Spiers, R. P., Subbaraman, C. V., Wilkinson, W. L., Chem. Eng. Sci., 29,389-396 (1974). Spiers, R. P., Subbaraman, C. V., Wilkinson, W. L., Chem. Eng. Sci., 30,379-395 (1975). Strang, G.. Fix, G. J., "An Analysis of the Finite Element Method," Prentice-Hall, Enalewood Cliffs, N.J.. 1973. Van dyke, M., "Perturbation Methods in Fluid Mechanics," Parabolic Press, Stanford, Calif., 1975. Van Rossum, J. J., Appl. Sci. Res. A, 7, 121-144 (1958).
Wehausen, J. V., Laitone, E. V., "Handbuch der Physik," Vol. 9, pp 466-779 Springer-Verlag. New York, N.Y., 1960. White, D. A., Tallmadge, J. A., Chem. Eng. Sci., 20, 33-37 (1965). Williamson, A. S.,J. FluidMech., 52, 639-659 (1972).
Received for reuiew February 16,1977 Accepted June 24,1977
Some Theoretical and Experimental Observations of the Wave Structure of Falling Liquid Films F. Wayne Plerson and Stephen Whitaker' Department of Chemical Engineering, University of California, Davis, California 956 16
The linear stability problem associated with a vertical liquid film flowing under the action of gravity has been solved in terms of a numerical solution of the Orr-Sommerfeld equation. Results have been obtained for both the temporal and spatial representations of growing waves and for a wide range of the physical parameters characterizing this type of flow. Experimental values of the wavelength and wave velocity have been determined for water films and the results are in reasonably good agreement with the theory. Calculated values for the wavelength, wave velocity, and growth rate of the most unstable wave indicate only a small difference between the temporal and spatial formulations for water films.
1. Introduction
The wave characteristics of thin, falling liquid films have long been of interest to engineers because of the prevalence of thin liquid films in industrial processes. In addition, there is a general interest in the phenomenon as it provides an attractive challenge for the comparison of stability theory with experiment. Our own interest in this problem stems from the sometimes dramatic effect of surface active agents on the wave characteristics of thin liquid films. Under certain conditions small amounts of surfactant can completely inhibit wave formation to the extent that the liquid film appears as motionless as a plate of glass. Davies and Rideal (1963, p 266) have commented on this phenomenon, and have provided some photographic observations. As a prelude to an investigation of some of the anomolous experimental results for surfactant solutions (Strobe1 and Whitaker, 1969; Whitaker, 19711, a detailed survey of linear stability theory for pure liquids is in order. Linear stability theory has severe restrictions regarding the range of application for falling liquid films. Such films may be formed by fluid issuing from a channel or flowing over a weir as illustrated in Figure 1.Both types of flow have been analyzed theoretically by Cerro and Whitaker (1971a, 1974) and the calculated values for the free surface velocity and the film depth are in good agreement with the experimental data of Lynn (1960) and of Cerro and Whitaker (1971b). When the original channel depth in Figure l a is equal to the final film depth, h a , the distance required for the surface velocity to reach 99% of the final value is 0 . 5 h - N ~ This ~ . is an order of magnitude larger than the entrance lengths for pipes or rectangular channels; however, the transition here is from a confined flow to a free surface flow and we do not expect the entrance lengths to be the same. For falling liquid films the entrance length is somewhat sensitive to the way in which the film is formed, but a reasonable estimate of the length is given by
Le
N
hmNRe
(1.1)
where N R is~the Reynolds number based on the free surface velocity of the uniform flow and h mis the film thickness for the given uniform flow. These quantities are defined by ha = (3q~/g)l/~ u, = hm2g/2v
(1.2)
(1.3)
where q is the volumetric flow rate per unit width, g is the gravitational constant, and v is the kinematic viscosity. The film thickness can be written in terms of the Reynolds number h, = ( ~ N R~,' / g ) ' / ~ so that the entrance length can be expressed as
Le
-
( 2v2/g)1/3N~e4/3
-
(1.4) (1.5)
For water we have ( 2 ~ ~ / g ) l 6/ ~X cm, and an estimate of the entrance length as a function of the Reynolds number is given by Le
-
[6 X 10-3N~e4/3]cm
(for water)
(1.6)
This provides us with an indication of the region in which the flow is nonuniform and the usual small disturbance analysis leading to the Orr-Sommerfeld equation may not apply. For small Reynolds numbers, say N R ~ 1,this region is negligible; however, for N R =~ 100 the entrance length is estimated to be about 3 cm and a t a Reynolds number of 500 we find Le to be on the order of 24 cm. Certainly the entrance region cannot be ignored in any comprehensive study of the stability characteristics of falling liquid films. The distance down the film at which waves are first visible is referred to as the wave inception line, and the location of this line has been tabulated by several investigators (Pierson, 1974; Cerro and Whitaker, 1971b; Strobe1 and Whitaker, 1969; N
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Figure 1. Formation of falling liquid films. Figure 3. Visual appearance ofwaves for N R =~ 20
I-1
2
1
r
8
10
20
10
so
80 100
200
1w
600
"It
Figure 2. Wave inception line as a function of the Reynolds number.
Tailhy and Portalski, 1962). As should be expected, the results are rather scattered and the range of observations is indicated in Figure 2 along with the estimate of the entrance length given by (1.6). It is clear that for Reynolds numbers of 400 and greater, the small disturbances grow to a finite amplitude entirely within the entrance region, while for Reynolds numbers of 20 or less the entrance region is a small fraction of the distance required for finite amplitude waves to appear. While the effect of nonuniform flow in the entrance region on the stability of falling liquid films is always ignored (and will he in this work), it is important to keep this aspect of the problem in mind. The presence of nonlinear effects also places a constraint on the region of applicability of the Orr-Sommerfeld equation. In Figures 3-6 we have shown photographs of water waves near the wave inception line for Reynolds numbers ranging from 20 to 150. At N R =~20 the waves appear a t a distance of ahout 10 cm from the top of the film and they are reasonably uniform over a distance of several centimeters. In this region one could hope that linear stability theory might satisfactorily represent experimental observations. In Figure 4 the situation for N a = 50 is shown and there we see that in a distance equal to one or two wavelengths nonlinear effects have become important. A short wavelength wave appears to he leading the main wave and the instability is no longer two dimensional. For this case we expect that linear stability theory is only applicable in a region near the wave inception line and in the 402
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Figure 4. Visual appearance of waves for N R =~50. region above it. In Figure 5 the results for N R =~1Mindicate increasing nonlinear and three-dimensional effects, and the results for N R =~ 150 shown in Figure 6 would suggest that experimental measurements must be made above the wave inception line if one is to compare the experimental ohservations with small disturbance theory. For high Reynolds numbers this would mean that measurements must he made in the entrance region, and there we cannot expect the OrrSommerfeld to he valid. Clearly the range of validity of linear stability theory for falling liquid films is severely limited by nonlinear effects and the length of the entrance region; however, a detailed survey of theoretical results based on the Om-Sommerfeld equation should prove to he of value for low Reynolds number flows and should provide an interesting comparison with future nonlinear theories and stability studies of the entrance region flow.
Figure 5. Visual appearance of waves for Nn. = 100.
Figure 7. Temporal disturbance
better agreement between theoretical and experimental growth rates for low Reynolds number and low surface tension number flows. In the spatial formulation the disturbance is represented by exp[i(px - ut)] (6 ~ 10. The values of cy, shown in Figure 13 are about 20%higher than our values in the low Reynolds number range, and slightly lower for Reynolds numbers larger than 500. The analytic expressions obtained by Anshus are
-
-
= 0.915N,-4/11N~e1/3
(3.1)
(3.2)
The values of the wave velocity and wave number given by Anshus and Goren (1966) were obtained from the Orr-Sommerfeld equation with = 1.0. Under these circumstances the coefficients in (2.3) are constant and an analytic solution is readily obtained. For N R > ~ 15 the calculated values of Anshus and Goren are somewhat smaller than our own and then become slightly larger for N R >~500. The wave numbers determined by Anshus and Goren are somewhat larger than our own for N R
