The Chemical Engineering Journal, 6 (1973) 273-275 @ Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands

Short Communications

273

A note on wave inception in film flow

R. W. ATHERTON and G. M. HOMSY

Department of Chemical Engineering, Stanford University, Stanford, California 94305 (USA)

(Received 29 May, 1973; in revised form 8 October, 1973)

The flow of a liquid film down an inclined plane is a problem with a large and sometimes controversial literature112. The original rectilinear flow solution by Nusselt3 possessed an appealing simplicity, which made the corresponding heat and mass transfer problems fairly easy to solve4. Unfortunately real flows do not exhibit the flat surface assumed by the model, and experimentally determined transport rates for a wavy film are from 20% to 200% greater than that predicted on the basis of steady flows6.

It is now well known that the non-wavy flow of a liquid film down a vertical plane is unconditionally unstable7-9, and one might expect that the film surface would be wavy for the entire length of the column. A tantalizing aspect of the problem, however, is that for low flow rates the film can exhibit both states of flow on the same device. For some length from the top of the plane, the wave inception distance x, the film appears flat. For the rest of the plane the surface is wavy. In terms of linear hydrodynamic stability theory, thus, the growth rate of a disturbance is sufficiently small that an inception time t is required for the disturbance to manifest itself. The use of extremely short columns is common, and allows the use of the Nusselt model as a means of calculating transport properties19 Recent experimental results indicate that waves are virtually absent from the upper- most part of the film1i3’*, in the sense that the ampli- tude of the disturbances is below the response of the measuring device. The wave inception distance is, thus, a function of the amplitude of the waves detectable by a given device, and is to that extent an ambiguously defined parameter.

The wave inception distance is of practical import- ance, since in the region where the waves are undetect- able, Nusselt’s model will allow transport properties such as diffusivities to be calculated from experimental measurements of transport rates.

It is now known that surface contaminants can damp the growth rate of the disturbances and increase

the inception distance 2 for a given Re. This stabilizing effect is a direct measure of the surface elasticityls314. It has been proposed to use wave inception data to determine rheological properties of contaminated surfaces15. However, the results are in poor agreement with surface elasticities obtained by other methods. This fact is not surprising considering how little is known about clean surfaces and wave formation. Thus, the design and effectiveness of two experimental tech- niques could be improved if an accurate method of predicting wave inception distance could be developed.

A linear theory of inception It has long been thought that although linear stability theory ceases to be valid when disturbances are measurable (and hence of finite amplitude), the main features of the evolving wavy structure can be described using the results of linear theory. This idea has been substantiated with regard to the wave length and wave speed of the first visible wave. A host of experimental determinations of those two quantities have shown that the observed wave number and wave speed corre- spond to that of maximum growth rate according to linear theoryz16. In this note we explore the corre- sponding conjecture with regard to the amplitude of evolving waves. We shall use the approach originally suggested by Benjamin in which the growth of a disturbance is assumed to follow linear theory up to the actual inception of visible waves.

According to linear theory the disturbance is a wave of the form

T&C, t) = A0 exp xT exp (x - crt) 1 1

77 is related to the dimensionless film height h, by h = 1 + q(x, t). Here x is a downstream distance made dimensionless with respect to the Nusselt film thick- ness, Mt and cr are the dimensionless growth rate in time and wave speed of the wave of maximum growth rate as determined from linear theory. Note that we have made the usual transformation to express growth in distance in terms of the temporal growth rate of most stability calculations16. The amplitude of the wave is thus A0 exp {X~t/cr} where A0 is a measure of the initial amplitude, and is undetermined in this analysis.

274 SHORT COMMUNICATIONS

We now define w the amplification factor, as the ratio of the first visible wave to its initial amplitude, Viz.

A lnw=ln-==~ Ao cr

Introducing a dimensional distance x, we have

X crho -=- lnw (ycr

(2)

(3)

where ho is the Nusselt film thickness. The right hand side of eqn. (3) is now entirely determined using linear theory, and is a function of Re, <, the physical proper- ties group and the angle of inclination. For a vertical column,

-

& =fl(Re, 5). (4)

Now if the hypothesis that waves grow according to linear theory up to the point at which they become visible is true, then the amplification experienced by a disturbance in becoming manifest should be a constant, depending at most on fluid properties. Equation (2) would then yield a relation between onset distance and well-known results from linear theory. Using experimentally determined values of x we may deduce the amplification factor, and hence test our hypothesis.

Fig. 1. The quantity x/In w vs. Re for water according to linear theory.

Water is presently the only fluid for which sufficient data are available to test this hypothesis. The value of t is fixed, and we need only consider fr a function of Re. Using the theoretical results of Yihssg, Whitaker14, and Anshus17, the functionfr was constructed and is given in Fig. 1. The shape of the curve is quite reminiscent of experimentally determined onset datatLra,ra.

The test of the hypothesis comes by examining the deduced values of w, shown in Fig. 2. The results are inconclusive at best. Stainthorp and Allen’s dataa” were influenced by surface contaminants, and Strobel’s

lnwl I

Fig. 2. The amplification factor w, according to linear theory as deduced from published inception data for water. - - - o Stainthorp and Allen”; - -- n Strobe1 and Whitakert9; ___ A Cerro and Whitaker’t;- - q Tailby and Portalskit ; a Portalski and Clegg. l2

datalg were admittedly approximate21. But the remaining sets show only reasonable agreement. What is apparent however, is the fact that the hypothesis, often assumed a priori, does not hold. This is corrobor- ated by the recent experimental determinations of w for glycerine solutions16. We therefore conclude that wave onset is a manifestly non-linear phenomenon whose description must await development of a non- linear hydrodynamical treatment. Although much work has been done on the problem of prediction of the non-linear equilibrated wave forma~aa~as, the initial-value problem for non-linear wave inception has yet to be considered.

SHORT COMMUNICATIONS 275

Nomenclature REFERENCES

A ACI ci cr fl

h ho

Re

t

UO W

X

x

wave amplitude, dimensionless initial amplitude, dimensionless imaginary wave speed, dimensionless real wave speed, dimensionless function derived from linear stability theory, see eqn. (4) film thickness, dimensionless with regard to ha Nusselt film thickness, cm

film Reynolds number = uoho -, dimensionless

V

time dimensionless with regard to ho/u0 average velocity of film flow, cm/set amplitude ratio, dimensionless distance, dimensionless with respect to ho wave inception distance, cm

1

2

3 4

5

6

10

11 12

G. D. Fulford, Advances in Chemical Engineering, Vol. 5, Academic Press, New York, 1964, p. 151. Kambiz Javdani Milani, Ph.D. Thesis, University of California, Berkeley, 197 1. W. Nusselt, Ver. Deut. Ingr. Z., 60 (1916) 549. R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. S. Banerjee, E. Rhodes and D. S. Scott, Chem. Eng. Sci., 22 (1967) 43. D. R. Oliver and T. E. Atherinos, Chem. Eng. Sci., 23 (1968) 525. T. B. Benjamin, J. Fluidhfech., 2 (1957) 554. C. S. Yih, Phys. Fluids, 6 (1963) 321. C. S. Yih, Fluid Mechanics, McGraw-Hill Book Co., New York, 1969. P. L. T. Brian, J. E. Vivian and S. T. Mayr, Ind. Eng. Chem. Fundomentols, 10 (1971) 75. R. Cerro and S. Whitaker, Chem. Eng. Sci., 26 (1971) 742. S. Portalski and A. J. Clegg, Chem. Eng. Sci., 27 (1972) 1257.

Greek Symbols

(Y wave number = 2nholA, dimensionless

rl surface disturbance function, dimensionless

;

wave length of disturbance, cm physical properties group = a3 1/3/pg1/3@/3

V kinematic viscosity, cmZ/sec

P density, g/cm3 CJ interfacial tension, dynes/cm

13 14 15 16

17

T. B. Benjamin, Arch. Mech. Slos., 16 (1964) 615. S. Whitaker, Ind. Eng. Chem. Fundamentals, 3 (1964) 132. R. Cerro and S. Whitaker, J. Coil. Int. Sci, 37 (1971) 33. W. B. Krantz and S. L. Goren, Ind. Eng. Chem. Fundo- mentols, 10 (1971) 91. B. E. Anshus, Ind. Eng. Chem. Fundomentols, 11 (1972) 502.

18

19 20

21 22

S. R. Tailby and S. Portalski, Trans. Inst. Chem. Engrs., 38 (1962) 324. W. J. Strobe1 and S. Whitaker, A.I.Ch.E.J., 15 (1969) 527. F. P. Stainthorp and J. M. Allen, Trans. Inst. Chem. Engrs., 42 (1965) 185. S. Whitaker, Private communications, 1971, 1972. R. W. Atherton, Engineer Thesis, Stanford University, 1972.

23 S. P. Lin, Phys. Fluids, 14 (1971) 263.