Capillary waves on a falling film

This paper shows that the wavelength and rate of spatial attenuation of the small ripples accompanying the large waves that form on a liquid film flowing down an inclined plane are dictated by the requirement that their velocity equal the velocity of the large wave. For this purpose numerical simulations of film flow are carried out and validated by comparison with published work. The wavelength and rate of spatial attenuation of the computed capillary waves are shown to equal, to an excellent approximation, the corresponding quantities obtained from an approximate form of the Orr-Sommerfeld equation in which the phase velocity is set equal to the velocity of the large waves. Thus the overall structure of the falling film waves is similar to that encountered in many other problems exhibiting a wave-hierarchical structure, such as waves in dusty or reacting gases, hydraulic jumps, undular bores, and others.

Figure 1

The open circles show the measured film height (top) and streamwise velocity at a distance of $120\mu \mathrm{m}$ from the wall (bottom) reported by [33]. The solid lines are the results of the present simulations, and the barely distinguishable dotted lines are the numerical results reported in [23]; see text for parameter values.

Figure 2

Real (left) and imaginary parts of the dimensionless wave number $hk=h{k}_r+ih{k}_i$ as functions of the dimensionless wave velocity $hc/\nu$ for ${\mathrm{Re}}_h=0$ (upper pair of panels) and ${\mathrm{Re}}_h=10$. The lines are for $\mathrm{\Sigma }=200$ (solid, red), 400 (dashed, green), 1000 (dotted, blue), and 2000 (dash-dotted, purple). Details near the minimum points of $k_r$ are shown in Figs. 3 and 4.

Figure 3

Real (left) and imaginary parts of the dimensionless wave number $kh$ as functions of the dimensionless wave velocity $hc/\nu$ for small $c$. Here ${\mathrm{Re}}_h=0$ in the upper pair of panels and ${\mathrm{Re}}_h=10$ in the lower pair. The lines are for $\mathrm{\Sigma }=200$ (solid, red), 400 (dashed, green), 1000 (dotted, blue), and 2000 (dash-dotted, purple).

Figure 4

Real (solid lines, red) and imaginary (dashed lines, blue) parts of the dimensionless wave number $kh$ as functions of the dimensionless wave velocity $hc/\nu$ in the neighborhood of the second minimum of $k_r$ for $\mathrm{\Sigma }=200$. The left panel is for ${\mathrm{Re}}_h=0$ and the right one for ${\mathrm{Re}}_h=10$. The scales for $k_r$ and $k_i$ are on the left and right vertical axes, respectively.

Figure 5

Computed shapes of some of the waves used for the comparisons of Fig. 7; the pertinent parameters are given in the first three lines of Table 1. The sections shown, with a length equal to 460 times the mean film thickness $\overline{h}$, have been obtained by splicing together repeated single waves. The color indicates the velocity parallel to the plate normalized by the flat-film mean velocity (2); the range for each case is specified in Table 1. The vertical scale is magnified by a factor of 15 with respect to the horizontal scale.

Figure 6

Computed shapes of some of the waves used for the comparisons of Fig. 7; the pertinent parameters are given in the last three lines of Table 1. The length of the sections shown equals to 230 times the mean film thickness $\overline{h}$. The color indicates the velocity parallel to the plate normalized by the flat-film mean velocity for the vertical case; the range for each case is specified in Table 1. The vertical scale is magnified by a factor of 15 with respect to the horizontal scale.

Figure 7

Comparison between the predicted and calculated values of the wave number (left) and rate of spatial attenuation of the capillary ripples accompanying the large waves shown in Figs. 5 and 6 (squares) and those characterized by the parameters shown in Table 2; the cases denoted by circle, diamond, and cross are identified in the table. The nondimensionalizations used are given in (11).

Figure 8

Real (left) and imaginary parts of the complex frequency $\omega$ normalized as in (A18) of capillary waves on the surface of a viscous liquid layer as functions of $kh$. The curves correspond to different values of the parameter $\nu /\sqrt{(\sigma /\rho )h}$, which, in the direction of the arrows, takes the values 0.02, 0.05, 0.1, 0.2, and 0.4. With the nondimensionalization used here, ${\omega }_r^{*}$ equals the phase velocity nondimensionalized by division by the inviscid phase velocity. In the left panel the dashed portion of the lines shows the parameter range in which the group velocity is smaller than the phase velocity. In the right panel the dashed curves are ${\omega }_i^{*}$ in the parameter range where ${\omega }_r=0$.