Universal self-similar van der Waals breakup of ultrathin liquid films

A. Martínez-Calvo A. Sevilla Área de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain. [email protected]

I Supplemental Material

In this Supplemental Material we show a direct comparison of the exponential thinning regime for two different $Bq$ and $\Theta=1$ , obtained with the complete Stokes and Boussinesq-Scriven equations, and the one-dimensional model derived in MartinezSevilla2018, and the numerical integration of the same model by Wee2020.

Refer to caption — Figure 1: Radius of the filament as a function of time obtained from our numerical simulations of the 1D model and the one performed by Wee2020, with the same values of parameters reported therein: $k=0.7$ , $\epsilon=0.4$ , $\mbox{{Bq}}=10^{-3}$ and $\Theta=1$ . The inset shows a direct comparison between the 1D model and the complete Stokes and Boussinesq-Scriven equations but for $\mbox{{Bq}}=3.4$ , fitted with the slope given by the function $F(\Theta)$ .

I.1 Linear stability and the limit of highly viscous interface

I.1.1 Passive ambient: modified Rayleigh-Chandraskhar dispersion relation

In the Stokes limit reads

\omega=\frac{(1-k^{2})[\mbox{{Bq}}(\Theta+1)(F(k)(F(k)-2)-k^{2})-2]}{4[1-F(k)^{2}+k^{2}+\mbox{{Bq}}^{2}\Theta(F(k)(F(k)-2)-k^{2})]-2\mbox{{Bq}}[1+4F(k)(F(k)-1)k^{2}(\Theta-3)\Theta]}

(1)

When surface viscous stresses dominate, $\mbox{{Bq}}\gg 1$ , the dispersion relation becomes

\omega=\frac{(1-k^{2})(1+\Theta)}{4\Theta}

(2)

Universal self-similar van der Waals breakup of ultrathin liquid films

I Supplemental Material

I.1 Linear stability and the limit of highly viscous interface

I.1.1 Passive ambient: modified Rayleigh-Chandraskhar dispersion relation

I.1.2 Liquid-liquid configuration: modified Tomotika’s dispersion relation

I.2 Comparison with the leading-order one-dimensional model