-i.e., a t R e > 400 for ‘water (32)-a more complicated flow pattern emerges, the waves showing great tendency either to coalesce into ‘.surge w,aves“ with more complicated Lvave forms or to break u p into ripples exhibiting a less regular behavior ( 6 , 22, 32, 3 4 . Conclusions
A phvsical picture of the film f l o ~ vagreeing lvith both theory and experiment has been presented. T h e mechanism of flow may best be described as a characteristic motion of approximately periodic lvaves traveling on a falling liquid film and generating circulating eddies in the direction of flow Lvithout causing lateral spread. The formation of eddies promotes vigorous bulk mixing and surface renaval in the film. Nomenclature
varying film thickness, cm.
a
=
a.
= mean film thickness in wavy flobv, c m .
= acceleration d u e tc, gravity, cm.,’sec.* = phase velocity, c m /I sec. = 2X’ x Q = liquid flow rate per unit length of wetted perimeter, cc.: sec. c m . mean stream velocity. cm. ‘sec. V” = mean stream velocity a t film thickness, a,, cm.:’sec. ratio of phase velocity to mean stream velocity a = amplitude of wave motion 1 7 = dynamic surface tension, dynl; c m . 6 = kinematic surface tension = u p I J = dynamic viscosity. P kinematic viscosity = fi ‘ p . S l J = A = wavelength. c m . P = free surface functiomn of flowing film P = density. grams cc.
g k n
v = z=
Literature Cited
(1) Belkin, H . H.. MacL.eod. A . A , , Monrad. C. C., Rothfus, R.R . . A . I . C h . E . J . 5, 245 (1959). 12) Benjamin. T. B.. J . FlziidMech. 2. 554 11957). (3) Binnie. A . M.. Zbid., 2, 551 (1957). (4) Brauer. H.. Chen/.-Ingr Tech. 30, 75 (1958). (5) Brauer, H.. KaltetPchni,i- 9, 274 (1957).
(6) Brauer; H., VDZ-Forschungsh. No. 457 (1956). (7) BrBtz, I$’.. Chern.-Ingr.-Tec’i 2fiq470 (1954). (8) Davies, J. T.. Chem. Znd. London 1962, 906.
(9) Davies. J. T., Dept. Chemical Engineering, The University, Birmingham. England, private communication. October 1963. (10) Davies, J. T., Trans. Inst. Chem. Engrs. London 38, 289 (1960). (11) Davies, J. T., Bell, G..’Law, P. J. S.; Research Project, Dept. Chemical Engineering, Cambridge Cniversity. 1960. (12) DaL-ies: J. T.; Bradley, P. J.; Research Project. Dept. Chemical Engineering, Cambridge University. 1960. (13) ~, Davies. J. T.. Rideal. E. K.. ‘.Interfacial Phenomma.” ~, pp. 317-19. Academic Press. New York, 1961. (14) Ibid.. 2nd ed.: pp. 266-8, 1963. (15) Downing. A . L.. Truesdale, G. .-JI .. ‘4ppl. : Chem. 5, 570 /,“c:\ (LYJJ,.
(16) Dukler, A. E.:Chem. E n g . Progr. Symp. Sei. 56, No. 30. 1010 (1 960). (17) Dukler. A . E., Bergelin, 0. P.. ChPm. E n g . Progr. 48, 557 (1 952). (18) Emmert. E. E.. Pigford, R. L., Ibzd., 50, 87 (1954). (19) Fallah; R.: Hunter, T. G.. Nash, A. LV., J . Sac. Chem. Znd. London 53, 369T (1934). (20) Friedman. S. J.: Miller, C . O., Ind. E n g . Chem. 33, 885 (1941). (21) Grimley. S.S..Ph.D. thesis. London University; 1947. (22) Grimley. S.S.: 7‘rans. A m . Inst. Chem. Engrs. 23, 228 (1945). (23) Jackson. M, I.. d . I . C h . E . J . 1, 231 (1955). (24) Jeffrvys. H.. Phil. .Z4a,g. 49, 793 (1925). (23) .Jeffreys. H.. Proc. Canibridge Phil. Soc. 26, 204 (1930). (26) Karnri, S..Oishi. J . . Mem. Fac. En?. X j o t o t’niL8. 17, 277 (1 955). ( 2 7 ) Kapitza. P. L.. Zh. EXspt. Tear. Fzz.18, 3 (1948). (28) Kirkbride, C . G . , Trans. Am. Inst. Chem. Engrs. 30, 170 (1933-34). (29) Laning. J. H., Battin, R. H.. “Random Processes in Automatic Control.” pp. 105-1 5, McGraw-Hill. New York. 1956. (30) Mantzouranis. B. G.. Ph.D. thesis, London University, 1959. (31) Nussrlt. LV.. Z . Ver. deut. Ing. 60, 541 (1916). (32) Portalski. S.,Ph.D. thrsis. London University. 1960. C:.. Hurt. D. M . . A.I.Ch.E. J . 1, 178 (1955). S.R.. Portalski. S..Chem. En,?. Sci. 17, 283 (1962). S. R.?Portalski, S..Trans. Inst. Chem. Engrs. London 38, 324 (1960). (36) Thomas. I V . J.. Portalski. S..Ind. Eng. Cheni. 50, 1081 (1958). (37) \Viener. N.. “Extrapolation, Interpolation and Smoothing of Stationary Time Series.” pp. 15-20, 46-7. Chapman and Hall: London, 1949.
~
RECEIVED for review March 18, 1963 ACCEPTED November 12. 1963
INITIAL INSTABILITY OF A VISCOUS FLUID INTERFACE W I L L I A M
E.
R A N Z A N D W I L L I A M
M . D R E I E R , J R . ]
Department of Chemical Engineering, L’niLersiti of .Minnesota, .WinneapoIzs, .Mznn. Initial instability of the interface between two viscous fluids, one being moved impulsively past the other a t a constant velocity, i s investigated theoretically b y an analysis of the growth of small disturbances and experimentally b y mic:rosecond photomicrographs of a liquid jet injected into another liquid. For various limiting cases, theory is developed relating W e b e r number based on the wavelength of the fastest growing disturbance to a dimensionless viscosity number, viscosity ratio, and density ratio. This theory is then used with experimental d’uta to construct an empirical correlation valid for a general planar system. This correlation can b e shown to b e valid also for liquid jets of sufficient diameter and viscosity.
of a liquid jet in another liquid appears to be a chaotic process. Upon a closer look. hoxvever, some order appears. Tl’ithin several jet diameters of the orifice there is a somewhat regular pattern of transverse surface disturbances, These disturbances can be characterized by a n average tvavelength in the direction of motion and by rate of HE BREAKL-P
1
Present address, 27904 Rexford, Bay Village 40, Ohio.
gro\vth. Ultimately. they- grow so large that they are torn from the jet to form drops. Since \rhat happens to the jet during the period of breakup should be influenced b>- initial instabilities. a theoretical and experimental investigation of such disturbances \vas undertaken. Because short tvavelengths uninfluenced by jet size are of the most general interest. instabilit). of a plane interface was made the pubject of study and appropriate data were VOL. 3
NO. 1
FEBRUARY
1964
53
obtained from jets. Microsecond photomicrographs were obtained oca liquid column bring injected into another liquid. Figure 1 is typical of the behavior of most common liquid pairs, while Figure 2 shows the behavior of a very dense liquid being injected into water. T o explain these observations, a theoretical treatment by Taylor was extended, and ultimately a semiempirical madel was developed. T h e resulting correlation and theory are valid only far viscous jets and other viscous fluid surfaces where the waves formed have wavelengths smaller than a n y other dimension of the fluid surface and where a forced instability does not supersede the fluid mechanical instability. In Figure 2, transverse disturbances appear first, while in Figure 1, parallel disturbances appear equally prominent. In any case of actual instability, one is faced with the problem of forced disturbances-for example, those caused by orifice roughness or periodic phenomena upstream-as well as several competing natural instab es. In the present case, experience indicates that a transverse disturbance is the dominant natural disturbance, and this is the one investigated. Theory
Imagine two phases separated by a n interface having a certain interfacial tension, T. Suddenly, at zero time, the upper phase is impulsively moved tangent to the lower phase a t a constant velocity, U. T h e system is dynamically unstable. Waves will form; when they are large enough, their crests break off and become dispersed droplets. In the case of a liquid jet in a liquid, impulsive action exists as long as the boundary layers are much thinner than the lengths of the waves formed. Under a constant stress, where no viscous dissipation occurs, one expects a wave to grow exponentially after some zero time when a wave of initial amplitude yo somehow appears. This yo can he very small. I t could he the result of vorticity left over from a flow obstruction ahead b f the o r i f i c e s a y , wall roughness. In each case, however, yo is much smaller than any length seen on the photomicrographs. Figure 3 represents a mathematical model for all these breakup processes. T h e amplitude y a t time t of one of the small disturbances of original amplitude y, is given by: y = yo e"t = yo(l
+ + ut
- - + u -bu+ u - au =
bU
at
ax
ay
1 &P 3.x
---++vu
P
Here gravity is presumed to act downward on a horizontal surface. Solution of these equations is extremely complex. A special case, receiving the most attention (hut still without a complete solution), is that of an inviscid gas passing over a viscous or inviscid liquid whose surface tends to he restored by the action of surface tension and gravity. This is the problem of windblown waves and ripples (8). For applications to jet breakup, however, one is more interested in cases where wavrlengths are so small that gravity is not important. Interaction is between two dense, viscous phases where only surface tension tends to restore the surface. Taylor's Analysis of Gas-liquid Systems
Taylor (7) simplified Equations 2 to 7 by assuming the upper Ufory> 0 phase to beinviscid, U(y) = and the wavelengths 0 for Y < 0' short enough to make the effect of gravity negligible. For thelatter to he true, one must have:
&
