Waves on a thin film of vicous liquid

WAVES ON A THIN FILM OF VISCOUS LIQUID

John Byatt-Smith

REPORT NO. AS-69-4

CONTRACT NO. Nonr-222(79) FEBRUARY 1969

COLLEGE OF ENGINEERING

UNIVERSITY OF CALIFORNIA, Berkeley

ARCHIEF

Technische Hogeschool

Deift

WAVES ON A THIN FILM OF VISCOUS LIQUID

John Byatt-Smith

This document has been approved for public release and sale; its distribution is unlimited

FACULTY INVESTIGATOR:

M. HOLT, Professor of Aeronautical Sciences

DIVISION OF AERONAUTICAL SCIENCES UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720

CONTRACT Nonr-222(79) SPONSORED BY

REPORT NO. AS69-4 U. S. OFFICE OF NAVAL RESEARCHI

In this paper we consider the problem of the flow of a viscous in-compressible fluid down an inclined wall. A solution is obtainedby assuming that the free surface is a wave of low frequency. The

solution is numerical and the results are compared with existing theories and available experimental results.

1

1.0 INTRODUCTION

In this paper we will consider the flow of a viscOus liquid in a thin film. Such flows are often observed in everyday life; for example, when rain runs down a window pane or when paint drains from some solid

object which has been dipped in it. This is also a subject of importance in chemical engineering and has been studied by. experimenters in that field. The character of the flow has been shown to depend largely on the Reynolds number, although surface tension is important in most cases. For example, experiments show that in flow down a vertical wall, the motion is turbulent when R is greater than 300 (Jeffreys, 1925). When

R is less, the mean flow is governed by a law of laminar friction, and

the mean depth is approximately that given by the simple theory of Nusselt (1916) and Jeffreys (1925) who assume uniform Pouseuille flow. Nevertheless, waves of various amplitudes are observed in all laminar flows, except those at very small R.

The stability of the uniform solution given by Nusselt and Jeffreys has been studied by several authors (Kapitza, 1948) ,Yih (1954), Benjamin (1957), and Whitaker (1964). These theoreticians have shown that there is a critical value of R that depends on the value of the surface tension and the angle of the inclination of the wall, such that only flows that have R less than this critical value are stable. When the

wall is vertical, Benjamin has shown that the critical Reynolds number is zero; that is, all flows are unstable. However, he also shows that for small Reynolds number the amplification of the most unstable wave is very small, but that it increases rapidly as . R increases above 4. This

Thus the onset of instability of the uniform Poiseuille flow is well understood. The problem of finding the free surface profile after the onset of instability has not received much attention. In fact, the. various authors that have turned their attention to the problem have studied only infinitesimal disturbances of the uniform solution. This theory gives undamped sine-waves. There are many observations that are not explained by this theory and there are .many reasons to doubt the validity of this theory. For example, in most observed waves the solu-tion is not a sytmietric curve like a sine wave., even when the amplitude is small. Also in several cases, observed waves can have amplitudes comparable with the mean depth of the water.

Here we will use a non-linear theory rather similar to the cnoidal wave theory of inviscid flow in the hope that the results of the theory will apply to finite amplitude waves.

2.0 The Significant Results From Prior Work

The first major contribution was the 'solution given by Nusselt

(1916). As mentioned before, he assumed.the flow to be uniform and obtained Poiseuille flow. His results take on the form

where

u=- (y-0)

h0

gh2

=i

_f

udy---

sine,

0 0

(2.1)

(2.2)

where u is the velocity component parallel to the wall which is

3

This theory is invalid when waves start to appear, but is widely used as a first approximation. Later investigators have, in fact, started from the assumption (2.1) in a local sense where and h0 vary slowly; that is, they assume

u = 3 (y - -y2/h0(x,t) . (2.3)

They then use the x momentum equation to find an equation for the height. To eliminate the pressure, they also have to assume that the pressure gradient perpendicular to the wall is the same as that given by the hydrostatic approximation. The height is linearized with re-spect to the departure r from the mean height and a third order

differential equation for n is obtained. We shall not assume that the solution has the form (2.3) nor that the pressure across the film is hydrostatic. This gives rise to extra terms in the final equation for

r which are of the same order as those already obtained. However, we

shall assume that the free surface varies slowly. This assumption will be explained fully later. We will also keep the significant non-linear terms and discuss how they affect the linear solution.

3.0 The. Basic Equations

The motion is assumed to be two-dimensional flow of an incompressible viscous fluid down a wall of inclination 0 . The x axis is taken

in this direction and the y axis perpendicular to it. This system of coordinate axes is shown in figure 1.

The equations governing the motion are the Navier-Stokes momentum equations, with the equation of continuity

+ av

ax ay - 0 (3.1)

+ u + v

= -

I ..2.

+ vV2 u-g sinO , (3.2)

ax ay p ay

+u- +vg

+

vv2v-gcose

. (3.3)

Equation (3.1) implies the existence of a stream function p such that

u

= ap/ay

v

= -

a'p/ax . We now look for a solution where the height

h is a function of the single variable

X=x+ct

(3.4)

where c is the velocity of wave propagation.

We now, assume that the dependence of on X can be replaced by a dependence on h and its derivatives.

Hence,

= p (y, h, h', h" .

. .) . (3.5)

Anticipating that the solution is a wave of low frequency for which the X derivatives are in decreasing order of magnitude, we neglect

all products

of

derivatives and also all derivatives higher than 'the

third.

Therefore p can be expanded as

= 1(yh) + ip2(yh) - + ip3(yh)

h

We now substitute for i in equation (3.3) and integrate with respect to y to find the pressure.

To our order of approximation, equation (3.3) becomes

i _- a2

(__+c)_

₂

_py

y=h -p p R 2 - - 2 2. -(3.7)

The equation expressing the continuity of normal stress across the free surface may be written as

aty=h ,

(3.8)

where , a constant, is the pressure just outside the film and R the radius of curvature.

To our order of approximation

2

-.

(3.9)

Therefore

1h

h g(y-h) coso h

)L

3.10) cose,

Now we substitute for P in equation (3.2) to get ah 2 a2h 3

ah

(

-+

c ay ayah + + ah + + }

-

a2

ah

ah

3 4

=-ycosO-+v{

₃

+.

₃₊

₃ } X ax ay ax _ay 2 a3h +2v [ _{ayah 2} + ;;;:3- ]

ah

p 1 a2

+ -

2 33h 3 h

-s-

(--c)_

dy - ayah ;;;2-ax y=h ax

a3

+

r

(3.11)

The boundary conditions are

=..=

_ay 0

aty=0

(3.12)

and the condition for zero tangential stress at the surface is

P:ns

= 0 (3.13)

where P is the pressure tensor, n the unit normal and s the unit

That is

ah h 2

1/2

= ( 1

)/ {1 + (F))

$ =

(1 ,) /

1/2

ax

So equation (3.13) becomes

_

+p

-p

=

xx

ax xy

_yy

or

ay2 ax2 0

at y=h

(3.14)

We now choose

2 , , and

so that the coefficients

of

ah/x,

a2h/ax2

and

a3h/3x3

in equations (3.11), (3.12) and

(3.13) are identically zero.

The final equation fOr h

coming from the.

conservatiOn

of mass, namely

y=h +

c h

=

const.

(3.15)

Equation (3.11) thus gives

V

g sine

,

_(3.16)

ap

aip1 v ay3

-9 cose

+ (

c ) _ahay 3y2 .

(3.17)

a3p a atp a2ip a2p

v

.

-

c

-

-

-

v

-ay ay ahay

a

+ V (

-V ₍ ay3

+c)

+V (s)

y=h 2 2

Equation (3.15) now gives

gsinO

_{( hi}

-v ' 2 3! h

f

( c )

-

dy , (3.19) y 0

-

T-2v-We now choose a typical height h0 as that attained by uniform Poiseuille flow with a given flow rate _Q . This also introduces a Reynolds number R.

at y = h . (3.2.1)

(3.22)

The boundary conditions then become

=

- 1

-ay ay y

at y = 0 _(3.20)

Therefore -ij (he) _{= Q} and 1/3 h = {3Q/(y sine)} 0

This gives a typical scale

= ._. = _{g sine h Iv}

and hence the Froude number as

- =

(QSiflO)'

.

RsinO1"2

-u0

g0

₃₎

- 3

The final parameter we introduce is a non-dimensional wave speed,

a. = (3.26)

Substituting for in equation (3.17) and integrating gives:

= . [ ( - t!. ) cose 3cF2 h2y2 h0 4!

22!

+

YF2h

- - -) ] (3.?7)

Similarly i3 and i4 are found by integrating equations (3.18) and (3.19).

In evaluating p4 we make the further assumption that the film is

so thin that the term involving surface tension is the most important in equation (3.19)

9

(3.23)

(.3.24)

We make the final equation for h non-dimensional by putting X = h0 , h = h0 (1 + ri).

We also approximate by taking the terms up to order 2 in

up to order in and by taking only the constant terms in

and

Hence at y = h

+ch

gh

- sinO [ a(l i-11) - (1 + 3

+

32)

] gh2 15a F2 (1 + 4) = - [cose (1 + 3) - 8 + - F2 (1 + 611)

]

gh0

r

27 F2 222 9F2 431

-L -sine

_Siflo

COSO _{+ 8!}

a , 16 cosO + 9 F2 2409

3'

5! 8! rh3 h3 1/3 I 0

-

0. ' "4 - 3v -

gv

where is defined as 1/3 = r/(gv)

and depends only on the liquid.

The final equation for n is then obtained from equation (3.15) as

+ 2

F2 3416

11 , sin.e 27F 227 + 9F2431 c , 16 9F22407 (3F)4 ' - 1 sjn2O 7T 8! ' COS e 8!

+ ci2F2 3416 -

I

cote (1 3ri)

l5F2çl+4n) + 81F2(i+6nJ 8 sin e 40 sin.O

+ CL (l+n) - (l+3r +

32)

= CL - 1 . (3.28)

4.0 The Existence of Periodic Solutions

The final equation for the departure, n , from the basic height is

a third order non-linear differential equation with constant coefficients. Equation (3.28) can be written in the form

A'"+Bn"+(Cn+D)n'+Eri+Fn0

(4.1)

where A, B, C, D, E and F are constants

If the equation is linearized about r = 0 , the equation becomes

A n'" + B

ri" +

0

ii'

+ E n = 0 , (4.2) This equation has as solutions

3

1=1

A e

_(4.3)

where A are constants and the are roots of the cubic equations

A3

+

B2

+ D + E = 0 (4.4)

The only periodic solution that is bounded occurs when two of the roots of equation (4.4) are pu imaginary. Then the so1utio is just a sine wave. We also require that all solutions of the form (4.3) tend to

a sine wave as x so we must have the real root greater than

zero. Massot, Irani, and Lightfoot (1966) obtained an equation similar to equation (4.1) and proposed that this was the only boUnded solution. They gave a relation between certain physical parameters so that the

roots of the equation corresponding to (4.4) had two pure imaginary roots, that is AE = BD . However these solutions have an arbitrary amplitude and no acceptable method for determining the amplitude was found. More over these solutions are only valid when the amplitude is small. If,

however, the real parts of the two complex roots of equation (4.4) are negative then the oscillating part of the linear solution will have a growing amplitude as the waves proceed downstream. This is analogous

to inviscid irrotational waves where the linear sine wave is only a good approximation to the non-linear cnoidal wave when the amplitude is

small. Also the non-linear cnoidal wave provides a definite amplitude. from the initial conditions. Again when the stream is super-critical, the linear solutions are exponentials which are modified by the non-linear terms to form waves. So we hope that the non-linear terms in equation (4.1) will modify the linear solution of (4.2) in such a manner as to give a finite amplitude bounded solution with a given amplitude.

5.0 Discussion of the Numerical Results

Equation (3.28) was integrated fOr the case of water flowing down a vertical wall. We first notice that the coefficient of

i"

in equation (3.28) is negative. This means that when the parameters F and are chosen so that a solution of the linear equation (4.2) is an undamped sine wave, all solutions tend to this solution as x -'- - .

13 5.1 Wave Description

The parameters F and were first chosen so that the

corresponding cubic equation (4.4) had two complex roots whose real part was zero. When the differential equation was integrated with arbitrary

initial conditions, it was found that all waves were damped out as the integration advanced down-stream. This indicated that the undamped sine wave of the linear equation (4.2) was indeed only a solution for

in-finitesimal amplitudes.

The parameters were then altered so that the real part of the complex roots of the equation corresponding to (4.4) was negative. Then when the resulting equation was integrated, it was found that all solutions tended to the same solution as x - - . When the amplitude was small the solution was a single-peaked asymmetric oscillation which was steeper' on the downstream side of the crest than on the upstream.

As the parameters were altered so that the real part of the complex roots of the equation corresponding to (4.4) became more negative, it was found that.in general the solution fell into two categories. For high values of the Weber number, that is thick films, the solution remained a single peaked asymmetric oscillation whose amplitude and wave length increased until finally it was impossible to find a bounded solution. For

low values of the Weber number, that is thin films, the single peaked solution split up into a double peaked solution. Then the double peaked oscillation split up into a four peaked oscillation. Next the four peaked solution split up into a very irregular solution in which no periodicity could be found. Finally, as in the case where only a single peaked oscillation existed, no bounded solution was found. These solutions are all shown in figures (2) 'and (3).

5.2 _{Wave Length and Wave Velocity}

Figure (4,) gives the non-dimensional wave velocity of infini-tesimal sine waves on water The curve, however, varies from fluid to fluid not as in the theory of Massot, Irani, and Lightfoot (1966). The present theory shows that when 0, and F are given, there exists a range of wave velocities, less than the critical wave velocity for infinitesimal sine waves, for which a steady periodic solution to equation (3.28) exists. For each wave velocity the wave length and amplitude are unique.

Also on the graph are the results of the numerical integrations of the Orr-Somerfeld equation. These results were obtained by Whitaker (1964) and represent the wave velocity of the most unstable infinitesimal

wave. We would expect that the present theory for the case of zero amplitude and the theory of Whitaker would be in reasonable agreement. However, the present theory is essentially a long wave theory and the discrepancy could be accounted for by including the remaining linear terms, that is, the higher derivatives, that should occur in equation

(3.28). The two theories differ most when the Weber number, and hence the Reynolds number, is large. It is in this case that the neglected terms have a bigger effect. Equation (3.28) shows that the coefficient of n' has a term proportional to F2 and that the coefficient of n"

has a term proportional to F4 . It is easy to deduce that the coefficient of the higher derivatives that have been neglected in the derivation of equation (3.28) have a term that is proportional to a corresponding power of

F2.

15

with that of Whitaker (1964), then we could explain the scatter of ex-perimental results. The theory shows that for given Reynolds and Weber numbers there is a range of values of the wave velocity that gives a steady wave train.

The wave number (2h0/wave length) is given as a function of the Weber number in figure (5). The results are qualitatively in agreement with experimental results but again they are not as good as those obtained by Whitaker (1964).

6.0 Concluding Remarks

We have derived an equation for the steady state of the free surface of a thin film of a liquid. We have shown that the equation has finite amplitude solutions which are qualitatively in agreement with observed wave forms. The equation derived is, however, only valid provided

that Reynolds number is less than about 20. When the Reynolds number is above 20, we would have to include more of the higher derivative terms to obtain more accurate results.

REFERENCES

Benjamin, 1. B. 1957 J. Fluid Mech., 2, 554. Jeffreys, H. 1925 Phil. Mag. (6) 49, 793.

Jones, L. 0. and Whitaker, S. 1966 Amer. Inst. Chem. Eng. Journal, 12,

525.

Henratty, 1. J. and Hershman, A. 1961 Amer. Inst. Chem. Eng. Journal, 7, 448.

Kapitza, P. L. 1948 Zh. Eksper.i Teor. Fiz. 3,18. Kapitza, P. L. 1949 Zh. Eksper.i Teor. Fiz. 19, 105.

Massot, C., Irani, F., and Lightfoot, E. N. 1966 Amer. Inst. Chem. Eng.Journal. 12, 445.

Nusselt, W. 1916 Z. VereinDeutcher Ingenieure, 60, 541.

Whitaker, S; 1964 Industrial and Engineering Chemistry Fundamentals Quart., 3, 132.

Yih, L. S. 1954 Proc. 2nd U. S. Congr. Applied Mech., 623. Yih, L. S. 1963 Physics of Fluids, 6, 321.

0.2

0

-0.2

-0.4

-

0.2-

-0.2--j

:

0.2-0

-0.2

-0.4

0

10

20

30

40

50

DISTANCE DOWNSTREAM

70

FIG. 2

TYPICAL: WAVE PROFILE

I I I I

0

10

20

30

40

5&

60

70

DISTANCE DOWNSTREAM

FIG. 3

TYPICAL WAVE PROFILES

0.20

a:

w

a)

0.16

z

w

> 0.12

U)

Cl)

w

-J

z

0

i

0.0:

D

PRESENT. THEORY

FOR WATER

ESULTS OF NUMERICAL

INTEGRATION OF ORR

-SOMMERFELT EQUATION

FOR WATER

WHITAKER

EXPERIMENTS OF JONES AND WHITAKER

io

102

WEBER. NUMBER

3

2

1.6

0

0

0

0

PRESENT THEORY

FOR WATER

EXPERIMENTS OF

A MASSOT AND

IRAN I

O

KAPITZA

JONES AND WHITAKER

HANRATTY

ANI!

0

0

RESULTS OF NUMERICAL

INTEGRATION OF

ORR-SOMMERFELD EQUATION

FOR WATER

WH ITAKER

0

A

0

A

0

THEORY OF MASSOT,

IRANI AND LIGHTFOOT

0

0

A

I I I

A

102

WEBER NUMBER

'0-I

Securit3i Classification

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Waves on a Thin Film of Viscous Liquid

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5. AUTHOR(S) (Last nan,.. JJ,.t name, initial) Byatt-Smith, John

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13. ABSTRACT

In this paper we consider the problem of the flow of a viscous incompressible fluid down an inclined wall. A solution is obtained by assuming that the free surface is a wave of low frequency. The solution

is numerical and the results are compared with existing theories and available experimental results.

linci assified Security Classification 4. KEY WORDS Thin Film Surface Waves Viscous Flow LINK A ROLE WY LINK B ROLE Wi. LINK C / ROLE WY

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