Spontaneous rupture of surfactant-covered thin liquid sheets

Rupture of thin free films of Newtonian fluids is analyzed when the sheets' two free surfaces are covered with insoluble surfactant and surface rheological effects are important. The analysis relies on a long-wavelength model composed of a system of one-dimensional evolution equations for film thickness $h(z,t)$, lateral velocity $v(z,t)$, and surfactant concentration $\mathrm{\Gamma }(z,t)$ ($z$: lateral coordinate, $t$: time). As the dynamics near the space-time singularity in sheet rupture is asymptotically self-similar when surfactants are convected away from the rupture point, the partial differential equations are also reduced to a set of ordinary differential equations in similarity space. For both highly viscous fluids in the Stokes limit and moderately viscous fluids, it is shown that the dominant balance involves van der Waals pressure and bulk viscous as well as surface viscous stresses while surface tension pressure and Marangoni stress are negligible. In the Stokes limit, self-similarity is of the second kind and $h\sim \mathrm{\Gamma }\sim {\tau }^{1/3}, z\sim {\tau }^{{\alpha }_z}$, and $v\sim {\tau }^{{\alpha }_z-1}$ where $\tau$ is time remaining until rupture and ${\alpha }_z$ is the lateral scaling exponent. Although ${\alpha }_z$ cannot be determined by dimensional arguments, it is shown to equal 0.249. For moderately viscous fluids, inertia also enters the picture and self-similarity is of the first kind ($h\sim \mathrm{\Gamma }\sim {\tau }^{1/3}, z\sim {\tau }^{1/2}$, and $v\sim {\tau }^{-1/2}$). Scalings determined from theory are confirmed by numerical solution of the evolution equations. Closed-form expressions for the sheet's thinning rate, which have heretofore been lacking, are also reported.

Figure 1

A free liquid film or liquid sheet: perspective view (top) and cross-sectional view depicting the problem domain (bottom). All the variables in the figure are dimensionless so that the half-thickness of the undisturbed sheet equals unity as shown. As discussed in the text, the same variables but with tildes over them are the dimensional counterparts of those shown here.

Figure 2

Simulation results for the instantaneous profiles of the free-surface shape $h(z,t)$ and surfactant concentration $\mathrm{\Gamma }(z,t)$ that depict the initial and late stage dynamics of a Stokes sheet, and the evolution in time of the function $c(z,t)=\mathrm{\Gamma }(z,t)/h(z,t)$ determined from numerical solution of the evolution equations. (a) Interface shape $h(z,t)$ and (b) surfactant concentration $\mathrm{\Gamma }(z,t)$ at the initial instant (curves marked as $t=0$) and at the instant when $h_{\text{min}}=2.84\times {10}^{-4}$ (the unmarked curves). (c) Temporal evolution of the function $c(z,t)=\mathrm{\Gamma }(z,t)/h(z,t)$ between $t=0$ and the time when $h_{\text{min}}=2.84\times {10}^{-4}$. As discussed in the text, as $t\to t_R$ or $h_{\text{min}}\to 0, c(z,t)$ asymptotically tends to a fixed profile that is invariant in time and, therefore, despite the entirely different initial conditions for $h(z,t)$ and $\mathrm{\Gamma }(z,t)$, the interface shape and surfactant concentration profiles become virtually identical in form in the vicinity of the singularity as the sheet approaches rupture. Here, the reference Boussinesq-Scriven number is $B_r=1$ (or equivalently the normalized reference Boussinesq-Scriven number $B_0\equiv B_r/{\mathrm{\Gamma }}_0=2$).

Figure 3

Scaling behavior of key variables during rupture of liquid sheets undergoing Stokes flow: evolution in time remaining to rupture $\tau$ of the minimum sheet half-thickness $h_{\text{min}}$ and surfactant concentration ${\mathrm{\Gamma }}_{\text{min}}$ at that location and lateral length $z^{\prime }$ and lateral velocity $v^{\prime }$. The data points represent simulation results and the straight lines are the theoretical predictions. (a) Computed variation with $\tau$ of $h_{\text{min}}$ (green square $\square$ symbols) and ${\mathrm{\Gamma }}_{\text{min}}{\equiv \mathrm{\Gamma }|}_{h_{\text{min}}}$ (blue gradient $▿$ symbols). In accord with theory, the simulations predict that the scaling exponents ${\alpha }_h={\alpha }_{\mathrm{\Gamma }}=1/3$ as indicated in the figure. Inset: Zoomed-in view of the variation of $h_{\text{min}}$ with $\tau$ near rupture that highlights the excellent agreement between the simulations (green data points) and the theoretical predictions—the top equation in the legend [Eq. (53)] and the black line. The bottom equation in the legend and the straight red line represent the theoretical prediction for a clean interface or when surface rheological effects are absent. (b) Computed variation with $\tau$ of $z^{\prime }$ (orange diamond $\diamond$ symbols) and $v^{\prime }$ (red circle $\circ$ symbols). In accord with theory, the simulations predict that the scaling exponents for the lateral length and lateral velocity are given by ${\alpha }_z=0.249$ and ${\alpha }_v=-0.751$ as indicated in the figure. In both this figure and the next one, the reference Boussinesq-Scriven number is $B_r=1$ (or equivalently the normalized reference Boussinesq-Scriven number $B_0\equiv B_r/{\mathrm{\Gamma }}_0=2$).

Figure 4

Evolution in time of the free-surface profile and self-similarity of the interface shape during rupture of sheets undergoing Stokes flow. The main figure shows the film profile $h(z,t)$ determined from simulations at ten instances in time and confirms that sheet rupture is symmetric as predicted from theory. Inset: appropriately rescaled interface shapes obtained from solution of the transient partial differential evolution equations in physical space (curves) agree well with the normalized similarity profile obtained from solution of the ODEs in similarity space (symbols). The individual curves obtained from the simulations are hard to distinguish from one another because of the excellence of the collapse and/or the closeness of the collapsed solutions to one another.

Figure 5

Scaling behavior of the minimum sheet half-thickness $h_{\text{min}}$, surfactant concentration ${\mathrm{\Gamma }}_{\text{min}}$ at that location, lateral length $z^{\prime }$, and lateral velocity $v^{\prime }$ during rupture of liquid sheets undergoing inertial-viscous flow or when inertia is not negligible: evolution in time remaining to rupture $\tau$ of $h_{\text{min}}, {\mathrm{\Gamma }}_{\text{min}}, z^{\prime }$, and $v^{\prime }$. The data points represent simulation results and the straight lines are the theoretical predictions. (a) Simulation results for the computed variation with $\tau$ of $h_{\text{min}}$ (green square $\square$ symbols) and ${\mathrm{\Gamma }}_{\text{min}}{\equiv \mathrm{\Gamma }|}_{h_{\text{min}}}$ (blue gradient $▿$ symbols). The simulations predict that the scaling exponents ${\alpha }_h={\alpha }_{\mathrm{\Gamma }}=1/3$, in accord with theory and as indicated in the figure. Inset: Zoomed-in view of the variation of $h_{\text{min}}$ with $\tau$ near rupture that highlights the excellent agreement between the simulations (green data points) and the theoretical predictions—the top equation in the legend [Eq. (67)] and the black line. The bottom equation in the legend and the straight red line represent the theoretical prediction [Eq. (66)] for a clean interface or when surface rheological effects are absent. (b) Computed variation with $\tau$ of $z^{\prime }$ (orange diamond $\diamond$ symbols) and $v^{\prime }$ (red circle $\circ$ symbols). The simulations predict that the scaling exponents for the lateral length and lateral velocity are given by ${\alpha }_z=1/2$ and ${\alpha }_v=-1/2$, in accord with theory and as indicated in the figure. Here, the Ohnesorge number $\text{Oh}=0.5$ and the reference Boussinesq-Scriven number is $B_r=2$ (or equivalently the normalized reference Boussinesq-Scriven number $B_0\equiv B_r/{\mathrm{\Gamma }}_0=4$).

Figure 6

Self-similarity of the interface shape during rupture of sheets undergoing inertial-viscous flow. Appropriately rescaled interface shapes obtained from solution of the transient partial differential evolution equations in physical space (curves) agree well with the normalized similarity profile obtained from solution of the ODEs in similarity space (symbols). The individual curves obtained from the simulations are hard to distinguish from one another because of the excellence of the collapse and/or the closeness of the collapsed solutions to one another. Here, the Ohnesorge number $\text{Oh}=0.5$ and the reference Boussinesq-Scriven number is $B_r=4$ (or equivalently the normalized reference Boussinesq-Scriven number $B_0\equiv B_r/{\mathrm{\Gamma }}_0=8$.)

Figure 7

Scaling behavior of the minimum sheet half-thickness $h_{\text{min}}$, surfactant concentration ${\mathrm{\Gamma }}_{\text{min}}$ at that location, lateral length $z^{\prime }$, and lateral velocity $v^{\prime }$ during rupture of liquid sheets undergoing (a) Stokes flow and (b) inertial-viscous flow at large but finite Peclet number $\text{Pe}=100$: evolution in time remaining to rupture $\tau$ of $h_{\text{min}}$ and ${\mathrm{\Gamma }}_{\text{min}}$ (main figures) and of $z^{\prime }$ and $v^{\prime }$ (insets). The data points represent simulation results and the straight lines are the theoretical predictions when $\text{Pe}=\infty$. Main figures: simulation results for the computed variation with $\tau$ of $h_{\text{min}}$ (green square $\square$ symbols) and ${\mathrm{\Gamma }}_{\text{min}}{\equiv \mathrm{\Gamma }|}_{h_{\text{min}}}$ (blue gradient $▿$ symbols). The simulations predict that the scaling exponents ${\alpha }_h={\alpha }_{\mathrm{\Gamma }}=1/3$, in accord with theoretical and simulation results when $\text{Pe}=\infty$ and as indicated in the figure. Insets: simulation results for the computed variation with $\tau$ of $z^{\prime }$ (orange diamond $\diamond$ symbols) and $v^{\prime }$ (red circle $\circ$ symbols). The simulations predict that the scaling exponents ${\alpha }_z=0.249$ and ${\alpha }_v=-0.751$ in Stokes flow and ${\alpha }_z=1/2$ and ${\alpha }_v=-1/2$ in inertial-viscous flow, also in accord with theoretical and simulation results when $\text{Pe}=\infty$ and as indicated in the figure. In panel (a), the reference Boussinesq-Scriven number is $B_r=1$ (or equivalently $B_0\equiv B_r/{\mathrm{\Gamma }}_0=2$). In panel (b), the Ohnesorge number $\text{Oh}=0.5$ and the reference Boussinesq-Scriven number is $B_r=4$ (or equivalently the normalized reference Boussinesq-Scriven number $B_0\equiv B_r/{\mathrm{\Gamma }}_0=8$).

Figure 8

Publications in which the thinning and rupture of free films of Newtonian fluids have been analyzed in situations in which the sheets' two free surfaces are covered with insoluble surfactant and surface rheological effects are important when $\text{Pe}\gg 1$ (surfactant convection is dominant over surfactant diffusion). In this figure, Matar and Choudhoury et al. denote Refs. [41] and [42]. Listed here are three key results and/or conclusions that have been obtained in these papers: (i) the power-law dependencies of key variables (in black font), i.e., the variation with time remaining to rupture $\tau$ of the sheet half-thickness $h$, surfactant concentration $\mathrm{\Gamma }$, lateral length $z$, and lateral velocity $v$, (ii) the dominant balance of forces (in red font), and (iii) the theoretically predicted thinning rate which is given by the time derivative of $h_{\text{min}}$ (in blue font). In this and the next figure, vdW, I, V, SV, and ST denote van der Waals, inertial, (bulk) viscous, surface viscous, and surface tension (capillary) forces. The Stokes limit has not been considered in Refs. [41] and [42]. Furthermore, theoretical expressions for the sheet's thinning rate are not reported in Refs. [41] and [42] when the sheet is undergoing inertial-viscous flow.

Figure 9

Summary of the dynamical responses of liquid sheets undergoing van der Waals driven thinning and rupture: phase diagram in $(B_{\text{loc}},\text{Re})$ space. In this figure, Matar, Choudhoury et al., Vaynblat et al., and Thete et al. denote Refs. [41], [42], [38], and [40]. Theoretical scaling laws exist when $\text{Re}=0$ ($\text{Oh}=\infty$), $\text{Re}\sim 1$ ($\text{Oh}\sim 1$), and $\text{Re}=\infty$ ($\text{Oh}=0$), and are now known in these limits for sheets with both clean interfaces ($B_{\text{loc}}=0$) and ones that are covered with a monolayer of insoluble surfactant in the presence of surface viscous effects. The inertial-viscous scaling regime in the limit of $B_{\text{loc}}\to \infty$, highlighted in red, constitutes a mathematically correct solution of the governing equations but may be unattainable in practice because of the aphysical nature of the relationship between surface viscosity and surfactant concentration that was assumed in those papers, i.e., ${\gamma }_s=\mathrm{constant}$ or ${\gamma }_s\ne 0$ when $\tilde{\mathrm{\Gamma }}=0$. The star in the box in the middle of the diagram (“This ${\mathrm{paper}}^{*}$”) indicates that the power-law dependencies for the regime where $(B_{\text{loc}}\sim 1,\text{Re}\sim 1)$ for which ${\gamma }_s\propto \tilde{\mathrm{\Gamma }}$ have also been obtained in Ref. [42] but the authors of that paper have incorrectly concluded that surface viscous force is negligible at the incipience of sheet rupture and have not reported an exact analytical expression for the sheet's thinning rate as has been done in this paper. Reynolds (Ohnesorge) number cannot equal zero or infinity in real systems. Thus, for systems for which $\text{Re}\ll 1$ ($\text{Oh}\gg 1$) or $\text{Re}\gg 1$ ($\text{Oh}\ll 1$), the dynamics would transition from an initial Stokes or inviscid scaling regime to a final inertial-viscous scaling regime as indicated by the arrows in the figure.