measurement of liquid-film thickness by laser interferometry

Measurement of liquid-filmthickness by laser interferometry

Wali M. Nozhat

Here the variation of a liquid-film thickness at small Reynolds numbers is discussed. The film thicknessmeasurement by laser interferometry corresponds to the liquid flowing on the inner surface of a small-bore glass tube. An adequate theoretical background for the techniques used in this experiment isdiscussed to demonstrate the capability of the experimental technique. An advantage of this method isthat it shows the shape of the thin film on the inner surface of the vertical tube at a point in a horizontalcross section. The results obtained from this experiment show that the flow of liquid films on verticalsurfaces is inherently unstable and three dimensional even at a Reynolds number smaller than 1.© 1997 Optical Society of America

1. Introduction

The gravity-driven flow of viscous liquids in thin lay-ers on vertical surfaces involves an important class ofproblems in fluid mechanics. Many properties of thefilms are a function of their free surface shape, thick-ness, and longitudinal distance of growth in the flowdirection.1 The applications of falling liquid filmsextend over a variety of mechanical and chemicalengineering fields. These films have been used ef-fectively in devices such as condensers, evaporators,wetted-wall columns, packed towers, film reactors,chemical etching, electrochemical plating, crystalgrowth, and chemical conversion in liquid–gas cata-lytic reactors. In coating technology the behavior ofthe liquid film can affect the quality of the final coatedsurface. The hydrodynamic behavior of liquid filmsand their thickness distribution is also of crucial im-portance in many engineering applications.

One of the interesting features of thin-liquid-filmflows is the appearance of an instability in the form ofsurface waves whose wavelengths are much largerthan the thickness of the film. Since the instabilityoccurs at very small values of the Reynolds numberRe ~i.e., Re ,1!, it is important for it to be understood

The author is with the School of Aerospace and MechanicalEngineering, University College of the University of New SouthWales, Australian Defence Force Academy, Northcott Drive,Campbell, ACT 2600, Australia.

Received 17 January 1997; revised manuscript received 14 May1997.

0003-6935y97y307864-06$10.00y0© 1997 Optical Society of America

7864 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997

thoroughly. Although numerous different avenues,both theoretical and experimental, have been ex-plored in the past eight decades, a complete and con-sistent understanding in this field still eludes us.

Theoretical predictions still depend on constraints,such as the assumption of periodic waves or the as-sumption of infinitesimally small wave amplitudes.Experimental methods reported in the literaturehave been more successful, but still the difficultiespresented by random three-dimensional surfacewaves are formidable. Previous experiments2–6

have shown that it is possible to observe the twodimensionality of steady periodic flow only when con-ditions ensuring the two-dimensional character of thefluid flow are produced. These conditions are com-plete symmetry for the production of the flow withrespect to the perimeter and also the similarity ofperturbing factors. Previous experiments2–6 haveshown that, under the conditions therein, the flow ofthin liquid films on vertical surfaces without specialprecautionary measures does not occur as two dimen-sional, but rather the periodic flow assumes a three-dimensional irregular character.

Most solutions to the film problems have been con-fined to planar film flows, although many industrialapplications involve flow over cylindrical surfaceswith small radii of curvature. The flow of liquidfilms in a vertically standing cylindrical tube ~see Fig.1! is chosen in this study since many industrial andchemical devices use this type of geometry. Thewaves in a two-dimensional, periodic, wavy flow am-plify further downstream and then deform aroundthe circumference of the tube. Such deformation isreferred to as a circumferential flow of wave crests

and results in self-centering and three-dimensionaldisturbances where the wave crests randomly as-sume an oscillatory helicallike flow.1,7

In the experiment discussed in this paper we mea-sure the local film thickness bounded by two bound-aries, namely, a solid wall ~i.e., glass tube! and agas–liquid interface. Figure 1 is a diagram of theexperimental arrangement. The technique and itsvalidity were first applied to the measurement of thewall thickness of a glass tube.8 The laser interfero-metric method9 consisted of a glass cylinder whosewall thickness was measured by interferometry andcompared favorably with values in the range 1.55–1.60 mm obtained with a tube micrometer. Someinterferometric methods suitable for the measure-ment of surface characteristics of liquid films flowingon flat surfaces have been reported in Refs. 3 and 4.Considering flat liquid films, single-wavelength in-terferometry has been applied in Refs. 10 and 11 formeasurement of the thickness gradient as well as therate of thinning discussed in Refs. 12–14. The in-terferometric method described in this paper showsthe variation of thickness and free surface-inherentcurvature, i.e., curvature in the longitudinal directionas well as in the circumferential direction.

2. Theory

Figure 2 shows a portion of a horizontal cross sectionof the vertical glass tube shown in Fig. 1. A laserbeam is incident from the right. After refractionand reflection, two rays within the beam, ACHKJGYand BEIFY, are brought to coincidence at point Y ona photographic camera ~Canon F-1!. The beammakes an angle of incidence a and an angle of refrac-tion b when passing from the glass tube to liquid film~H2O!. The bore of the glass tube is cylindrical, al-though a portion of the exterior surface of the glasstube where the beam is incident is planar between Cand G. This helps to eliminate unwanted spuriousinterferences that would otherwise arise from reflec-tions off two circular glass surfaces.

Fig. 1. Diagram of the optical arrangement.

The method is insensitive to R provided that RyD,, 1. We additionally use an approximation of dyR,, 1 ~thin-liquid-film approximation!. A simplifiedtheory based on small-angle and thin-film approxi-mation is introduced where the circular inner sectionof the tube between J and H is assumed planar.With reference to Fig. 2, the line OJ makes an anglea with OX, whereas JY’s angle with OX is ;2a. Onecan then write

y 5 JM 1 MX tan 2a

5 R sin a 1 ~D 2 R cos a!tan 2a. (1)

Using the small-angle approximation, one can findthat visibility for small a is

y 5 Ra 1 ~D 2 R!2a (2)

so that

a 5 yy2D 2 R. (3)

From Snell’s law

ng sin 2a 5 na sin G, (4)

where na and ng denote the refractive index of air andglass, respectively,

y*ytan 2a 5 yytan G. (5)

For small angles tan 2a 5 sin 2a 5 2a and tan G 5 sinG 5 G, so

y* 5 y~nayng!. (6)

The measured y is larger than the value ~y*! thatwould be measured in the absence of refraction. Toa good approximation the quantity to be substitutedinto Eq. ~3! is no longer y but y*. Now one cancalculate the optical path difference ~OPD! betweenrays AY and BY. Since the OPD is a unique function

Fig. 2. Portion of the horizontal cross section showing a fallingliquid film flowing down the inner surface of the glass tube. Twointerfering rays are shown, namely, AY and BY ~representing re-flection from the inner and outer surfaces of the film, respectively!.

20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 7865

of the angle of incidence a, we denote this as OPD~a!.With reference to Fig. 2,

OPD~a! 52nwd

cos b2 2d tan~b!sin~a!ng. (7)

From Snell’s law

sin aysin b 5 nwyng, (8)

where ng and nw denote the refractive index of aglass tube and of water, respectively. Substitutionof Eq. ~8! into Eq. ~7! leads to

OPD~a! 52nwd

cos b2 2nwd

sin2 b

cos b, (9)

OPD~a! 52n2d

cos b~1 2 sin2 b!, (10)

OPD~a! 5 2nwd cos b. (11)

It follows from Eq. ~11! that

OPD~0! 5 2nwd, (12)

OPD~0! 2 OPD~a! 5 2nwd~1 2 cos b!. (13)

Since sin by2 5 ~1 2 cos by2!1y2, one can use a small-angle approximation to find that b2y4 5 1 2 cos by2.By applying this to Eq. ~13!, one can obtain

OPD~0! 2 OPD~a! 5 2~2nwd!~b2y4! 5 nwdb2, (14)

and using Eq. ~8!, one can discover that b2 5 ng2a2y

nw2; therefore

OPD~0! 2 OPD~a! 5 d~ng2ynw!a2. (15)

A bright ~dark! fringe on the screen occurs when

OPD~am! 5 mL, (16)

where m represents one or a successive number offringes, which is an integer, and L is the wavelength.Since OPD~0! 2 OPD~am! 5 DmL, from Eq. ~15! onehas the liquid-film thickness

d 5 ~DmLyng2!~nwya2!, (17)

where D indicates a range of the fringe. Equation~17! is used to calculate the values of film thickness d.The value ~y*! from Eq. ~4! is used in Eq. ~3! tocalculate the angle a after the measurement of y andD. The value of Dm is obtained from the interfero-grams.

3. Experiment

In the experiment we used a 15-mW He–Ne laserbeam ~wavelength, 632.8 nm!. Distilled water wasused as the working liquid film flowing down theinner surface of the cylindrical, Pyrex glass tube.The outer radius of the film ~the same as the innerradius of the tube! was R 5 4.5 mm. Reflection fromglass–water and water–air interfaces produced aninterference pattern on a 35-mm photographic film


~Kodak Tri X Pan! located inside the camera. Thecamera was situated at a distance of D 5 115 mmfrom the center of the glass tube. The film thicknessvariation was monitored over the small circular areadetermined by the diameter of the laser beam, whichin this case was 1.6 mm. The photographic cameraenabled the pictures to be recorded with an exposuretime of as short as 1y500 s. We eliminated un-wanted specular reflections and multiple interfer-ences from two circular glass surfaces by using amask on the outer surface of the tube, as shown inFig. 1. In addition, rays such as BY were incident onthe planar surface ~between C and G! at an angle ~of16°! designed to eliminate specular reflection fromfalling on that portion of the interference pattern ofinterest.

Under laboratory conditions in still air we recordedthe interferograms for steady flow at Reynolds num-bers ranging from 0.5 to 100 at 22 °C. The Reynoldsnumber was defined by Re 5 Qyy, where Q is thedischarge of the stream per unit perimeter of theglass tube and y is the kinematic viscosity. Withthis method we observed and photographed the filmthickness only at one point. From time to time aCCD camera ~no lens! connected to a monitor wasused to provide a continuous, easily viewable pictureof the film behavior.

4. Interferometric Results

Figure 3 shows two interferograms recorded for suit-ably steady films at flow rates corresponding to Re 522 and 53, respectively. Measurements give an in-stantaneous film thickness of 210 mm @Fig. 3~a!# and255 mm @Fig. 3~b!#. Figure 4 was obtained at a smallflow rate corresponding to Re 5 2.66, and the filmthickness was 100 mm. Figure 5 is unlike Figs. 3~b!and 4, suggesting that the film was convex ~wavecrest! and its thickness was recorded at 365 mm.

Fig. 3. Two interferograms recorded for suitably steady films at~a! Re 5 22 and ~b! Re 5 53. ~The values of the thicknesses arediscussed in Section 4.!

Fig. 4. Interferogram recorded for suitably steady films at Re 52.66.

Figure 5 shows that the fringe pattern has a horizon-tal as well as a vertical component. The horizontalcomponent indicates a variation of film thickness inthe vertical direction. The absence of a verticalfringe component suggests that an increase in filmthickness away from the center line may almost ex-actly compensate for the decrease in OPD~a! dis-cussed in Section 2. Note that the inner radius ofthe Pyrex glass tube used in the experiment was 4.5mm, so that dyR is obviously sufficiently small tojustify the thin-film approximation. The angle a in-volved in the experiment was between 2° and 3° ~andb ' 3°!, thus justifying the small-angle approxima-tion. The length HJ over which this approximationis applied is 2d tan b.

The film thickness was calculated from Eq. ~15!after determining D ~i.e., range of the fringe number!from the photographs. The film thickness measure-ment with this method was between 3 and 5% largerthan that measured by the shadow photographicmethod.1 This comparison was made for Re $ 3when both methods ~i.e., measurement of the localfilm thickness in the flow direction by shadow photo-graphic and laser interferometry! were employed sep-arately for the same Reynolds numbers. In thisexperiment the accuracy of quantitative thicknessmeasurement by the interferometric method was be-tween 4 and 8%. Since there was considerable dif-ference in wave crest and trough thickness, a goodand meaningful result was obtainable only for steadyflow and continuous monitoring and photographing.By analyzing a series of photographs, we obtained theminimum and maximum thicknesses at a point for agiven flow rate. Mean film thickness found by inter-ferometric methods is plotted for the different Reyn-olds numbers in Fig. 6. As seen from this figure, the

Fig. 5. Interferogram recorded for suitably steady films at Re 592.

Fig. 6. Mean film thickness obtained from a series of interfero-grams measured at a point for each Reynolds number.

mean film thickness increases sharply until itreaches a Re of 10; then its slope decreases as theReynolds number increases because of the self-centering effects of wave crests that flow toward theneighboring circumferential troughs and carry the

Fig. 7. Demonstration of how the fringe spacing at a circumfer-ential direction decreases ~increases! with a slight decrease ~in-crease! in film thickness: ~a! uniform thickness, ~b! film thicknessdecreasing away from the center line, ~c! film thickness increasingaway from the center line.

Fig. 8. Demonstration of how the fringe spacing at a circumfer-ential direction decreases ~increases! with a slight decrease ~in-crease! in film thickness: ~a! film thickness further increasingaway from the center line, ~b! film thickness not symmetric at thecenter line ~steep front at left!, ~c! film thickness not symmetric atthe center line ~steep front at right!.


liquid there. The self-centering action results fromthe capillary pressure gradient that exists between awave crest and its neighboring circumferentialtrough of a flowing liquid film.15 As a consequence ofself-centering action, an even film is formed aroundthe inner circumference of the tube and flows down-ward under the influence of gravity.

A notable characteristic of this method is that itshows the shape of the thin film on the inner surfaceof the vertical tube at a point in a horizontal crosssection, as seen from the photographs in Fig. 7 and 8.From Eq. ~15! one can see that the angular spread ofa fringe pattern @i.e., the value of a corresponding toOPD~0! 2 OPD~a! incrementing by L# varies as d21y2.If an image has no vertical fringe component such asin Fig. 8~a!, the film thicknesses increase away fromits center line. This can almost exactly compensatefor the decrease in OPD~a!. When measuring thethickness of the film, one can see that the cross sec-tion of the film is not uniform.

The photographs obtained by the interferometricmethod, as shown in Figs. 7 and 8, illustrate how thefringe spacing first decreases ~increases! with a slightdecrease ~increase! in film thickness and its variationaway from the center line. The photograph in Fig. 9of a draining film ~no in flow of liquid! at Re , 0.5shows that the cross section of the film is not smoothand uniform. This indicates that the film is wavyeven at much smaller flow rates.

To observe the fringe pattern in the longitudinaldirection ~i.e., along the length of the tube!, one mustconsider the circumferential spread of the fringe pat-tern. One can observe bright fringes at the center ofthe symmetrical fringe pattern from top to bottom.For normal incidence the spacing between two adja-cent fringes corresponds to a thickness variation ofLy2nw, bearing in mind that the interface pattern isformed by light that passes twice through the wallmaterial. This variation occurs over a longitudinaldistance of ;1.6 mm. The optical path decreaseswith increasing distance away from the center of thefringe pattern. Thus the OPD at a lower fringe issmaller than the one at the upper. It follows thatthe OPD at the center of the fringe ~on the samefringe! is also smaller than the upper. This wouldimply that the film thickness decreases from the topof a fringe to the bottom. Thus one can deduce themagnitude of the lengthwise variation of thicknessand whether this variation is an increase or decrease.

5. Conclusion

A noninvasive interferometric method for the mea-surement of the thickness of liquid films flowing

Fig. 9. Demonstration of the nonuniformity of the film thicknessobtained at Re , 0.5 of a draining film.


down the inside of a cylindrical glass tube has beendescribed. A simplified theory based on the thin-liquid-film approximation was introduced ~describedin Section 2! that is valid for dyR ,, 1. The methodis insensitive to R provided that RyD ,, 1. Theaccuracy of the determination of d is limited by sev-eral factors; however, a measurement as accurate as64% is achievable in small flow rates ~i.e., Re # 20!if a steady flow condition and precise determinationsof D and y are ensured.

The advantage of this method is twofold. First, itgives an indication of circumferential spreading ~i.e.,self-centering action! and variation of the film at apoint in a horizontal cross section. Second, we mea-sured the local and mean film thicknesses down theflow direction. The experiment demonstrated that,owing to the presence of waves on the free surface ofliquid film, the fringe spacing may first decrease ~in-crease! with a slight decrease ~increase! in film thick-ness away from the center line. A gross variation infilm thickness away from the center line always re-sults in a decrease in fringe spacing. Photographs ofthe liquid film obtained by this method show that theflow of liquid film on vertical surfaces is nonuniformin thickness, wavy, unstable, and three dimensionaleven at very small flow rates ~i.e., Re # 1!.

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measurement of liquid-film thickness by laser interferometry

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