Final state of a perturbed liquid film inside a container under the effect of solid-liquid molecular forces and gravity

We investigate theoretically the possible final stationary configurations that can be reached by a laterally confined uniform liquid film inside a container. The liquid is under the action of gravity, surface tension, and the molecular interaction with the solid substrate. We study the case when the container is in an upright position as well as when it is turned upside down. The governing parameters of the problem are the initial thickness of the film, the size of the recipient that contains the liquid, and a dimensionless number that quantifies the relative strength of gravity with respect to the molecular interaction. The uniform film is always a possible final state and depending on the value of the parameters, up to three different additional final states may exist, each one consisting in a droplet surrounded by a thin film. We derive analytical expressions for the energy of these possible final configurations and from these we analyze which state is indeed reached. A uniform thin film may show three different behaviors after a perturbation: The system recovers its initial shape after any perturbation, the fluid evolves towards a drop (if more than one is possible, it tends toward that with the thinnest precursor film) for any perturbation, or the system ends as a uniform film or a drop depending on the details of the perturbation.

Figure 1

An amount of fluid with an initially uniform thickness $H$ inside a rectangular recipient with width $2\ell$. The system may remain in the initial configuration or evolve towards a final state with the shape of a drop with precursor film $h_f$.

Figure 2

Thickness $H$ of the initial uniform film vs the thickness $h_f$ of the surrounding film of the possible drop for $d=0.02$. Thin lines correspond to equally spaced values of $\ell$ from 15 to 115. Thick lines correspond to $\ell ={\ell }_1=30.132$ and $\ell ={\ell }_2=39.427$. The dashed line is the locus of the critical points of the curves, whose ordinates are $H_1$ for the minimum and $H_2$ for the maximum. The minimum and maximum values of $h_f$ are $h_{f,\min }\approx 1.205$ and $h_{f,\max }\approx 1.559$.

Figure 3

Energy ${\varepsilon }_d$ of a drop minus the energy ${\varepsilon }_f$ of a film vs the thickness $H$ for $\ell =25$ and $d=0.02$. In this case (due to $\ell <{\ell }_1$) there is only one possible drop as the final state. The bold dot denotes the value of $H$ for which $[h\left(\ell \right)-h_f]/(h_m-h_f)={10}^{-3}$. The lower plot is a zoom of of the region $H⪆H_m$.

Figure 4

Energy ${\varepsilon }_d$ of a drop minus the energy ${\varepsilon }_f$ of a film vs the thickness $H$ for $\ell =35$ and $d=0.02$. In this case (${\ell }_1<\ell <{\ell }_2$) there are one or three possible drops as the final different states, depending on the value of $H$. The bold dot denotes the value of $H$ for which $[h\left(\ell \right)-h_f]/(h_m-h_f)={10}^{-3}$. In the lower plot we show the details of the graph for $H⪆H_m$. The segments $i$ (black line), $ii$ (blue line), and $iii$ (red line) correspond to the drop with the lowest, intermediate, and largest $h_f$, respectively.

Figure 5

Energy ${\varepsilon }_d$ of a drop minus the energy ${\varepsilon }_f$ of a film vs the thickness $H$ for $\ell =55$ and $d=0.02$. In this case (${\ell }_2<\ell$) there are one, two, or three possible drops as the final different states, depending on the value of $H$. The bold dot denotes the value of $H$ for which $[h\left(\ell \right)-h_f]/(h_m-h_f)={10}^{-3}$. In the lower plot we show the details of the graph for $H⪆H_m$. The segments $i$ (black line), $ii$ (blue line), and $iii$ (red line) correspond to the drop with the lowest, intermediate, and largest $h_f$, respectively.

Figure 6

Numerical solution of the evolution of a perturbation $Pcos(\pi x/\ell )$ on a film with thickness $H=1.582$ for $\ell =35$ and $d=0.02$: (a) $P=0.1$, (b) $P=0.2$, and (c) comparison of the final profile for the case (b) with the closed-form expression given by Eq. (A3) with $h_f=1.38586$.

Figure 7

To the left (right) of the solid red line a uniform film is stable (unstable) and to the left (right) of the dashed blue line a uniform film has less (more) energy than a drop. In region I a uniform film is the final state, while in region II a drop is the final state. The final state in region III is either a film or a drop, depending on the perturbation. In region IV the state with the least energy is a half drop, as discussed in the text.

Figure 8

The final configuration for large values of $\ell$ is a pancake-shaped drop. Here $\ell =400$, $d=0.02$, and $H=2.5$. With these values replaced in Eq. (4) one obtains $h_f=1.206$.