Thin liquid films appear in many real world phenomena: they are used in electronic components and naturally occur in a coating of the human lung. In this summer research project, we use numerical and computational methods to explore the behavior of these thin films under a variety of conditions.

Our project models the physical experiment shown in Figure 1. The experiment consists of an inclined plane coated with a thin layer of liquid and with a constant temperature gradient . Since temperature and surface tension are inversely related, this creates a constant surface tension gradient. In what is called the Marangoni effect, the surface tension gradient induces a force which drags the surface of the liquid (and consequently the liquid below) toward regions of higher surface tension. When the temperature gradient is large enough, the Marangoni force causes the fluid to flow slowly upwards in opposition to gravity. In addition, the experiment uses localized forcing (such as heating in the form of an infrared laser) which causes interesting wave structures to appear in the fluid by the same effect. Possible wave structures include classical compressive shocks (C) and rarefactions (R) as well as non-classical undercompressive shocks (UC). Some examples of these structures are labelled in Figures 3 and 4.

To obtain a model for the evolution of the film in time, we use the thin-film or lubrication approximation to the Navier-Stokes equations; this applies in the limit that the film height is very small compared to the characteristic in-plane length scale. The fluid is governed by a nonlinear, fourth-order partial differential equation:
(1).
In this nondimensionalized equation, h(x,t) is the height of the fluid, M represents the strength of the local forcing and θ(x) is a Gaussian profile of the heating laser:
.

Equation (1) is difficult to solve analytically, so we find an approximate solution using a computer-based numerical solver. Specifically, we use a Crank-Nicolson method in which we replace the derivatives in equation (1) with finite differences.

We use a double tanh function, shown in Figure 2, as our standard initial condition. This profile (the red dashed line) is an approximation to a double-wave structure (the solid black line) which consists of a leading undercompressive wave and trailing compressive wave. The double wave structure has evolved from a single tanh jump initial condition.

Depending on the initial fluid profile h(x,0) and on the strength of the localized forcing M, the fluid may move to either of two steady state solutions, shown in Figures 3 and 4. For weak forcing (small values of M), a type I steady state will occur, in which the fluid heights to the left and right of the forcing region are equal. For strong forcing (large values of M), a type II steady state will occur. For a type II steady state, the fluid heights to the left and right of the forcing are unequal, and are related by
(Haskett, et. al.).

Figure 5 shows two snapshots in time (t = 400, 600) of the fluid for two different widths (w = 20, 40) of the initial condition. We use a forcing strength M = 0.9, which is just strong enough to produce a type II steady-state solution. To the left of x = 0, the fluid forms an extremely slow moving compressive wave moving to the left. To the right, the fluid profile depends only on the time of evolution, not on the size or shape of the initial fluid bump. This demonstrates that such forcing could be used as a valve, allowing the fluid to pass through the forcing region at a specific rate. Such a "microfluidic valve" could be useful in industrial applications.

Figure 6 shows the fluid profiles that evolve in a variety of forcing amplitudes. It shows the transition between type I steady state solutions (M = 0.6, 0.8) and type II steady state solutions (M = 1.0, 1.2). The critical value of forcing, which depends on the initial conditions, is . This experiment demonstrates that the heights of the fluid to the left and to the right can be controlled by changing the strength of the laser forcing. Again, this could be applied in industrial processes which require precise liquid coatings.

Another interesting phenomenon which occurs in the thin film simulations is the N-wave, shown in Figure 7. The N-wave consists of two compressive shocks connected by a rarefaction. The N-wave is generally much smaller than the other types of wave structures; as a result, it has not been found experimentally. However, we are in the process of planning laboratory experiments to search for these N-waves.

For both type I and type II forcing on a uniform initial condition, an N-wave is given off from the forcing region. The velocity of an N-wave depends on height of the fluid in which it is propagating: . As a result, N-waves occuring at heights h < 2/3 travel to the right, and N-waves occuring at heights h > 2/3 travel to the left. The size (maximum height - minimum height) of an N-wave is proportional to the forcing parameter M, but this relationship breaks down for large values of M. For negative values of M, corresponding to local cooling rather than heating, an inverted N-wave is created, as shown in Figure 8.

This summer research project demonstrates the interesting behavior of a thin liquid film under gravity and Marangoni forces. Some aspects of the behavior may be particularly useful in industrial processes which require a thin film coatings with a precise thickness or mass. Others, such as the N-wave, are interesting in that they are not yet fully understood or experimentally verified. Further research could search for the N-wave experimentally or could explore the results of using different forces, such as the effect of adding a chemical surfactant to decrease the liquid's surface tension.