flow rate ljer unit width of plane! cni3/sec ' T ~ m, ~ i 2 'sec diameter of rotatiiig cylinder, crii = radial di-tancc iii cyliiidrical coordinate, cm = length of n-etted periphery, c m or Laplace's = = =
= = = =
= = = = = =
= =
c;Rl.LK
=
I9
P P 7
TlJ
yelocity of fluid, cm 'set d i p vplocity, e m '.ec .r cornii~neiitof fluid velocity, cni 'sec romi)oiieiit of fluid velocity, cni,:sec r cwnpoiient of fluid velocity! cm 0 c o i n i x m n t of fluid velocity, ern wordinate linlf ivitlth of channel, c m c~oorciiiiate thickne-5 of fluid layerj cm coortlinatc
:ingle shown in Figure 2 , radiaiis vuri:tble defiiied liy Equation 14 = angle ehowi in Figure 2 , radians = inc~liiintiouof fluid surface in rotatiiig cylinder. riitlisny =
Y
Y
o])Pr;lt(Jr toiqiie, dyi1c.q
LIXTCRS
cy
H
&'
c w
inclination of channel, radians or angle in cylindrical coordinate, radians = viscosity, poises = density, g/cm3 = shear stres.;, dynes, em2 = yield stresa, dynes/cni2 = stream function = angular velocity of rotating cylinder, sec-l =
literature Cited
.2starita, G., IIarrucci, G., Palumbo, G., IXD.1:s~. CHI-11. FvsD.i>r. 3 , 333 (1964). Bird, R. B., Stewart, W. E., Lightfoot, E. S., "Trntisport Phenomena," p. 8.5, Riley, Yew York, 1060. Boslev, J.. dchofield., C.., Shook, C . A , , Trans. Inst. Chcrn. Ena. 47.'T14i 119691. Bro6i1, 1%.L., -Ydiirc (London) 191, - - - , 4%- (1961). --, Holland, J., ililes, Hollai;d, Jliles, J. k. E. P., dchofield, C., dhook, C. &i.,Trans. I m t . Cheia. Eng. 47,T1>4 (1969). Kurihara, K., Kuiio, €I., .-lppl. Ph!/.s. ( J a p a n ) 34, 7'27 (1965). Laforge, I?. >I., Roriifi', B. K., Intl. Eng. C h o u . 5 6 , So. 2, 43 119641. I
\ -
Stability of Thin Liquid Films Flowing Down a Plane William B. Krantzl and Simon L. Goren Department of Chemical Engineering, rniversitu of Californin, Berkeley, Calif. 94720
Wave lengths, wave velocities, and amplification or decay rates were measured for waves on film flow of low Reynolds number. Disturbances of controlled amplitude and frequency were imposed on the flow and fluctuations in the film thickness were monitored b y a light-extinction technique. The data are compared with linear stability theories from the literature and two new momentum integral theories developed during this work. All theories are in semiquantitative agreement with the data for values of the parameters, such that the approximations made are valid. Wave velocities and wave lengths appear to b e independent of wave amplitudes.
L i r i u i i l f i l i i i - flowiiig ivith one .olid-liqriitl interface and one ga.--liquitl interface are found frequently iii engineering and natuiul ~ I ' I J C P ~ > ~Flow *. i i i vertical coiitie~~xers, film reactors. wcrtetl-\vall coliiniiis, n i i t i water l'lili\vay> are esaml'les. T h e geonietry for film fltm t l ~ n - ai ~plaiie wall incliiietl a t aii angle p to tlir hoi izoiital i- depicted iii Figure i. The theory for this f l ~ :l>.-iiiiiitig \ ~ :i 1)l:inar iiiterface \viis developed by Siisselt (191 6). Seglectitig tlie hytlrotlyiiarnic drag of the arts phase, ttie tt1cCJl.y I)i'etlict- :I +ernilmxliolic vrlocity profile i n the liquicl film. 'lX- predictioii, however, ib strictly applicable otily ;it very low Reynoltl- iii1rnl)er>, becaure of the almost iiiiivelwl ocwrreiice of ripl~le*.The ripple. that develop ii1iti:illy ~ i r rnio.-t often tn.o-din~eii.ioiia1, -It low Reynold:, nilintiers they may gron. t(J ail equi1it)rium amplitude or deV e k J i J iiito a three-tliiiieiisioiial pebbled flow. -It higher ReyiiIJhls iiiiniliei.- i > 2 j 0 ) , fully developed turbulent flow ucually rc.ult.. Prebent addre-s, Departmetit of Chemical Engineering, L.iiiversity of C o I ~ r a d o ,Boulder, Colo. 80302. To whom corresporideilce hhuuld be sent.
Figure 1 hhon.:, a >l)ati:llly growiiig tn-c)-tli~nell-ioll~~l riplJle of wave length A , wave velocity c T . anitl instantaneou-: anipliturle ii', sul)erinil)o,-erl 011 a falliiig filii1 ~f iiieiiii thickiie>s ii. iiutnt)er of theoretical aii:ily.*es have lieell atlv~iicetlto predict the oilbet of .such w v e i , and their growth rates. wave velocities, nntl wave leiigths. Esperiniental confirniatioii of these theorie.. i- iii large 1)nrt lackiiig. The great majority of meawrementc xre of the \vave lengths aiitl Ivave velocities of wave,. ai,i*iiigfrom room (li?tici~I)ance? (most highlj- amplified waves) a t moderate Heyiiolds iiiinil)er> ( 5 to 100). Thorough reviews of thi. literature have beeii give11 liy Fulford (1964) and more recently by I\;r:iiitz (1968). Tables I and I1 cuinniarize the theoretical and esl)erimental literature, re>l)ectively,and 1)lace the 1)rebelit ~ v o r kill persliective. This paper de.cribes ai1 esperiinelital investigxtioli of t'he groivth rates, wave velocities, tiiid n a v e lengths of twodimensioiial \ ~ a v e sresulting from imljosed disturlxuice:, of controlled amlilitude and frequency. In dimeiisioiiless form, these pro1)erties depend oii four indepeiideiit dimeiisionless parameters: a Reynolds iiutnber, Re E ah v ; a surface Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
91
Geometry of rippled film flow
Figure 1.
tension parameter, 3 3 1 / 3 ~ / p g 1 ' 3 v 4 / 3the ; angle of the column to the horizontal, p ; and the wave number or frequency parameter, a E 2?rh,/h = whit,. i n the above, and h are t h e mean velocity and film thickness according to Susselt (1916), v is the kinematic viscosity, p the density, u the interfacial t,eiision, g the acceleratioii of gravity, aiid w the frequency (radians per zecoiid) oi t,he imposed waves. --Idinieiision1e.s:. group of considerable utility for the present probleni is the Keber number, We E u/plaiL2: which is related to the Reynolds number and the surface tension parameter by K e = (Re- 6 / 3 . I n the present experiments Re ranged from -0.13 to -1.3, { averaged 2.3 aiid 10.1 for the two oils ubed, 0 was 74.5' or go", aiid the impo>ed wave number, a : varied from -0.1 to -0.7. Equilibrium :implitudes were a150 mea.;uretl (Iirantz and Goreii, 1970). Review of Theoretical Investigations
T w o very similar theoretical approaches to the problem of the stability of t'hin films exist: the monieiituni integral approach and classical stability theory. Both assume that the Fourier compoiieiitu of arbitrary small disturbances are dyiiainically independent. i n the first, infinitesimal perturbations with an x component of velocity of the dimensionless form S(y)eiaz-iwt = f(ll)eiarx-iwt e - a x j ( 2 / ) e i a r x - i ~ te -aa,cd
differential equation for i p ( y ) , known as the Orr-Sommerfeld equation. This equation, together with the linearized boundary conditions, yields a n eigenvalue problem. L-sually the complex wave velocity. c: is .solved for a i a functioii of the wave number, a(=2nh:X, a real numher in thi. case), and t,heother parameters:
If the imaginary part of the ware velocity, c z >is positive, the disturbance of wive number (Y will grow in time; if c i is negative, the disturbance will decay in time. 111 thi. approach a = F ( R e , j-! p ) for c i = 0 defines the neutral stability curve. i n most experiment.; dibturbances grow in tli?taiice. not in time. If c i I +
+ i(252wRe)la + [ 2 8 8 ~
dRe
cy2
3-
i(-96w2Re)] = 0
(2)
The first approach described above considers the twodimensional equations of motion directly and bears clove resemblance to the momentum iiitegral solut'ions of Kapit'za (1948) and other iiivest:igators listed in Table I. The second integral approach considers the (inkgral) Orr-Sommerfeld 94
Wave Velocity vs. Wave Number F i x e d Re,
(, p
Schematic of wave characteristics predicted by linear stability theories
I n the second method of solution, a forni for the stream fulic.tioli of the disturliaiice i-; assumed. iiivolviny five adjuhtable wii3tiiiita:
7
lciigths ant1 w t v e velocities of m0.t highly amplified u.avc f'or the oils ( j 2 . 3 :inti -10.1) measured in this study.
-
inch in diameter a t the photoconductive cell located 5 inches from the film was obtained. The beam was thus thin enough a t t h e film to provide point measurements (typical wave lengths were -1 inch) and yet completely cover the sensitive area of the photoconductive cell (0.14 x 0.20 inchj. T h c illuminator and photoconductive cell were rigidly mounted on a U-shaped bracket which could be traversed down the column. The fluctuating resistance of the CdS photoconductive cell was converted into a voltage signal and di-played on one channel of a dual-channel recorder (Panborn, l l o d e l 320) ; the second channel displayed the velocity transducer feedback. Since the resistance of the photoconductive cell was greater than the input impedance of the recorder, a cathode follower circuit was used to match impedances. Calibration showed that the changes in voltage were linearly prol)ortion:il to the changes in film thickness. provided B E 1 ii 5 0.65, where B is a parameter characteristic of the optical system, ,t is the extinction coefficient of oil, and AIL is the change in film thickness. B and [ aln.ayh appear in the combination B t , which war determined empirically prior to each i'un by measuring the voltage as a function of mean film thi(,kne.s on the column-i.e., changing the Reynolds number. With suitable choice of the extinction coefficient, wave amplitudes of the order l o d 4crn could be detected.
Experimental Design
In order t o increase the extilictioii coefficiciit of the oils to obtain the desired ,*en$itivity. n snitill :\irioiuit (if carbon black (Sher\\-iii-~~illiani., X13S-E'3 coiitaiiiing 257; i)ign-~eiit, 6470 mineral *l)iritsv, and 11% rehiii .olitl* :is :iii :iiitic*o:igulutThe flow surface was 1;8-inch-thick plate glass 5 feet long ing agent) \vas atldetl. .4ltliough ndditioii of the cnr1)oii hlnck and 8 inches \vide. This width \vas sufficient to prevent wall effect.* from penetrating to the center line of the flowv,where containing :i sin:ill tiniouiit of *iuface-:ictivc :igciit h i d no film thickiieh., measurement> were taken. Liquid supplied effect on tlie htatic siirfnce teii.ioii to ivitliiii tlie :iccui'acy froni :I con>tnnt-head tank was continuously recirculated of niea.iirement. 0.1 tlyiie 1)ei' cni. tlie lio..sihiIity csi-th that aiid the flow rate w,- controlled by varying the distance ~ it could affect the ~vtiveproliertie* t h i w i g i i :I w i ~ f : i rrl:i..tic*ity between the flow .surface and a plate glass film gate which also acted a. the liquid distributor. The column way supported osity. The theoretical uiiuly-c~ of lieiijaniiii a t two 1)ointz in such a way that the flow surface could be (1964) and Khitaker (1964) have sliowi that .mf:ice rhstivity inclined to the horizontal a t angles from 74.5" to 92'. Reynplay' the i>rini:ii,y d e iiiodifyiiig n-ave roperti tics :it lonold. number. n-ere determined from the volumetric flow rate Reynoltl~~ iiuni1)er~.Surface elasticity :\ri.e oiily i t it is for the central 4 inches of tlie columii. l)o,-silile for gradient.: i i i surface coiiceiiti,atioii to e1i-t. Such The fluid< used ivere Chevron S o . 5 white oil ( p = 0.499 poise! p = 0.853 gram per cm3, and u = 29.5 dynes per cm gradient- are produced liy the cwml)re4oii. aiid r:ii,efactions a t 25"C), Chevron S o . 15 white oil (w = 1.46 poi.es, p = of the waves, but they are reduced liy Iiiolecuiar tliffu.ion 0.868 gram per cm3, and u = 30,s dynes per c m a t 25"C), between the bulk ant1 surface. I-siiig 1,easoii:il)lc e-tinintes and a mixture of these ( p = 1.25 poises, p = 0.868 gram for the equilibrium coiirtant tietween the bulk :nit1 hl1rf:ic-e per cni3, and u = 30.8 dynes per cni a t 2 5 ° C ) . Values of t h e dimensioiileas burface tension group, {- = ~ 3 ~ ' ~3, i p gconceiitrntioiis ~ ~ ~ ~ ~(10-j em) and the molecular tliffu-ivity of the changed from run to run for a given oil becauhe of changes surfactant (10-8 em2 per qecond). it nxiy lie shon-11 that the in the T.j,vo.*itj- with changing rooni temperature. Yi,>couh time for the esta1)lishnient ol' eyui1il)riuin l)y tliffu-ioii is iiiuch oils were chohen to give reasonable film thicknesses a t the smaller th:iii the time characterizing changes in sui,face colilow Reynold. numbers of interest and because they are centratioii at the Ion frequencie. ( < l o c p ) used i n thi.: study. affected by surface-active contaniinants to a much les.er degree than are aqueous solutions. (Even in the nii-ence of diffusion, it \v:i. eytimutctl th:it the Tno-tlinieii.iona1 disturbanceb of controlled amplitude growth rate ~v-ouldlie reducctl by only -40% liy wifare-elastic and frequency n-ere impressed on the flow by means of a effects for the most severe flow contlitioii. of thi. study.) 0.0071-inch-diameter n-ire, running the width of the channel, Thub, the ,surface qhould alwayh 1)e c1o.e to it. equili1)iiuin inserted into the flow and driven normal t o the flow surface conceiitratioii. and the presence of the surf'act:nit i i i the oil Iiy an electromechanical vibrator (Goodmans vibrator, 3lodel 5'47). The wire was supported on a tee whose shaft should have little effect on the measured wave 1)~01)erties. passed through a velocity tranbducer (Sanborn, Model 6LV1) K i t h the extinction coefficient of the dyed oil known, the and n-a.. affixed to the vibrator. Comparison of the signal wave amplitude and frequency were determined from t h e from the velocity transducer with the differentiated output of recorder traces. K a v e lengths were obtained by observing a sine Ivave generator (Heivlett Packard, Xodel 202.4) t h e Lissajous figures produced on an oscilloscope when the provided closed-lool) feedback control. T h e velocity transinput to the vibration generator \vas plotted against t h e ducer a1.o provided continuous monitoring of the impressed measured wave form eignal. The light source and photodisturbance. The circuit uced was a slight modification of that conductive cell were traversed down the column. and the described in detail by Zane (1966). With this design, the distance between successive in-phase points identified by probe displacement was sinusoidal within 27, rms and it,3 st,raight-line figures on the owilloscope corresponded t o the amplitude was continuously variable u p to 1 mm and was wave length. IVave velocities were obtained by multiplication irequency-iiidei,endeiit in the range 2 to 40 cps. The wire of the wave length by the frequeiicy. JTave velocities for niu5t be free from kinks or captured dust, as these would low-frequency waves were also obtained by visually following introduce three-dimensional disturbances. a wave crest and timing its passage between two points sepa\Tare amplitudes were obtained by a light-extinction techrated by a known distance. *Inother technique rvaq recording nique. -1 microscope lamp (.4merican Optical, Model 653) the wave form when the detector was positioned far downwith a O.l7, regulated power supply served as the light source stream from the disturbance generator and measuring from aiid a photoconductive CdS cell (Clairex, Model CL705HL) the trace the time for a change in wave form to reach the as the detector. Through the use of collimating holes a diverdetector after the generator was suddenly w i t c h e d off. gent light beam 0.0586 inch in diamet'er a t the film and 0.264 A det:iiled tlehci,iptioii of the experimental de.sign is given by Ii1,tiiitz (1968).
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
95
The three methods of determining the wave velocity were in agreement with each other within the experimental accuracy of t h e measurements. Experimental Results and Discussion
For Reynolds numbers less than -0.2, data were taken predominantly on the wave properties of decaying waves, since the amplification of unstable waves (act < 0.0005) was too small to be observable in the present system. In the Reynolds number range -0.2 to -1.3, data on the wave properties of bot,h stable and unstable waves could be obtained. hbility to observe decay rates diminished as the Reynolds number increased, because of the increased amplification of room disturbances. For Reynolds numbers greater than -1.3, the most highly amplified wave grew so rapidly that only room dist’urbaiices were observable. Very small amplitude waves ( 5 0 . 0 1 cm) were most often nearly sinusoidal. Larger amplitude waves usually exhibited varying degrees of “teardropping”-i.e., steeper slope on downstream edge of wave. Large-amplit’ude long n.aves on occasion exhibited “reverse teardropping”-i.e., steeper slope on trailing edge of wave-iyhich eventually gave way to a type of push wave. Under certain coiiditioiij when the first’ harmonic of the wave beiiig impres3ed also wai; uiidahle, bimodal waves were obseri-ed. These were composed of two waves whose wave rinmbei..: and x t v e velocities were predicted well by linear stability theory. Yiiice the wave velocity decreased with increasing \\-ave number under the flow conditions studied, the fundanieiital wave:, 1) ed the harmonic waves a. they traveled tloivii the column. Our ob.servations on the occurrelice of bimodal waves caii be summarized by the following rule of thumb: If the firht harmoiiic of the impresed fuiidanieiital ih much more highly ainplified than the fundamental, only the first harmonic is ohserved. If the fundxmeiital and its first harmonic have comparable amplificatioii rates, both waves are observetl. If the first harmonic is a ,stable wave, only the fundamental is observed. Data on amplification rates were taken only where the fundamental was the dominant wave. Bimodal or “multiple frequency” naves have also been reported by Tailby and Portalski (1962), Jones (1965), and Hallett (1966). x
Wave Amplification and Wave Velocity. Traversing the illuminator-photoconductive cell assembly down the column for input, disturbances of fixed amplitude and frequency gave data for t h e change of wave amplitude ivith distance. Figure 3 shows typical wave growth in distance for a 1.8-cps wave ( a = 0.147) and a 2.8-cps wave ( a = 0.233) aiid decay for a 4.0-cps wave ( a = 0.346); these data are for Re = 0.289, < = 2.77, aiid p = 90”. I n all cases, waves of sufficientlJ- small amplitude were seen t o grow or decay exponentially in distance, as would be predicted for ail infinitesinial wave. Exponential growth persisted until t,he wave amplit.ude approximated the film thickness or equilibrium amplitude. One important observation n-as that as long as the waves remained two-dimensionals the wave length and wave velocity remained constant, within experimental error, with distance downstream even when amplitudes approached the equilibrium amplitude. This observation may have some important implications for the nonlinear stability theories for t,his flow. From the .-lope:: of wave profile plots obtained for varying input frequencies a t fixed Reyiiolds number, surface tension parameter, niid column angle, the amplification factor, act, as a fuiictioii of n-ave iiuniber was deterniiiied. The flow conditiolis aiid range of wave number.? studied in the present experinleiits are summarized in Table 111. Represeiitative amplification factor curves along n-ith the wave velocit’y data are compared to the theories in Figures 4 through 8. For these plots, t.he wave velocity h:is heen made dimeiisionless by division by the mean velocity, a. The vertical arrow 011 the wave iiurnber :axis iii several of these figures corresponds to the experimentally det,ermiiied wave number for waves arising from room disturbances. Yih’s (1963) low wave number theory used here is accurate to t,erms of order a 2 ;Yih’s (1963) low Reynolds number theory is accurate t’o zero order in Re; Ueiijamin’s (195i) theory (Benj) is accurat,e to sixth-order in a aiid to third-order in CY R e ; the t’heory of Ankhus and Goreii (1966) is referred to as .i-G; the momentum integral theories here are referred to as K-G (both methods of solving the momeiituni iiitegral problem gave nearly identical results a t the low Reynolds numbers involved in t,his study).
lo-’
-
-5 a 5 Q
IO‘
6
10
22
14
26
30
34
x (in)
Figure 3. Typical data showing exponential growth or decay of wave amplitude in distance 96 Ind.
Eng. Chem. Fundarn., Vol. 10,
No. 1, 1971
Table 111. Flow Conditions and Range of W a v e Numbers Studied in Present Experiments Re
0.127 0,191 0.269 0.289 0.379 0.489 1.29 0.144 0.547 1.19 0.476 0.708 0.441 0.925
P 1.57 2.00 2.11 2.77 2.47 2.47 2.89 1.88 2.46 2.66 11.2 10.3 9.60 9.40
We-'
0.0204 0.0316 0.0533 0.0455 0.0805 0.123 0.527 0.0211 0.148 0.500 0.0259 0.0548 0.0266 0.0346
P 90" 90" 90' 90" 90' 90' 90" 74.5' 74.5" 74.5" 90" 90" 74.5" 74.5'
Figure a-Range
No.
0.209-0.405 0.170-0.344 0.157-0.516 0.121-0.506 0.191-0.543 0.201-0.686 0.293-0.726 0.098--0.291 0.127-0.520 0,148-0.666 0.141-0.353 0.143-0.422 0.122-0.332 0.106-0.501
, ,
,
, ,
.
., , 4 , ,
.
5 6 7 8
_.. .., ... .,. .. .
-0 - 0 0960
0 IO
020
030
040
050
a
L
u
0
0.10
0 20
0.30
0.40
050
a Figure 4. Comparison of data for amplification factor and wave velocity as functions of imposed wave number with various theories Re = 0.289, = 2.77, We-' = 0.0455, @ = 9 0 '
-0.08
-0.12-
The influence of increasing Reynolds number on the amplification factor for vertical columns and nearly constant surface teiision parameter, [, is seen in Figures 4, 5, and 6. The maximutn amplification factor observed varied from 0.00084 a t a = 0.21 for Re = 0.127 to 0.13 a t a = 0.41 for R e = 1.29. The wave number and wave velocity for neutral stability were 0.23 and 2.9 (estimated), respectively, for Re = 0.127 and 0.75 and 2.3 for Re = 1.29. The 101%wave number theory of Yih (1963) is in good quantitative agreement with the data for wave numbers below about 0.2. For wave numbers greater than 0.2, Yih's theory gives progressively poorer predictions as the Reynolds number is increased; a t Re = 1.29 the theory is in serious error. The theories of Benjamin (1957) and Anshus and Goren (1966) and the two integral theories described above all gave very similar predictions and were in good qualitative agreement with the data over the entire range of wave numbers and Reynolds numbers investigated. A t the highest Reynolds number, Re = 1.29 (Figure 6), the integral theories give a somewhat better prediction of the amplification factor for the most highly amplified wave; this may be due to large amplification rates, since there the wave is doubling its amplitude in a single wave length and the approximation of growth in time used in the other theories may be breaking down. These theories consistently underestimated the amplification factor. The theories, and especially that of Anshus
0
I
I
0.2
0.4
0.6
a
a Figure 5. Comparison of data for amplification factor and wave velocity as functions of imposed wave number with various theories Re = 0.489, { = 2.47, We& = 0.1 23,
P = 90'
and Goren, predict the wave number of maximum growth and of neutral stability well. The wave velocity dat'a shown in Figures 4, 5, and 6 are not nearly as sensitive for testing the various theories as the amplification factor data. A11 the t'heoriea, except Yih's lowwave-number t,heory, predict. the observed wave velocities within the experimental error of 4y0 over the eiitirc range of Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
97
- 0.004 -0.008
.-
0
-0.012
-0.016
-0.020
-0.024
oL 3.0
2.8
0.1
I
02
0
I
1
04
1
06
0
OB
a Figure 6. Comparison of data for amplification factor and wave velocity as functions of imposed wave number with various theories Re = 1.29, = 2.89, We-' = 0.527, p = 90'
r
experimental conditions investigat'ed. The wave velocity was independent of the wave amplitude. A referee supplied an unpublished equation for the complex wave velocity accurate to the third order in wave number derived by Yuan in 1966 using the method of Yih's (1963) power series expansion in wave number. For most of our experimental conditions, bhis equation differs only slightly from the second-order approximation of Yih, and a t the higher Reynolds and wave numbers still seriously overestimates the amplification factor. There is no change in the predicted wave velocity curves. This referee has also indicat'ed that Graef (1966) has made ext'ensive numerical computations for this stability problem. The effect on the amplification factor of varying the column angle a t nearly constant Reynolds number and surface tension paramet,er is seen in Figure 7. Whereas an unstable wave was observed a t Re = 0.127 for a vert,ical column, no unstable wave 1Tas observed a t Re = 0.144 for an angle of inclination p = 74.5'. This is to be expected, for t'he bifurcation point in the neutral stabilit'y curve occurs a t Re = 5/6 cot, p = 0.198; no infinitesimal disturbances which grow should be found for Reynolds numbers less than 0.198. Figure 8 shows the amplification factor us. wave number for an inclined column a t a higher Reynolds number, Re = 0.547; here there are unstable waves, but their growth rates 98
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
a
0.2
0.3
Figure 7. Comparison of data for amplification factor and wave velocity as functions of imposed wave number with various theories Re = 0.144, I = 1.88, We-' = 0.021 1 , p = 74.5'
are considerably reduced from that for vertical columns a t the same Reynolds number. The agreement of the theories with the data is comparable to that for vertical columns. The data for higher { (not shown here) demonstrate that increasing surface tension a t constant Reynolds number reduces the amplification rates considerably, in general agreement with the theories. The data presented here are not a definitive check of the various stability theories due to the restricted range of Reynolds numbers. It is doubtful that a detailed check of the various stability theories will be possible, since the rapid growth of room disturbances a t Reynolds numbers greater than -1.3 precludes the possibility of accurately measuring wave numbers and wave velocities for all but the most highly amplified wave. Wave Number and Wave Velocity for Neutral Stability. The wave number and wave velocity of neutrally stable waves on vertical columns for No. 5 and S o . 15 Chevron white oils (j- = 2.33 and j- = 10.1, which represent average values of j-) are compared in Figures 9 and 10 to the momentum integral theories developed here, the theory of Anshus and Goren (1966), the theory of hIassot et al. (1966), and t h e numerical solutions of Whitaker (1964) and Sternling and Towel1 (1965). These d a t a were obtained by interpolating the measurements to zero amplification. No other data are available from the literature for comparison, since this represents the first study of the amplification of disturbances of controlled amplitude and frequency. The data have been correlated using the reciprocal Weber
74.5'
0.032
I
,
!
i
0.016
0
-0.016
7
-0.032
I--+
-0.040
M
-0.064
0
0.I
0.2
0.3
I
a
10.3
I
IO"
10'2
a
10
102
We-'
Figure 8. Comparison of data for amplification factor and wave velocity as functions of imposed wave number with various theories Re = 0.547, < = 2.46, We-' = 0.1 48, p = 74.5'
Figure 9. Comparison of wave number for neutral stability as a function of reciprocal Weber number for vertical columns with various theories 0 No. 15 white oil; 0 No. 5 white oil
3.0
2.8
2.6
2.4
cr 2.2
2.0
I .8
1.6
I O3
Id2
10-1
I
10
I 02
We-' Figure 10. Comparison of wave velocity for neutral stability as a function of reciprocal Weber number for vertical columns with various theories 0 No. 1 5 white oil; 0 No. 5 white oil Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
99
0.7
0.E
0.5
0.4
U
0.z
0.2
0.I
0
however, cI decreases slowly with increasing wave number for the flow conditions studied here. The wave number and wave velocity of the most highly amplified wave for No. 5 and No. 15 Chevron white oils ({ = 2.33 and = 10.1) are compared in Figures 11 and 12 to the predictions of the momentum integral theory developed here and the theory of hnshus and Goren (1966), as well as the numerical solution of Whitaker (1964). Also shown are data for water ({ = 4887) taken from the lit'erature. I n distinction from t'he properties of neutrally stable waves, the wave number and wave velocity of bhe most highly amplified wave are (two-variable) functions of both the Reynolds number and surface tension group. To compare these results wit'li those of neutrally stable waves, we again have plotted a and cr us. We-' with { as a parameter. The literature data for water were taken on waves observed in the absence of imposed disturbances of controlled amplit'ude and frequency. The considerable scatter in the data taken by different investigators may well be due to the suscept,ibility of mater to surface-active agents, the irregular nature of room disturbances, or our use of a n average value for j-. The wave velocity predictions of the two momentum integral solutions are nearly identical and only one curve is shown in Figure 12. The wave number data are again a more sensitive test of the t,heory than the Ivave velocity data, as all the theories fall within the scatter of the wave velocity data. A11 the t'heories predict the most highly amplified wave number data for the oils well. The integral-eigenvalue solution, Equation 2, is in closer agreement with the data than is Equat.ion 1. Assuming a modified parabolic velocit'y profile or fourth-order polynomial for the st,ream function (cubic polynomial for the velocity) bccomes a progressively poorer assumption as the reciprocal Weber iiuniber or surface tension parameter increases because of the corresponding increase in the frequency of the most highly amplified wave. Whereas t,he velocity profile of the long-wave rippled flow may be slightly distorted from the parabolic velocity profile of undisturbed lamiriar flow a t low frequencies, it becomes highly distorted a t higher frequencies and a higher order polynomial is needed to describe it accurately. Whitaker's numerical solution is in good agreement over its raiige of applicability. The solutioii of Anshus and Goreii is in remarkable agreement' over the elitmirerange of paramet.ers.
