. The insets show the shape of the base state and the edge state (solid line) and their shapes after the modes are superimposed (dashed line). The interface motion is depicted by the arrows.

Figure 4

Edge tracking orbits in state space $(f_2,f_4)$ for the cases shown in Fig. 2, where $f_2$ and $f_4$ are the droplet radius projections onto the second and fourth Legendre polynomials. The star and the square indicate the locations of the edge state and stable base state, respectively.

Figure 5

Elongation of the droplet versus the capillary number for $\lambda =1$. The solid and dashed lines indicate the stable (base state) and unstable (edge state) branches, respectively. Stable and unstable droplet shapes are plotted for $\mathrm{Ca}=0.05$ and $\mathrm{Ca}=0.1$ (solid line), superimposing the most unstable mode (dashed line). The eigenvalue spectrum is also reported. Circles indicate eigenvalues corresponding to symmetric modes and crosses to asymmetric modes. The translational symmetry is canceled by constraining the droplet center of mass in the origin.

Figure 6

Step change from a supercritical flow ${\mathrm{Ca}}_{\text{super}}=0.125$ (solid line) to a subcritical flow ${\mathrm{Ca}}_{\text{sub}}={\mathrm{Ca}}_{\text{super}}/2$ (dashed lines) for $\lambda =1$. (a) Elongation of the droplet versus time and corresponding shapes. The horizontal lines indicate the elongation of the unstable equilibrium (edge state) $L_u$ and stable equilibrium (base state) $L_s$ for ${\mathrm{Ca}}_{\text{sub}}={\mathrm{Ca}}_{\text{super}}/2$. (b) State-space orbits. The edge state selects the path towards the base state or end pinching. The edge state and the base state are plotted and indicated by the star and the square, respectively.

Figure 7

Breakup of a droplet whose initial elongation $L_0$ is smaller than the edge state elongation $L_u$ for $\mathrm{Ca}=0.0625$ and $\lambda =1$. (a) Droplet elongation versus time (solid line) and edge state elongation (dotted line). The inset shows the initial shape at $t=0$ and the edge state. (b) State-space orbit plotted together with the orbits in Fig. 6. The inset shows snapshots corresponding to the markers.

Figure 8

Droplet elongation $L$ versus capillary number $\mathrm{Ca}$. Each line corresponds to a different viscosity ratio $\lambda$; crosses indicate the critical capillary number. The inset shows a log-log plot of ${\mathrm{Ca}}_{\text{crit}}$ versus $\lambda \in [0.02,1]$ (dashed line) compared with the theoretical prediction of [28] (solid line).