Thin liquid film resulting from a distributed source on a vertical wall

We examine the dynamics of a thin film formed by a distributed liquid source on a vertical solid wall. The model is derived using the lubrication approximation and includes the effects of gravity, upward airflow, and surface tension. When surface tension is neglected, a critical source strength is found below which the film flows entirely upward due to the airflow, and above which some of the flow is carried downward by gravity. In both cases, a steady state is established over the region where the finite source is located. Shock waves that propagate in both directions away from the source region are analyzed. Numerical simulations are included to validate the analytical results. For models including surface tension, further numerical simulations are carried out. The presence of surface tension, even when small, causes a dramatic change in the film profiles and the speed and structure of the shock waves.

Figure 1

A sketch of characteristic lines with $S_0=4/27$; note that the vertical axis is the distance $x$ increasing downward, and the horizontal axis represents time $t$. The red line is the shock curve formed through the intersection of the blue and green characteristics. The blue characteristics emanate outside the source region and are horizontal. The green characteristic curves emanate from the source region and upon passing $x=0$ become straight lines. The yellow characteristics represent an expansion fan emanating from $x=1$.

Figure 2

The family of steady-state solutions with $S_0$ ranging from 0 to $8/27$. The bottom curves are for source strengths below the threshold and the top curves for those above the threshold.

Figure 3

Plot of the numerical solution at time $t=20$, with $S_0=\frac{5}{27}$ and $h_0=0$. The horizontal axis represents the $x$-coordinate along the vertical wall, with the positive direction being downward. The vertical axis is the height of fluid film. Since this source value is larger than critical value $\frac{4}{27}$, we can see two shock waves, one going upward and one downward.

Figure 4

Evolution of the film profiles from $t=0$ to 20 for $S_0=\frac{5}{27}$ and $h_0=0$. The film height is plotted as a function of $x$ and $t$.

Figure 5

Comparison of numerical simulation results at large times with the steady-state solution calculated from Eq. (2) over the range $x\in [0,1]$.

Figure 6

Weak source simulations indicate the propagation of waves in one direction only: to the left due to airflow. Top picture: $S_0=4/35$, $\alpha =0.001$; middle picture: $S_0=4/50$, $\alpha =0.001$; bottom picture: $S_0=4/50$, $\alpha =0.0001$. Refer to the text for more detailed descriptions.

Figure 7

Strong source simulations indicate propagation of waves in both directions: left-going due to airflow and right-going due to gravity. Top picture: $S_0=4/15$, $\alpha =0.001$; middle picture: $S_0=4/15$, $\alpha =0.0001$; bottom picture: $S_0=4/25$, $\alpha =0.001$. Refer to the text for more detailed descriptions.

Figure 8

The plot of the left front wave speed versus its height is on the left and the plot of the left front wave height versus values of $\alpha$ (surface tension coefficient) is on the right.

Figure 9

Comparison of models without (top panel, $\alpha =0$) and with (bottom panel, $\alpha =0.001$) surface tension.