Stability of similarity solutions of viscous thread pinch-off

In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. The stability of the first two solutions (those with the largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of breakup producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime.

Figure 1

Schematic of a thin thread of viscous fluid. The system is modeled with two variables—the streamwise velocity $v(z,t)$ (uniform across the thread in the slender body approximation) and the thread radius $h(z,t)$—both functions of the streamwise coordinate $z$ and time $t$.

Figure 2

(a, c) The first (stable) similarity profile and (b, d) the second (unstable) similarity profile; solutions to (7) found by the numerical continuation procedure described in Sec. 3a. In (c) and (d), the scaled radius $\varphi$ is given by $e^{\omega }$.

Figure 3

Selection of eigenvalues $\sigma$ on the real line: (a) the first solution and (b) the second solution. We see the three trivial eigenvalues at ${\sigma }_T=1, {\sigma }_X=1/2$, and ${\sigma }_V=-1/2$, as well as the singular values ${\sigma }^{*}$, (18), where a coefficient in the power series expansion for the perturbation, (16), becomes singular and, therefore, does not correspond to an analytic solution.

Figure 4

Curves defined by the real part of the error parameter $\mathrm{Re}\left(\delta \tilde{c}\right)=\delta {\tilde{c}}_R=0$ and eigenvalues (where both real and imaginary parts of $\delta \tilde{c}$ vanish) for (a) the first solution at $\sigma =-0.670+2.246\mathrm{i}$ and (b) the second solution at $\sigma =-0.552+3.015\mathrm{i}$ and $\sigma =3.695+4.905\mathrm{i}$. In both cases, the eigenvalues are found by continuing $\sigma$ from real values in the positive imaginary direction, until the real part of the error $\delta {\tilde{c}}_{0R}$ is 0, then continuing into the complex plane (holding $\delta {\tilde{c}}_{0R}$ at 0) to find the curves defined by this equation. At points on these curves where the error $\delta {\tilde{c}}_{0I}$ is also 0, the corresponding value of $\sigma$ is an eigenvalue.

Figure 5

The complex eigenfunction for each of (a, c) the stable first similarity solution at the complex eigenvalue $\sigma =-0.670+2.246\mathrm{i}$ and (b, d) the unstable second similarity solution at $\sigma =3.695+4.905\mathrm{i}$. (a) and (c) plot the perturbations $\tilde{\psi }$ to the self-similar velocity profiles $\psi$, while (b) and (d) plot the perturbations $\tilde{w}$ to the self-similar profile of the log-radius $w$. The unstable eigenfunctions on the second solution are localized in the region near 0, which includes both the singular point and the region in which the base solution [depicted in Figs. 2 and 2] has large gradients.

Figure 6

Convergence of a generic initial condition to the stable similarity solution: (a) By plotting $h_{\min }$ against its time derivative and comparing it to (22) the decay rate and frequency predicted by the dominant nontrivial eigenvalue $\sigma \approx -0.670+2.246\mathrm{i}$ can be clearly observed (the values ${\mathcal{A}}_1=0.25$ and ${\mathcal{A}}_2=2$ are fitted). (b) The profiles asymptote to the stable similarity solution as time tends to the pinch-off time (the profiles are shifted by the point ${\xi }_{\min }$ where $\varphi$ is at its minimum, so the shifted profiles all have their minimum at 0).

Figure 7

Departure of the numerical solution to (1) from an initial condition equal to the second similarity solution. (a) The plot of $h_{\min }$ vs its derivative shows the solution leaving the neighborhood of the unstable solution and settling to the stable solution (where $d{h}_{\min }/dt\approx -0.0304$). Inset: A short-lived, but reasonable, match to the linear stability approximation, (22). (b) The profiles show that the oscillations grow first near the minimum in $\varphi$.

Figure 8

Convergence of the system at the unstable complex eigenvalue $\sigma =3.695+4.905\mathrm{i}$. As $\delta \xi$ decreases, the solution is observed to converge, until $\delta \xi$ tends to below 0.1, where the numerical round-off error in the high-order power series starts to have an effect. Nevertheless, the eigenvalue has been accurately determined to at least three decimal places.