(Fluid Mechanics and Its Applications 57) Victor Ya. Shkadov, Gregory M. Sisoev (auth.), H.-C. Chang (eds.) - IUTAM Symposium on Nonlinear Waves in Multi-Phase Flow_ Proceedings of the IUTAM Symposium.pdf

2. IUTAM SYMPOSIUM ON NONLINEAR WAVES IN MULTI-PHASE FLOW

3. FLUID MECHANICS AND ITS APPLICATIONS
Volume 57
Series Editor: R. MOREAU
MADYIAM
Ecole Nationale Superieure d'Hydraulique de Grenoble
Bolte Postale 95
38402 Saint Martin d'Heres Cedex, France
Aims and Scope ofthe Series
The purpose of this series is to focus on subjects in which fluid mechanics plays a fun-
damental role.
At> well as the more traditional applications of aeronautics, hydraulics, heat and mass
transfer etc., books will be published dealing with topics which are currently in a state
of rapid development, such as turbulence, suspensions and multiphase fluids, super and
hypersonic flows and numerical modelling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive intense
scientific attention, bringing them to the forefront of technological advancement. Fluids
have the ability to transport matter and its properties as well as transmit force, therefore
fluid mechanics is a subject that is particulary open to cross fertilisation with other
sciences and disciplines of engineering. The subject of fluid mechanics will be highly
relevant in domains such as chemical, metallurgical, biological and ecological enginee-
ring. This series is particularly open to such new multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts are mo-
nographs defining the current state of a field; others are accessible to final year under-
graduates; but essentially the emphasis is on readability and clarity.
For a list ofrelated mechanics titles, see final pafies.

4. IUTAM Symposium on
Nonlinear Waves in
Multi-Phase Flow
Proceedings of the IUTAM Symposium
held in Notre Dame, U.S.A.,
7-9 July 1999
Edited by
H.-C. CHANG
Department ofChemical Engineering,
University ofNotre Dame,
Notre Dame, Indiana, U.S.A.
.....
''
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

5. A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5517-0 ISBN 978-94-017-1996-4 (eBook)
DOI 10.1007/978-94-017-1996-4
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
No part of the material protected by this copyright notice may be reproduced
or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.

6. CONTENTS
Preface
Committee and Sponsors
Thin~Film Waves
Wavy Falling Liquid Films: Theory and Computation Instead
of Physical Experiment
V. Ya Shkadov and G. M. Sisoev
Nonlinear Theory of control of Instability in
a Liquid Film Flow
S. P. Lin
Coarsening Dynamics of Roll Waves
H.-C. Chang and E. A. Demekhin
Oscillatory Shear Stress Induced Stabilization of
Thin Film Instabilities
D. Halpern and J. B. Grotberg
Experimental and Modeling Studies of Wave Occlusion
and Evolution on Free Falling Viscous Films in a Vertical Pipe
E. K. Dao, L. T. Nguyen and V. Balakotaiah
Nonlinear Waves and (Dissipative) Solitons in Thin-Liquid Layers
Subjected to Surfactant Gradients
M. G. Velarde and A. Ye. Rednikov
On Saturation of Rayleigh-Taylor Instability
A. L. Frenkel and D. Halpern
Two-Layer Waves
Weakly-Nonlinear Waves in Two-Layer Flows:
Subcritical Bifurcations
M. J. McCready, M. R. King and B. D. Woods
Interactions Between Interfacial Traveling Waves and
a Long Wave Mean Flow Mode in Two-Layer Coutette Flow
F. Charru
Wave Formation and Drop Emission in a Hele-Shaw Cell
M. Rabaud, L. Meignin and P. Grondret
Numerical Simulation of Two-Fluid flows of Viscous
Immiscible Liquids
Y. Y. Renardy and J. Li
11
21
33
45
57
69
81
93
105
117

7. vi
The Linear Stability of a Core Annular Flow in
a Corrugated Tube
H.-H. Wei and D. S. Rumschitzki
Bubbles and Jets
Cusp Formation and Tip-Streaming Instabilities for Time-Evolving
Interfaces in Two-Dimensional Stokes Flow
M. Siegel
Bubble Propagation in a Flexible-Walled Channel
0. E. Jensen, M. K. Horsburgh and D. P. Gaver lli
Bubble Rising in an Inclined Channel
K. M. Debisschop and M. J. Miksis
Mobility Control of Surfactant-Retarded Bubbles
at Small and Order One Reynolds Numbers
D. T. Papageorgiou, C. Maldarelli and Y. Wang
Spinning Jets
J. Eggers and M. P. Brenner
Wetting/Dewetting
Three Attempts on Dry Wetting
D. Quere, J. Bico and D. Richard
Dewetting of a Hot Coated Solid Surface
S. G. Bankoff and A. Oron
Thermocapillary Droplet Migration and Instability
M. K. Smith and S. W. Benintendi
Bubbly .and Suspension Flows
Pressure Waves and Voidage Waves in Bubbly Liquids
L. van Wijngaarden
Modeling Gas-Liquid Flow Through Pipes of Variable
Cross-Section
P. S. Hammond, A. W. Meredith and J. R. A. Pearson
One-Dimensional Waves in Liquids Containing
Bubbles Flowing Along a Tube with an Elastic Wall
A. Crespo, J. Garcia and J. Jimenez-Femandez
Effective Equations for Flows With Drifting
and Oscillating Bubbles
N. Wang and P. Smereka
127
139
149
159
175
185
195
205
217
229
239
251
261

8. Preface
The IUTAM Symposium on "Nonlinear Waves in Multi-Phase Row" took place from July
7 to July 9, 1999, at the University of Notre Dame in Indiana, U. S. A., with 50
participants from 14 countries.
The symposium focused on the latest developments in multi-phase hydrodynamics. Topics
covered include jet breakup, voidage wave/instability in suspensions and bubbly flow, roll
wave formation, air entrainment, cusp formation, ribbing, viscoelastic and surfactant
effects on interfacial dynamics, rivulet and filament dynamics, wetting/dewetting and wave
dynamics on thin films. The enthusiastic participation from a very youthful and diverse
group of researchers attests to the robust state of multi-phase flow research.
It is my pleasure to acknowledge the efficient and tireless help of Mrs. Karen Jacobs and
Mrs. Jeanne Davids of the Chemical Engineering Department at Notre Dame.
Hsueh-Chia Chang
Notre Dame
April, 2000
vii

9. International Scientific Committee
H.-C. Chang (USA, Chairman)
A. Crespo (Spain)
S. Davis (USA)
J. Fabre (France)
T. J. Hanratty (USA)
M. J. McCready (USA)
M. Renardy (USA)
L. van Wijngaarden (Holland) (ex officio)
Local Organization Committee
H.-C. Chang(Chairman)
M. J. McCready
Sponsors of the IUTAM Symposium on Nonlinear Waves
in Multi-Phase Flow
International Union ofTheoretical and Applied Mechanics (IUTAM)
Center for Applied Mathematics, University of Notre Dame
The Provost's Office, University of Notre Dame
Department ofChemical Engineering, University of Notre Dame
College of Engineering, University of Notre Dame
Bayer Chair Funds
Kluwer Academic Publishers
ix

10. WAVY FALLING LIQUID FILMS: THEORY AND COMPUTATION
INSTEAD OF PHYSICAL EXPERIMENT
VICTOR YA. SHKADOV and GREGORY M. SISOEV
Department of Mechanics and Mathematics,
Lomonosov Moscow State University,
Moscow, 119899, Russia
We discuss a correlation of theory and experiments for classical problem of
regular two-dimensional waves in vertically falling liquid films. The base mathe-
matical model was developed in paper [1), the main mathematical problems were
formulated in [1-4]. The papers [5- 11] contain detailed investigations of strongly
nonlinear wavy solutions of the base model (1). There are many examples of the
successful theoretical explanation of separate experimental observations obtained
up to now in frame of the base model. At the same time the wide dispersion of
experimental points for slow waves have not been explained. We intend to give
not only explanation for that phenomenon but to construct also the procedures to
compute fitting functions for fast and slow waves by the means of theory.
1. Mathematical Model
Exploited system known as Shkadov model has been formulated in (1]. In
dimensionless form two evolutionary equations for film thickness h and local flow
rate q depending on time t and a coordinate axe x directed along solid plane are
ah oq _ 0
at +ax- ,
aq 6 a (q2 ) 1 ( a3 h q)
at +5ax h =5J h8x3 +h- h2
and contain single external parameter of similarity J introduced in [4]
J = _1_(pH!lg4)1/3
45v2 rr
(1)
(2)
Here p and v are density and viscosity of liquid, rr is surface tension, g is the
gravity and H. is mean film thickness. Relations of dimensional values denoted
by upper wave and dimensionless ones are determined according to
- H. _ H. -
t =--t, x = -x, h = H.h, ij =H.U.q,
KU. K
H.-C. Chang (ed.),IUTAM Symposium on Nonlinear Waves in Multi-Phase Flow, 1- 10.
© 2000 Kluwer Academic Publishers.
(3)

11. 2
H2 (202582 )
1111
u
u. =93,;, "'= -:ys- ,"f = p(v4g)l/3.
(4)
Liquid properties and film thickness H. effect on wave development through pa-
rameter 8 simultaneously. Scales of time and space depend on r;,(8, "f) where "f is
determined by liquid properties only. If spatial-periodic solutions are sought the
system contains also inner parameter that is a period.
To found stationary waves having period L and moving with velocity c non-
linear eigenvalue problem for wave shape h(~), ~ = x - ct has been derived in
[1]
h3 h'" + o[6(q0 - c)2 - c2 h2] h' + h3 - Qo- c(h -1) =0,
h(~) = h(~ + L), h'(~) = h'(~ + L), h"(~) = h"(~ + L), (5)
11€+L
I € hd~ =1.
where q0 is flow rate averaged on period L. Last condition in (5) fixes constancy
of mean film thickness. Statement of the eigenvalue problem for h(~), c, qo is
completed taking into account its invariance referring to transformations ~ --+
~ + L~.
In the case of computation of nonstationary periodic waves an initial conditions
at t =0 have to be added to (1).
Solutions of (1) and (5) are presented as finite Fourier series
N N
h = L hk(t)expiakx, q = L Qk(t)expiakx,
k=-N k:-N
hk = h*_k, Qk = q*_k> ho = 1
where wavenumber a =211"/ L. ·Equations for unknown coefficients with help of
Galerkin procedure are formulated.
As it was shown in many papers the system (1) is turned to be the main tool to
obtain numerous solutions for reaJ nonlinear waves observed in experiments. The
reason for this is that all simplifications during system derivation are well grounded
on the principle of uniform approximations. The full mathematical formulation of
the problem in dimensionless form has been justified in [1,2]. Briefly, full Navier-
Stokes problem in dimensionless form is written taking into account relations (3) :

12. 3
70
(a)
0.4 (b)
Re, Kl
R~
35 0.2
0 2000 4000
Figure 1: (a) Functions Re('i') at some <5 values and intervals of ReA in experiments:
1 - 'i' = 3906, 2 - 'i' = 3298, 3 - "Y = 1129, 4 - "Y = 1098, 5- "Y = 491, 6
- "Y = 465, 7 - "Y = 195, 8 - "Y = 105, 9 - "Y = 722, 10- "Y = 1147, 11 -
"Y = 1116, 12- "Y = 2850. (b) Functions ~~:2 ("Y).
y =0 : u =0, v =0, (6)
oh oh
y= h(x,t) : ot +u ox =v,
au + K:2 ( ov + _4_ oh ov) =0 b=K: oh
oy ax 1 - b2 ax ay ' ax'
2~~::2 1+ b2 ov ~~::2 1 82 h
p - - - - - - - -
~~:Re 1- b2 oy- We (1 + b2)~ ox2
where Re = U.H.jv,We= pU'!H./rr, Fr2 = u;fgH•. To select two free param-
eters U. and K: the equalities following from the fact that capillary, gravity and
viscous forces are of the same order in thin film flows are used
3 1 1
(7)
We - ~~:Re - ~~:Fr2 - 5o·
It leads to derivation ofthe formulas (2) and (4). Then our problem contains only
two free parameters <5 and 'i'· Now let us introduce the main simplification -
we intend to consider flows for which ~~::2 « 1. After omitting all terms of order
~~::2 we obtain system of the boundary layer type with self- induced pressure. This
system will be regular if coefficient <5 ::; 1. After applying Galerkin method with
one approximating function we obtain our base system (1).
The main question arises if there are experiments in which conditions ~~::2 «
1 and <5 ::; 1 are realized simultaneously. In Figure 1 experimental conditions
quoted in [13] on the plane ("'f,O) are represented. There are used in Figure 1a
two Reynolds numbers, namely, Re = (59 37"'(3 89) 1111 and ReA from [13] and these
values are related by ReA = Req0 . Since for nonlinear waves q0 E (1, 1.4) it is seen

13. 4
from Figure 1 that two main conditions for the most experiments with regular
waves are satisfied except region of small -y. Therefore our system is justified
and the problem is to investigate its manifold of solutions. Except experiments of
Kapitsa [12) (curve 12 in Figure la) in what o> 0.04 most experimental conditions
are exposed above curve o= 0.1. In connection with this fact some remarks on
the case of small ovalues must be done.
Weakly nonlinear asymptotics of (1) is derived below in accordance with [3].
After introduction of new variables t1 = ant, x1 = anx, an ::: .JIM and their
substitution in second equation of (1) we derive for o« 1
Then the equations (1) lead to
After substitution last relation in second equation of (1) one can receive
_ 3 3 ( 3 fPh 6 oh) ( 6 )
q - h +an h fJxy +h fJxl +o an .
and the first equation of (1) leads to within o(a~) to Benney equation
-+ - + a - -+ -
fJh 3h2 fJh 3 f) (h3fJ3h h6 fJh ) - 0
fJtl OX! nOXj fJxr OX! - .
(8)
With help of replace of variables
we obtain from (8) the equation [3] to within o(a~)
fJH fJ ( 2 83 H fJH)
fJt2 + 0X2 3H - cH + ox~ + OX2 =O. (9)
Nonlinear wave solutions of (9) were constructed in [3) in the first time; full
theory was developed in [14]. Popular weakly nonlinear equations (8) and (9)
which follow from Shkadov system (1) as o--+ 0 don't contain physical parameters
and their solutions describe mathematical waves only.
2. Bifurcations of Regular Waves
Now let us go to discuss the regular solutions of our system (1). Parallel with
main flow solution h = 1, q0 = 1 the eigenvalue problem (5) has extremely rich
set of solutions that investigations have been started in [1 ,5]. These solutions
compose two- parametric manifold studied in detail in [11]. Continuous traces of

14. 5
0.6 0.8
3
2.65
2.3
- --
2 L----L--------~~--------L---------~---------L--------~
0.2 0.4 0.6
Figure 2: Steady waves at 6 =0.15: Ia- the first family, lb - the second family,
/c,l, lc,2 - the intermediate families. Solid and dashed curves denote families of
one- and two-humped waves correspondingly.
this manifold on cutsets 6 =const are named families of waves. As an example
there are presented some families projections on plane (s, c) where s ::: a/an in
Figure 2. All families arise in bifurcation points placed in the vicinity of values
sn =1/n, n =1, 2, ... as nearly harmonic waves with period nL and are continued
to s -+ 0. The limiting wave profile as s -+ 0 is solitary wave with one or two
main humps [6]. All families are divided on fast or slow ones with respect to value
c =3 that is velocity of linear perturbations of main flow. To goal of this paper it
is sufficient to consider families of three types only.
The first family Ia go on from s =1 and there are not solutions with smaller
velocity c at given 6. The second family /b consists of waves with maximal veloc-
ities which belong to the small wavenumber domain. There are two intermediate
slow families /c,l, /c,2 which waves are fastest in two intervals of s in Figure 2.
Now from all solutions we'll consider only those ones which have extremely
properties, namely, their phase velocity c, amplitude a and mean flow rate q0 are
maximal at every fixed wave number s. We refer to this solutions as dominating
waves. From Figure 2 it is clear that at 6 = 0.15 the set of dominating waves
includes 4 pieces belonging to described families /a, lb, /c,l, lc,2 · There are jumps
of the dominating waves parameters at values of s dividing one piece from the
other.
As 6 grows the point of bifurcation of the second base family lb moves to
s -+ 0 and the quantity of intermediate families increases [8,9]. According to this
movement a number of the pieces constituting set of dominating waves is grown.
For example, this number is equal to 2,3,5,6,7 at 6 =0.04, 0.1, 0.2,0.225,0.247 and
so on.

15. 6
2.2
(a)
1.9
(b)
h hm
m
1.8
1.5
1.4 4
100 50 100
Figure 3: Formation of limiting regimes from small disturbances: (a) main flow at
J = 0.15: 1 - s = 0.15, 2- s = 0.3, 3 - s = 0.4, 4- s = 0.466, 5 - s =0.8;
(b) steady nonlinear waves of family 'i'c,l at J =0.1: 1 - s =0.2, 2- s =0.25 ,
3 - s =0.318, 4- s =0.477, 5 - s =0.5.
3. Dominating Waves as Global Attractors
Nonuniqueness of solutions of (5) at given governing parameters J and s leads to
necessity to select such ones that are realized in experiments. Initial study has been
carried out in [2] where linear stability analysis of the first family waves has been
done. With computations of new families a selection problem was complicated.
Now we proceed to prove that from all regular wave solutions existing at given
governing parameters the dominating waves are suitable only to compare with
experimental observations. Validity of this statement is based on fact that the
dominating waves are stable and have capacity to attract nonstationary solutions
of (1) from small vicinity of another regular solution. To clear it numerous inte-
grations of Coshy problem for equations (1) have been carried out with two types
of initial data (7- 10].
Firstly, in case of initial data as small harmonic or stochastic perturbations of
main flow h = 1, q = 1 it has been shown that limiting in time solutions belong
to set of dominating waves [8,9] and also this result doesn't depend on a form of
initial data. In addition the oscillating regimes have been discovered at vicinity of
values s and J closed to jumps between different pieces of set of dominating waves.
Some examples of computations are shown in Figure 3a where hm is maximal wave
height.
Secondly, instability of solutions (5) that don't belong to set of dominating
waves has been studied too [15] by their using as initial data for Coshy problem
of equations (1). Because accuracy of computation of steady waves is controlled
then these initial data correspond to small random perturbations of solutions (5).
Results of numerical experiments confirm that limiting regimes belong to set of
dominating waves. Examples of computations are presented in Figure 3b. It may

16. 7
15~--------------~-------(
7
b~)
s
10
1.8 L_______L_______.________,_______j
0 0.8 a 1.6 3 Q 6
Figure 4: Dominating waves: (a) 1 - <5 = 0.04, 2- <5 = 0.1, 3 - <5 = 0.2, 4 -
<5 =0.4; (b) 1- <5 = 0.1, 2- <5 = 0.15, 3- <5 =0.247, 4- <5 = 0.4.
be noted that initial and dominating waves at s = 0.477 and s =0.5 present the
same family "Yc,l ·
Thus it was shown by direct numerical experiments that main flow and all wave
regimes except the dominating ones are unstable. They are transformed to the
dominating wave for chosen wave number s at given value <5. Hence dominating
waves are global attractors of initial value problem for (1) and have to be used for
comparison with experiments.
Attractive properties of dominating waves are weakened for very long waves
(s « 1) and with growth of <5. In the first case the waves are solitonlike and
their shapes have lengthy unperturbed part that is cause of instability similar to
one of main flow. In particular an alternation of coherent solitonlike structures
and nonstationary periodic waves has been revealed in [10]. Thus attractors of
equations (1) ass---* 0 are degenerated.
In the case of large <5 a weakness of attractors appears as a rule in form of
nonstationary waves with parameters that are varied in narrow intervals closed to
values of the dominating waves. At any case we'll use a name the dominating wave
in accordance with definition introduced above but its global attractive capacity
must be studied in numerical experiments in frame of (1). If the limiting wave is
formed then it will be referred as the tested dominating wave.
Now we have tables of dominating waves for several <5 values to compare with
experimental observations. It allows to formulate numerical experiments instead
of physical ones.
4. Dominating Waves and Experiment Correlations
In Figure 4a the sets of dominating waves for several values of <5 is demon-
strated in plane (a, c). There are 3 main groups of regular waves. First group

17. 8
is composed by the waves which belong to the first base family at different value
of <5 and fall to one separate correlation curve. Pieces of intermediate slow fam-
ilies are placed separately and form second group. Third group of fast waves is
presented by solutions of the second base family. It is seen that at fixed value
of <5 the waves belonging to the intermediate families and the second family are
in narrow band which boundaries are rough straight lines closing with growth of
wavelength. It corresponds qualitatively to experimental functions c(a) [13) which
linear correlation with some dispersion is clearly demonstrated.
In Figure 4b the comparison of theory and experiments are shown in plane
(Q,S), Q = (45<5)9/ 11 q0 /3, S = 1Mq~/~. These parameters for representation
of experimental observations were used in [13) . There are depicted the domains
restricted by curves l1, l2, la,l4 in which waves are observed in experiments. The
accordance of experimentally established dividing curves with theoretical bound-
aries of mentioned above groups of regular waves is very accurate. The curve 11
coincides with neutral curve of equations (1) S =3113Q1116. The waves of the first
base family belong to the domain between curves l1 and l2. Only Kapitsa [12) by
special selection of the disturbance frequency fulfilled experiment with nonlinear
waves of the first family /a from interval <5 E (0.04, 0.2). All experimental points
goes to one correlation curve of optimal regimes 15 revealed in [1). Also regular
waves have been observed in two domains between curves h, La and la, l4 . Artificial
exiting waves may be generated in both domains but natural developing ones are
formed in domain l2, La only. It is very interesting to see that slow dominating
waves of intermediate families strictly belong to domain between curves l2 , l3 and
fast dominating waves of the second base family /b belong to domain between l3 , l4 .
Also there is depicted curve l6 corresponding to waves with maximal amplification
factors in frame of linear stability analysis of main flow for equations (1) .
We proceed now to discuss numerical experiments to model the regular waves
by comparing of concrete quantitative data from computations and observations.
In Figure 5a results for slow waves in different liquids in plane (Q, V), V =cfq~13
are shown. The lower boundary of experimental data in Figure 5a corresponds to
nearly harmonic waves of [12) and coincides with the optimal regimes [1). Also
for few values of <5 we marked curves of the dominating waves. As there are an
accordance of theoretical and experimental points in principal we see also that
the unique correlation curve does not exist for these waves. There is dispersion of
points as inherent property of the regular waves set. Accounting Q =ReAh3f1 1
it is surprising example when two wave velocities care registered for one Reynolds
number ReA, i.e. in equal experimental conditions. Moreover the phase velocity
of developed nonlinear regime is close to one of linear disturbance that is typical
property of the dominating waves belonging to the intermediate families.
Let us proceed to discuss fast waves. Contrary to slow waves there are more
clear correlation curves in plane (a, c) for the fast dominating waves set, see Fig-
ure 4a. The dispersion of points is decreasing in plane (a, c) as we move from
branch to branch towards the second base family /b or in other words from slow to
fast waves. As an example dimensional values of amplitude aand phase velocity

18. v
2.35
1.7 L______...J______,_______L..::'-'---='-'
0 2.5 Q
4~r-------------------------~
c,
mmls
410-
I
0.34
0
0
-
a, mm
0
0
(b)
0.59
9
Figure 5: (a) Regular waves in experiments (denoted by symbols) and the dom-
inating waves in computations: 1 - <5 = 0.04, 2 - <5 = 0.1, 3 - <5 = 0.2, 4
- <5 =0.4. Curves of linear waves with maximal amplification factors (l1) and
optimal regimes (l2 ) are depicted too. (b) Comparison of experimental (o) and
numerical (•) results at 'Y =3274.
cin experiments with liquid having 11 = 0.0103 cm2/s, ujp = 72.9 cm3/s2 are
indicated by empty circles in Figure 5b where the tested dominating waves are
denoted by black points for several <5 values.
Now we can reproduce numerically points which have been observed for arti-
ficially disturbed waves. Let us assume that liquid properties and flow rates are
known. Based on this data the similarity parameter <5 could be calculated. Then
from the list of the dominating waves for this <5 value we pick up the dimensional
parameters of waves in acceptable physical plane. Thus numerical calculations
could be exploited to obtain base points on the properties of regular waves instead
of physical experiments.
5. Conclusions
To give theoretical explanation of reqular real nonlinear waves in films we
apply adequate mathematical model (1) in frame of that we have found manifold
of regular wave solutions and investigated its properties (bifurcations, attractors).
Then we have revealed the dominating waves that compose the subset of this
manifold and possess extremely values of main wave parameters. Tables of the
tested dominating waves have been computed. On the basis of these tables it is
possible to reproduce data of physical waves in falling liquid films.
The base system (1) enables the investigation to be extended to nonstationary
evolution and interactions of the wave structures in films such as deterministic-
stochastic transitions [7,10] . Note that extension of system (1) to films on inclined
planes, axisymmetric bodies, to flows with tangential forces up to now are accom-

19. 10
plished (3,7]. For every such case the weakly nonlinear asymptotics (8,9) from
extended system (1) could be deduced.
Acknowledgments
This work was supported by the Russian Foundation for Basic Researches,
project numbers 97- 01- 00153 and 98- 01-03559.
6. References
1. Shkadov, V.Ya.: Fluid Dynamics 2 (1967), 43.
2. Shkadov, V.Ya.: Fluid Dynamics 3 (1968), 20.
3. Shkadov, V.Ya.: Problems of Nonlinear Hydrodynamic Stability of Viscous
Liquid Layers, Capillary Jets and Internal Flows, Diss. D. Sci. , Moscow
State University, Moscow (1973).
4. Shkadov, V.Ya.: Fluid Dynamics 12 (1977) , 63.
5. Bunov, A.V., Demekhin, E.A. and Shkadov, V.Ya.: PMM USSR 48 (1984),
495.
6. Bunov, A.V., Demekhin, E.A. and Shkadov, V.Ya.: Moscow University Me-
chanics Bulletin 41 (1986), 73.
7. Demekhin, E.A., Tokarev, G.Yu. and Shkadov, V.Ya.: Teor. Osn. Khim.
Tekhn., 21 (1987), 177.
8. Sisoev, G.M. and Shkadov, V.Ya.: Phisics-Doklady, 42 (1997), 683.
9. Sisoev, G.M . and Shkadov, V.Ya.: Fluid Dynamics, 32 (1997), 784.
10. Sisoev, G.M. and Shkadov, V.Ya.: Phisics-Doklady 43 (1998), 785.
11. Sisoev, G.M. and Shkadov, Y.Ya.: Phisics-Doklady 44 (1999), 56.
12. Kapitsa, P.L. and Kapitsa, S.P.: Zh. Exper. Teor. Fiz. 19 (1949) , 105.
13. Alekseenko, S.V., Nakoryakov, V.E. and Pokusaev, B.T.: Wave Flow of
Liquid Films, Nauka, Novosibirsk (1992).
14. Demekhin, E.A., Tokarev, G.Yu. and Shkadov, V.Ya.: Physica D, 52 (1991) ,
338.
15. Sisoev, G.M. and Shkadov, V.Ya.: Moscow University Mechanics Bulletin
(in press).

20. NONLINEAR THEORY OF CONTROL OF INSTABILITY IN A LIQUID
FILM FLOW
S.P. LIN
Department ofMechanical and Aeronautical Engineering
Clarkson University, Potsdam, New York 13699-5725, USA
Recent developments in the linear theory of control of instability in a falling liquid
film over an inclined plane is briefly introduced. The linear theory predicts that the onset
of two- and three-dimensional waves can be suppressed within certain windows in the
parameter space by imparting appropriate amplitudes and frequencies of oscillation to the
incline. Whether these windows of stability stay open for finite amplitude disturbances
remain unknown. A system of partial differential equations is derived for describing the
weakly nonlinear evolution of the two-dimensional disturbances inside and outside of the
windows of linear stability.
1. Introduction
The onset of instability of a liquid film flow down a stationary inclined plane was
analyzed by Benjamin [1], and Yih [2]. In particular, they showed that the film on the
vertical plate is always unstable. The instability manifests itself as gravity driven surface
waves with a wavelength much longer than the film thickness. When the angle of
inclination of the plate is smaller than 0.56' but greater than zero the instability in a thick
liquid layer may manifest itself as relatively short shear waves [3,4,5]. When the inclined
plane has zero angle of inclination, the horizontal layer of liquid is stable. However,
when the horizontal plate is made to oscillate in its own plane, unstable waves synchronous
to the plate oscillation can be generated [6,7]. Control of instability by use of external
forcing has attracted attention recently. Coward and Renardy [8] studied a modulated two-
layered problem, Or and Kelly [9] studied the control of thermocapillary instabilities in
a heated layer by external forcing, and Woods and Lin [1OJ investigated the interaction of
the shear waves, the surface waves and the Faraday waves in a liquid film on the inclined
plate that vibrates in a direction perpendicular to the plate. They found that the
amplification rate of the unstable surface wave may be reduced by such an oscillation, but
cannot be completely suppressed.
Bauer and Kerczek [11] looked into the possibility of suppressing the unstable two-
dimensional waves in a liquid film flow down an inclined plane by imparting an in-plane
oscillation to the wall. Their results suggest that such a possibility may exist. However,
as they pointed out in their concluding remarks, the results obtained with long wave and
oscillation amplitude expansions did not have a sufficiently large parameter range of
validity to allow them to reach a definitive statement. Lin and Chen [12] obtained the
solution to the same problem for arbitrary wavelength and oscillation amplitude. They
showed that unstable two-dimensional waves can be suppressed by use of appropriate
amplitudes and frequencies of the plate oscillation. The windows of stability depend on
II
H.-C. Chang (ed.),JUTAM Symposium on Nonlinear Waves in Multi-Phase Flow, 11-20.
© ?000 Kluwer Academic Publishers.

21. 12
the flow parameters. The question of whether these windows will be closed by the three
dimensionality of the waves was answered by Lin and Chen [13]. For steady parallel
flows, the Squire [14] theorem states that three-dimensional infinitesimal disturbances are
more stable than two-dimensional ones for all wave numbers. Lin and Chen showed that
this is not true for the present problem. However, they showed with numerical results that
the three-dimensional disturbances do not close the stability window opened for the two-
dimensional disturbances, although three-dimensional disturbances can be more dangerous
than the two dimensional ones. Lin and Chen [15] elucidated from an energy consideration
the physical mechanism of film flow stabilization. They showed that the stabilization is
achieved mainly by creating a thin free surface boundary-layer which is much thinner than
that at the solid-liquid interface. The kinetic energy of the disturbances is very effectively
dissipated in the free surface boundary layer through the agent of viscosity.
Theoretical and experimental works on the nonlinear waves in a liquid film flow over
a stationary plate are quite extensive [16,17]. The main difficulty associated with the
nonlinear theory of instability control by plate oscillation arises from the highly time-
sensitive interaction between the unsteady basic flow and the disturbances. The quasi-
steady approximation used for the steady basic flow case cannot be applied even for long
wave disturbances, unless the forcing frequency is extremely small. Unfortunately the
onset of instability cannot be achieved by such a small frequency. Fortunately the
stabilization is achieved by viscous dissipation which has a time scale of viscous diffusion
time. This time scale is much larger than the time scale associated with the forcing
frequency required for stabilization. Moreover the narrowest part of the stabili<:ation
window occurs at a - 0 in many cases. Hence we shall use the methods of two-time-scale
and long wave expansion to derive the governing equation of the nonlinear stability
control.
r
.
.
FJGUJU! 1. Diagram of the theoretical model.

22. 13
2. Formulation
Consider the stability of a Newtonian liquid film flow down an inclined plane that
makes an angle 8 with the horizontal as shown in Fig. 1. The rigid plane oscillates
sinusoidally with a constant frequency Q. The oscillation is parallel to the flow direction,
and its amplitude is ~- The liquid is assumed to be incompres&ible, and the effect of the
ambient gas is neglected. The governing equations in a dimensionless form are
(1)
'V·V=O, (2)
where V is the gradient operator nondimensionalized with the half film thickness D, Vis
the velocity normalized with the maximum velocity Um = 2g sin 8D2/v in a nonoscillatory
film flow, g and v being, respectively, the gravitational acceleration and the kinematic
viscosity of the liquid, Tis the time normalized with Q, a is the rate of strain tensor
nondimensionlized with Um!D, pis the pressure normalized with p Um2, p being the liquid
density, Re =U~/v, is the Reynolds number, St =QDIUm is the Strouhal number, and Fr·1
and a are, respectively, the dimensionless gravitational body force per unit mass and the
D'Alembert apparent body force per unit mass, i.e.,
Fr-1 = (F.-')(i sin 8-j cos 8), F. = Ujf(D, a = id.St 2 sin 't",
in which i and j are, respectively, the unit vectors in the direction of and perpendicular to
the inclined plane, and dx = ~- The corresponding boundary conditions relative to the
Cartesian coordinate system (X,y,z) in Fig. 1 are the no-slip condition at the inclined plane
and the dynamic and the kinematic conditions at the free surface y=h respectively given
by
V=O at y -1, (3)
( -Po+Re -1a)·n + nWe - I 'V·n = 0, (4)
Vj (Sta, + V·V)h, (5)
where ois the identity matrix, n is the unit normal vector at the free surface and We is the

23. 14
Weber number defined by We= U",ZDIS, S being the surface tension.
The basic flow velocity (U,V,W) in the Cartesian coordinates (X,y,z) and the pressure
field P that satisfy Eqs. (1)- (5) exactly are given by
V=W=O,
U(y;t)
3
-(3+2y-y 2)+Std cos r;+U0 (y,r;),
4 X
(6)
(7)
where P0 is a reference pressure, C.C. stands for complex conjugate, and
ft(P) = e 3P sin eP-e -P sin p, p = (ReSt/2) 112,
To investigate the stability of with respect to two-dimensional disturbances, we
consider the perturbed velocity- pressure field (U+u, v, 0, P + p) where (u,v) are the
(X,y) components of velocity perturbation, and p is the pressure perturbation. For two-
dimensional disturbances the Stokes stream function ljl exists, and the pressure can be
eliminated from the governing differential system. Nevertheless the dependent variable
of the system, ljl and h, are highly uns(eady and the quasi steady approximation employed
for the case of stationary plate cannot be applied. However, we notice that the narrowest
part of the window of stabilization alluded to earlier occurs at a =0 for many cases, and
that the disturbance can grow significantly only over a time much larger than the forcing
period. These facts allow us to rescale the distance in the flow direction with the
wavelength i.e., =aX, and define a slow time T=a'r; for the purpose of investigating the
nonlinear evolution of stable and unstable linear disturbances near the edge of the window
of stabilization. Here we take s to be 1 in this study. T and r; will be treated as
independent variables in a sense which will be made clearer later. In terms of ljl and the
above defined independent variables, the Navier-Stokes equations can be written as

24. 15
(8)
where the subscripts, x,y, r and T stands for partial differentiation. The no-slip condition
(3) becomes
lJI(r,T,x,- 1) = 0, (9)
(10)
The normal and tangential components ofEq. (4) at y=h(r,x,T) leads to
The kinematic boundary condition (5) will be applied in the last step.
3. The Fast-Slow Time Solution
The purpose of this work is to investigate if the disturbance remains stable when its
amplitude becomes finite near a - 0 where the window of stability is the narrowest. Thus
we expand the solution for lj1 in powers of a,
lj1 = I: anljln(r,T,x,y).
n=O
(13)
Substituting Eq. (12) into Eq. (8) and its boundary conditions Eqs. (9)- (12), and

25. 16
retaining only terms of zeroth order in a, one has
(14)
IJroC~:,T,.x,-1) = 0, (15)
ljr~(-c,T,.x,-1) = 0, (16)
(17)
Ill (l2 II _
ljr0 (t,T,.x,h)-2p ljr0,(t,T,.x,h) - 0, (18)
where the upper primes denote partial differentiation with respect toy, and the boundary
conditions at y=-1 and y=h are applied after differentiation with y. The solution of Eq.
(14) is given by
where ek = (1+I)p..fk, and summation over b 1 is implied.
The boundary conditions Eqs. (15)- (18) are evaluated at
h = 1+aH(-.;,T,.x). (20)
The exponential functions in Hare then expanded in Taylor's series. Only up to second
order in aH are retained in actual computation. Thus we are limited to considering
weakly nonlinear evolution. The amplitude function His further expanded in a Fourier
series

26. 17
H = l: HiT,x) exp(ik'r) (21)
k=- M
Substituting Eqs. (19) - (21) into the boundary conditions, one finds from each Fourier
component
(22)
where F01 (p) is a lengthy expression of exponential and harmonic functions of p, and will
not be given here.
Similarly the O(et) approximate solution can be obtained
I n. + _21Hl)/2n.2(l+e41
),
- ie (F311 (f-')H0 f-'
where the coefficient functions of H's are lengthy and will not be given here. Moreover
the coefficients for the higher harmonics are not presented.
The O(et2) solution is given by
tVz = Az + Bzy + Czyz + Dzy3

27. 18
_]_R420 + .l._R320,
2 2
3
--R420 + R320 - lj12P0(T,x,-1),
2
where lj12P is the particular solution, and
(25)
in which F301, F302, F40 as well as lj12p0 are lengthy functions of variables indicated in the
parantheses. They are omitted here. Similarly for k=l, one can obtain the expression of
a21, b21, c21 and d21• It suffices to point out that these are lengthy functions of dxSt, R,
p, HoT, Box• Hlx• Hlcx·
Thus we have obtained a regular perturbation solution up to O(a2), which satisfies
the governing equation and four of the five boundary conditions while the behavior in x
and T remains to be determined by use of the kinematic boundary condition Eq. (5).
Moreover the short time behavior is resolved only up to the first two components in the
Fourier series. Substituting Eq. (7) into Eq. (5), dividing through the resulting equation
by a and identifying a-rand aX with T and x respectively, one has the kinematic condition
which describes the slow modulation in time and space in the flow direction
(26)
Substituting the obtained solution into Eq. (26) one obtains from the appropriate Fourier
component the following nonlinear equations for the evolution of the disturbance
amplitude,

28. 19
H - IT = C.C., (29)
where C.C. stands for the complex conjugate of the right side of the previous equation,
and C's, T's and S's are lengthy functions of variables indicated in parantheses.
Linearized equations of (28) - (29) reproduces the known linear theory results. The
nonlinear evolution of stable and unstable disturbances near the edge of windows of
stability is currently sought. The results to be obtained will be compared with the
experiments ofDrahos eta!. [18] for the case of unstable waves. Experiments are being
planned for the case of wave suppression.
This work was supported by NSF Grant No. CTS- 9616135.

29. 20
4. References
[I] Benjamin, T.B.: Waves formation of laminar flow down an inclined plane, J. Fluid Mech. 2 (1957), 554-
575.
[2] Yih, C.S.: Stability of liquid flow down an inclined plane, Phys. Fluids 6 (1963), 321-334.
[3] Woods, D.R. and Lin, S.P : Critical angle ofshear wave instability in a film, J. Appl. Mech. 63 (1993),
1051-1052.
[4] DeBruin, G.J.: Stability ofa layer of liquid flowing down an inclined plane," J. Eng. Math. 8 (1974),
259-270.
[5] Roryan, J.M., Davis, S.H. and Kelly, R.E.: Instability of a liquid film flowing down a slightly inclined
plane, Phys. Fluids 30 (1987), 983-989.
[6] Yih, C.S.: Instability of unsteady flows or configurations. Part I. Instability ofa horizontal liquid layer
on an oscillating plane,]. Fluid Mech. 31 (1968), 737-751.
[7] Or, A.C.: Finite wavelength instability in a honrizontalliquid layer on an oscillatory plane, J. Fluid
Mech. 335 (1997), 213-232.
[8] Coward, A.V. and Renardy, Y.Y.: Small amplitude oscillatory forcing on a two-layer plane channel flow,
Bull. Am. Phys. Soc. 40 12 (1995), 1950.
[9] Or, A.C. and Kelly, R.E.: Maragorti convections ofa thin deformable layer of liquid subjected to finite-
amplitude nonplanar oscillations, Bull. Am. Phys. Soc. 40, 12 (1996), 1949.
[10] Woods, D.R. and Lin, S.P.: Instability of a liquid film flow over a vibrating inclined plane, J. Fluid
Mech. 294, (1995), 391.
[II] Bauer, R.J. and VonKerczek, C.H.: Stability of Liquid Film Row Down a Oscillating Wall, J. Appl.
Mech., 58, (1991), 278-284.
[12] lin, S.P, Chen, J.N. and Woods, D.R.: Suppressions of Instability in a Liquid FilmAow, Phys. Fluids,
8 (1996), 3247-3252.
[13] lin, S.P. and Chen, J.N.: Elrnirtination ofThree Dimensional Waves in a Film Row, Phys. Fluids A,
9 (1997), 3926-3928.
[14] Squire, H.B.: On the Stability of the Three-Dimensional Disturbances of Viscous Row Between Parallel
Walls, Proc. R. Soc. London, Ser A,142 (1993), 621-628.
[15] Lin, S.P. and Chen, J.N.: The Mechartism of Surface Wave Suppression in Film Flow Down a Vertical
Plane," Phys. Fluids, 10 (1998). 1787-1792.
[16] Lin, S.P. and Wang, C.Y.: Modeling Wavy Film Rows, Encyclopedia ofFluid Mechanics, Vol. I, Ed.,
Gulf Publishing, N.P. Cheremisinoff, Houston, (1986), 931-951.
[17] Chang, H.C. and Demekhin, E.A.: Wave Evolution on a Falling Film, Ann. Rev. Fluid Mech., 26,
Annual Review Inc. Pal Alto (1994), 103-136.
[18] Drahos, J., Tihon, J. Sobolik, V., Hasal, P. and Schreiber, 1.: Analysis of Wave Modes in Liquid Film
Falling Down a Vertical Oscillating Plate, Chem. Eng. Sci., 52 (1997), 163-176.

30. COARSENING DYNAMICS OF ROLL WAVES.
HSUEH-CHIA CHANG AND EVGENY A. DEMEKHIN
Department of Chemical Engineering
University of Notre Dame
Notre Dame, IN 46556 ,U.S.A.
1. Introduction
Our community will be forever grateful to Steve Davis for exposing us to the
myriad of fascinating interfacial dynamics that exist on thin films. One of us
(HCC) entered the field after reading Steve's clear exposition of the Benney equa-
tion for falling films. Due precisely to the film's thinness, which Benney exploited
in his longwave lubrication simplification, thin-film or shallow-water waves are
typically dissipative and strongly nonlinear. As such, they are distinctly different
from deep-water waves and their rich dynamics are beyond the classical inverse
scattering and inviscid wave theories. Fortunately, their strongly nonlinear and
dissipative nature allows a completely different coherent-structure approach that
exploits the prevalence of robust and localized solitary wave structures. Here, we
demonstrate this new approach on roll waves that appear on inclined shallow-water
channel flow.
Along straight stretches of canals with constant inclination angles and water
depths, roll waves often appear. The best examples are seen on cement urban
canals like the Santa Anita wash in Los Angeles, but they also appear on natu-
ral rivers (Brock, 1969). Such roll waves often intensify in amplitude and speed
downstream and cause river bank erosion when they reach sufficiently large scales.
Brock (1969, 1970) carried out the first channel flow experiments to study this wave
intensification dynamics and attributed them to irreversible coalescence events be-
tween roll waves. Such coalescence events reduce the number of roll waves and
hence increase their separation. To produce the same flow rate as before, these
liquid-carrying roll waves must then increase in amplitude and speed. Curiously,
the coarsening dynamics that yield larger roll waves are scale invariant. Brock ob-
served that the time-averaged wave period < t > at every station increases linearly
downstream as < t > increases by an order of magnitude. The rate of change in
< t >seems to be independent of< t >. Such scale-invariant coarsening was also
observed in our simulation of thin-film capillary waves on a vertical plane (Chang
et al., 1996a,b) and seems to be a universal phenomenon for waves on falling liquid
films.
21
H.-C. Chang (ed.), IUTAM Symposium on Nonlinear Waves in Multi-Phase Flow, 21-31.
@ 2000 Kluwer Academic Publishers.

31. 22
As elucidated in our earlier study (Chang 199Gb), a fundamental reason for
the coalescence-driven coarsening dynamics is the robustness and solitary struc-
ture of the waves. Unlike solitons of integrable systems like KdV and nonlinear
SchrOdinger equations, such solitary waves do not pass through each other but
rather coalesce and form a larger solitary wave. This larger solitary wave has a
higher speed than an average one in front and quickly captures it to perpetuate
a coalescence cascade. That such a solitary wave is stable, eventhough the fiat
substrate around it is strictly unstable in this active, dispersive and dissipative
system, has been explained with a convective stability theory of localized wave
structures (Chang et a!., 1998). Such stability and the lack of momemtum and
energy conservation as in integrable systems roughly explain why the coalesced
waves remain as one. It is energetically more efficient for the coalesced waves to
approach a stable equilibrium wave by slowly draining liquid out than splitting
rapidly into two equilibrium waves. What has not been clarified,however, is why
the coalescence cascade produces a constant rate of change in the wave texture.
This we explain here for the roll waves.
2. Hydraulic Equations and Roll Waves
The reason the coarsening dynamics are easier to quantify for roll waves is
that they are accurately described by a simple set of dissipative/active hyperbolic
hydraulic equations derived by Dresser (1949),
au +u &u +c&h = 1 - u2
&t &h&x 0 ox h
Ft + &x(uh) = 0
(1)
where a bulk friction factor formulation has been used to capture viscous dissi-
pation. The lone parameter G = cos eIFr2 is a modified Froude number with
Fr = ghNcosBju'Jv = cJitanB where subscript N denotes the Nusselt flat film,
cf the friction coefficient and ethe inclination angle from the horizontal.
Roll waves are travelling wave solutions h(x- ct) and u(x- ct) to (1) which
approach constant values at the two infinites, h ~ X and u ~ .JX where X is the
substrate thickness at infinity. This roll wave family is hence a 2-parameter familty
parameterized by G and X· Due tQ the symmetry of the hydraulic equations to an
affine stretching in x, one can actually normalize x to unity by the transformation
h = xH, X= x~, u = uvx c = c..;x or t = T..;x. The parameter G remains
invariant under this transformation. For a given channel, there is hence only one
normalized "equilibrium" roll wave with unit substrate thicknesS. This family is
depicted in Fig. 1 and its speed C and amplitude A have been derived explicitly
by Dressler (1949).
We have simulated (1) over a long channel with wide-banded disturbances at
the inlet. To simulate how such small-amplitude noise evolves into roll waves, we
added an artificial viscous dissipation term in (1) to allow length-scale selection
during roll wave inception. Ve find, however, that, once the roll waves are formed,
their coarsening rate is independent of the dynamics in the inception region and

32. COARSENING DYNAMICS OF ROLL WAVES
1.75 G::0.1
0.12
0.14
0.16
0.18
,L-~L-~--~--~--~--~
0 0.2 o.. 0.6 0.8 1.2
23
Figure 1: Normalized equilibrium roll wave family with unit substrate thickness
and parameterized by G.
hence independence of the artificial dissipation therm. A snapshot of one simula-
tion is shown in Fig. 2. The simulation is carried out for a sufficient duration to
allow meaningful statistics at every station. The time-averaged wave period < t >
is plotted as a function of x' = (c~0 )x (Brock's coordinate) in Fig. 3. The incep-
tion region occurs between x' = 0 and x' = 2000 where wave compression actually
occurs. Beyond x' = 2000, however, linear coarsening occurs as < t > increases by
a factor of 4. The inception length x' = 2000 and the < t > at that position are
achieved by adjusting the artificial dissipation term and the noise amplitude. It is
the only fitting done and the downstream coarsening beyond inception are insen-
sitive to either quantity. As seen in Fig. 3, excellent agreement with Brock's data
is obtained. The world-lines tracking the solitary roll waves beyond the incpetion
region are shown in Fig. 4. Cascade coalescence events are clearly evident as one
large excited roll wave seems to capture its front neighbors successively.

33. 24
3.5.------.----,----r------.----,--------,
2.5
1.5
0.5
0~--~---~---~--~----~--~
0 3.33 6.66 100 13.33 16.66 200
Figure 2: A snapshot of the simulated wave profile for G =0.04
3. Similarity and Scale-Invariant Dynamics
As evident in Fig. 3, the downstream roll waves are clearly solitary although
they are not identical: they tend to increase in amplitude, separation and speed
downstream. Nevertheless, they are locally stationary travelling waves as seen
from the world lines of Fig. 4. Since G remains constant over the channel, this
family of roll waves down the channel must be parameterized by X. Moreover, if we
normalize by the local substrate thickness, all roll waves on a given channel must
collapse into the corresponding member of the normalized equilibrium family in
Fig. 1. It is difficult to estimate x accurately since it does not change much as seen
in Fig. 2. However, the time-averaged local speed c increases rapidly downstream
and we use it to estimate x = (c/C)2 where Cis obtained from Dressler (1949).
When the time-averaged amplitude at every location is scaled by the extimated
x, a single value is obtained as seen in Fig. 5. This self-similarity downstream is
valid for 5 channels (G values). Moreoever, when the same normalization is carried
out with Brock's measured time-averaged speed and amplitude, values at upto 6
very different locations along any given channel nearly collapse into one point, as
seen in Fig. 5. All collapsed values are also close to Dressler's amplitude for the
normalized roll wave.
This self-similarity can also be exploited, in conjunction with the invariance
of time-averaged flor rate to downstream position or x, to obtain how the wave
separation < l > and wave period < t > are parameterized by X downstream.
However, the coarsening rate of interest is independent of the local wave texture
and these two quantities are irrelevant in this respect.
We examine the coalescence dynamics in the normalized coordinate with unit
substrate. For the coarsening to be scale invariant, the "excited" wave created
by the previous coalescence event must have the same normalized amplitude at

34. 25
...
3.5
•'
Figure 3: The time-averaged wave period < t > as a function of position from
simulations (curve) and Brock's experiments (circles and stars from two different
experiments) for G= 0.04.
every station. These excited waves are larger and faster than the equilibrium ones
in Fig. 1 and are, in fact, the driving wave in the coalescence cascades in Fig. 4.
However, our statistics show that the distribution in wave separation, t or l, does
not distinguish between excited and equilibrium waves at every station. We can
hence use a dilating periodic lattice in a mean-field theory for both excited and
equilibrium waves. The dilation occurs downstream because of the coearsening.
However, the ratio of the average separtion of equilibrium waves < t > to that
of the excited waves < T > must remain the same, eventhough both increase
downstream. Consider then, a train of waves spaced by < t > with every < r >
I < t > being an excited one at every position. The excited one travels faster and
will capture its fron neighbor after the train has travelled a distance of lc. This
increases the average wave period by ~;~ · < t > and hence
d < t >2
- <t >= - --:-
dx < T > lc
(2)
Since there are very few excited waves as seen in Fig. 4, < t >I< T ><< 1 and
the average wave speed is that of the equilibrium wave c = C.JX. Consequently,
< t >cv< L > c and
d<l>
dx
<t> <l>
< T > lc
(3)
We have argued that < t > I < r > is independent of position, it hence
only remains to show < l > Ilc is also constant along the channel to obtain a
scale-invariant coarsening rage. This requires an estimate of lc, or equivalently,
the differentail speed between the excited and equilibrium waves.

35. 26
Figure 4: Wi ld 1·
oc '""'of the 'hnuJation foe c~o04 . th
·
111 e x - t Plane.

36. <
10.---~--~---------r--------T-------~.--------,
4
11
A
G
A - Fr=3.45, sire= 0.05011
o - Fr=4.63, sire= 0.08429
• - Fr=4.96, sire =0.08429
o - Fr=5.06, sire= 0.08429
<) - Fr=5.60, sire= 0.1192
x - calculation
27
Figure 5: Collapse of all simulated and measured roll wave amplitude along 5
different channels.
<1
l > = < L > = !::J.C = (JX _ 1)
•e Le C
(5)
where X represents the ratio between the amplitude of the excited wave immedi-
ately after the previous coalescence event to the equilibrium one in the normalized
coordinate. This number must be identical from station to station. A resonable
number is 2, corresponding to an excited wave that is twice as tall as the equilib-
rium one and < l > lle = /2- 1 "'0.414 or lei < l >"'-' 2.4.
We have verified the invariance of lei < l > and < t > / < T > numerically
as seen in Fig. 7. The latter quantity is almost constant at every position while
the former seems to vary from 3.0 to 3.5 along the channel: slightly higher than
the predicted 2.4 and with a small variation. The simulations that, while < t >
I < T > is independent of X or position, it is a weak function of G as seen in fig.

37. 28
0.4
0.35
0.3
0.25
()
0.2
• <I
0.15
0.1
0.05
0
0
/
/
0.5
G=0.04
<IA0=1.5A
1.5
t.A
/
/
2.5
/
/
/
Figure 6: The decay of excited waves at G= 0.04 whose initial reduced ampli-
tudes .6.A0 are indicated. The dashed line is the theoretical C(G)IA(G) for the
correlation constant between .6-C and .6-A.
8. When these of< t > I < T > are inserted into (3) and if < l > lc is taken to
be 0.414 of the model or 113.5 from the simulation, the linear coarsening rates at
different G are satisfactorily approximated in Fig. 8.
4. Summary
The scale-invariant coarsening dynamics of roll waves are simulated and mod-
elled. Linear coarsening results because the excited wave has a differential speed
.6-C that scales linearly with respect to C in the normalized coordinates and that
the density ratio of excited and equilibrium waves < t > I < T > remains constant
down the channel. The existence of a normalized coordinate, on the other hand,
arises from the self-similarity of the hydraulic equations and greatly simplifies the

38. 3.5
2.5
0.14
0.13
0.12
0.11
18
..·······
.......
•
• • • •
•
20
29
.............
. ...
....·
• • •
• • • •
• ••••
<1>/<'t:>
22 24 26 28 30
Figure 7: The invariance of lc/ < l > and < t > I < r > along the channel for
G= 0.04 from our simulations.
modelling effort. The only remaining question is how < t > I < r > is determined
within the roll wave inception region.

39. 30
0
0 .05
0 .03 0 0
!
! !
<f>l<~>
! !
0 .06 G 0 .07
Figure 8: The quantity < t > / <: T > and the linear coarsening rate as functions
of G from our simulations. The two theoretical curves correpond to (3) with
< l > /lc = 0.414 and 1/3.5.

40. 31
Acknowledgement
This work is supported by an NSF grant and a NASA grant.
5. References
[1] R. R. Brock. Development of roll-wave trains in open channels. J. of Hy-
draulics Div., 4: 1401-1427, 1969.
[2] R. R. Brock. Permanent roll waves. J. of Hydraulics Dive., 12: 2565-2580,
1970.
[3] H.-C. Chang, E. A. Demekhin and E. N. Kalaiden. A simulation of noise-
driven wave dynamics on a falling film. AIChE J., 42: 1553-1568, 1996a.
[4] H.-C. Chang, E. A. Demekhin E. N. Kalaidin and Y. Ye. Coarsening dy-
namics of falling-film solitary waves Phys. Rev. E., 54: 1467-1471, 1996b.
[5] H.-C. Chang, E. A. Demekhin and E. N. Kalaidin. Generation and suppres-
sion of radiation by solitary pulses. SIAM J. App. Math. , 58: 1246-1277,
1998.
[6] P. S. Dressler. Mathematical solution of the problem of roll waves in inclined
open channels. Pure App. Math., 2: 149-194, 1949.

41. OSCILLATORY SHEAR STRESS INDUCED STABILIZATION OF
THIN FILM INSTABILITIES
DAVID HALPERN
Department of Mathematics, University of Alabama,
Tuscaloosa, AL 35487
JAMES B. GROTBERG
Department of Biomedical Engineering, University of Michigan,
Ann Arbor, MI 48109
1. Introduction
The airways of the lungs are coated with a thin viscous film. Often, especially
in disease, the liquid film can form a plug that obstructs airflow. The formation
of the liquid plug is due to capillary driven instabilities that can arise in the
lining, causing the lining to close up [1-4]. Airflow can also be obstructed if the
airway wall collapses. This occurs when the elastic forces of the tube are not large
enough to sustain the negative fluid pressures inside the tube. The phenomenon
of airway closure occurs in many normal and pathological situations. It can occur
in premature neonates who do not produce enough surfactant to keep the surface-
tension sufficiently low. A clinical treatment is to deliver exogenous surfactant
to these neonates in the form of a liquid bolus. In order to facilitate breathing
they are also placed on mechanical ventilators. The amplitude (tidal volume) and
frequency settings are two key parameters at the disposal of clinicians. Frequencies
may range from normal breathing to high frequency ventilation at 5 Hz or more [6-
7]. Airway obstruction and hypercapnea (build up of carbon dioxide in the lungs)
are two examples of the manifestation of chronic obstructive pulmonary disease
that may also occur in adults. In order to reduce breathlessness in patients with
obstructive airway disease, chest wall vibrators are placed around the patients rib
cage, which oscillate at frequencies up to 100Hz [8]. Airway closure is also an issue
in the microgravity environment of outer space since there is marked decrease in
ventilatory inhomogeneity. It has been shown using the single nitrogen breath
washout test that even though closing volumes are not affected by microgravity,
there is considerable variability [9]. Their investigations suggest that some airways
may close above residual volume, and that other factors besides gravity, such as
the mechanical properties of the airway walls, are important factors on whether
33
H.·C. Chang (ed.),IUTAM Symposium on Nonlinear Waves in Multi-Phase Flow, 33-43.
@ 2000 Kluwer Academic Publishers.

42. 34
airway closure occurs.
The effects of oscillatory flow on interfacial stability have also been investigated
by Halpern & Frenkel [10], who considered a two fluid system between horizontal
plates with the upper plate oscillating with zero mean velocity, consisting of a thin
viscous film which is bounded below by the stationary plate and above by a denser
fluid. A modified Kuramoto-Sivashinsky equation was derived for the limiting case
of small interfacial deflections. This type of equation has also been derived for
core-annular flows where the core pressure gradient is a constant plus a smaller
oscillatory term [11]. Oscillatory flow can stabilize the Rayleigh-Taylor instability
[10). At low frequencies, increasing the frequency reduces the maximum amplitude
of the disturbance since the fraction of the cycle during which the destabilizing
buoyancy effect dominates is curtailed. However, above a certain critical value,
which depends on the size of the domain, the amplitude increases linearly with
frequency. These findings indicate the existence of an optimal frequency that
minimizes the maximum amplitude.
Recently, a system of nonlinear equations for the positions of the air-liquid
interface and the wall-fluid interface was derived which includes the effects of an
oscillatory shear stress due to the airflow [12]. The governing equations were
linearized, and the growth of infinitesimal disturbances was studied using Floquet
theory. The analysis showed that a periodic shear stress could significantly dampen
the fastest growing mode provided the airway wall was sufficiently compliant, but
did not modify the range of unstable wavenumbers. In this paper, we extend our
previous work by considering the nonlinear instability of a thin film coating the
inner surface of a compliant tube. Lubrication theory is used to derive a system of
coupled nonlinear evolution equations for the positions of the air-liquid and wall-
liquid interfaces. The effect of breathing is modeled by applying an oscillatory
shear stress at the air-liquid interface which is derived from an uncoupled problem.
In section 2, we provide a brief description of the model equations. Results of
numerical simulations are given in section 3, and conclusions in section 4.
2. Governing equations
The evolution equations describing the positions of the air-liquid and wall-
liquid interfaces have been previously derived [4,12,13], and are based on applying
lubrication theory in the film region. Consider a flexible, cylindrical tube of average
radius a, with a uniform film of thickness a- b. To investigate the stability of the
film and the wall, we perturb the positions of both interfaces axisymmetrically to
induce a flow within the film. At time t the fluid-wall boundary position is given
by rw =1 +£1)(z, t) and the air-fluid interfac.e position is TJ =1 + £(h(z,t)- 1),
where£= 1- b/a « 1 is the dimensionless unperturbed film thickness. The radial
and axial coordinates are scaled with respect to the unperturbed wall radius,
(r, z) = (r* , z*)/a, the radial and axial velocity components with respect to a
capillary velocity Ucap, (u,w) =(w*,w*)/Ucap, time is scaled as t =t*/(a/Ucap)
and pressure as p = p*/(£a/a), where the starred variables refer to dimensional

43. 35
quantities. The remaining parameters are the surface tension a, the fluid viscosity
p,, and Ucap = t 3a / p, [4]. For thin films we define a local radial coordinate,
y =(rw- r)/t, which permits a lubrication approximation. Conservation of mass
in the film yields the following evolution equation for h(z, t):
(1)
where the terms within the curly brackets contain contributions due to the radial
wall motion, the fluid pressure, p(z,t), which is uniform across the film according
to the lubrication theory and is due to the surface-tension, and the shear stress
induced by the oscillatory core flow, r 8 , and the film thickness, y =Y =1 +7J- h.
Using a form of the Young-Laplace equation due to Gauglitz & Radke [14], the
pressure contribution is given by
(2)
The kinematic boundary condition, (1), is coupled to a wall-elastic equation with
fluid loading which is derived by applying a balance of normal forces [4,15]. This
provides an evolution equation for 17(z,t):
1 7]
r/>r!t =p +1i1Jzz - r (1 + i7J)2 (3)
where tf> is the wall damping parameter, T1 is the ratio of longitudinal wall tension
to surface tension, and r is a ratio of surface tension forces to elastic forces.
To model the interfacial shear-stress, r 8 , we solve the following uncoupled prob-
lem by setting a periodic core flow rate, Q*(t*), displacing a tidal volume Vr per
half period. Therefore, Q*(t*) = (wVr/2)sinwt•. For large enough Womersley
number {3 =(w/va) 112 b, where va is the air viscosity, the core axial velocity field
is assumed to take the following form:
• { U*(t*)
u = U"1t") (b- r•)
0:::; r• :::; b- 6
b- 6::::; r• ::::; b
(4)
where 6 = (va/w) 112 is a Stokes boundary layer thickness, and U*(t*) is determined
from Q*(t*):
U*( *) wVr . •
t = 27rb2(1- a) smwt (5)
where a= 6/b « 1. The following expression for r8 , scaled on t 2 aja, is obtained:
T8 = -A0312 sin Ot (6)
where A= eyVr/(21Tab)(aUcap/va)112 is a dimensionless tidal volume parameter,
'Y =J.ta/J.t is the viscosity ratio, and !1 =(p,aw}/(t3a) is a dimensionless frequency

44. 36 D. HALPERN and J.B. GROTBERG
representing the ratio ofthe capillary time scale to the time scale due the oscillatory
core flow.
Equations (1) and (3) are solved numerically using the method of Jines subject
to periodic boundary conditions and initial conditions of the form h = ho cos(>.z),
TJ = TJo cos(>.z), where ). is the wavelength of the disturbance and h0 and 'T}o are
the initial amplitudes.
3. Results
Closure occurs if t exceeds a critical value [14], tc ~ 0.12,that is a film lining
a rigid tube will, in finite time, undergo a topological change and become a plug,
thus blocking the passage of air. In Fig. 1 we show this evolution fort= 0.13 and
n = 0. An initial perturbation grows with time as a result of the destabilizing
curvature component in (2) causing the formation of a growing bulge at z = 0. The
drainage into this bulge can be quite slow as the film thins in certain regions (at
z ~ ±0.5). Once a critical volume of fluid is reached, corresponding to a minimum
core radius, Tmin, of approximately 0.5, closure ensues very quickly. The effect
of interfacial shear stress on closure is shown in Fig. 2, where the minimum core
radius is plotted as a function of time for different frequencies. The cases plotted
in this figure correspond to frequencies of 0, 0.5, 1 and 1.5 Hz. Note that there is
a term proportional to 0 312 in 7 8 so that the increasing the frequency causes the
amplitude of 7 8 to increase (significantly at large frequencies). At low frequencies,
the time for closure, tc, decreases slightly with increasing frequency. But as the
frequency increases beyond a certain value, tc begins to increase. And above some
critical value, Tmin reaches a time-asymptotic periodic state which results in non-
closure. A case illustrating how the position of the air-liquid interface varies over
one period of an oscillatory shear stress cycle for a stable configuration at large
times is shown in Fig. 3. The arrows in the figure denote the direction of motion
of the bulge. At early times, the instability grows along the lines described above
for the 0 = 0 case resulting in the formation of a bulge, with the exception that
this bulge can now translate axially due to the shear stress. At later times, there is
a strong interaction between the pressure gradient terms and the shear stress term
in equation (1). When the magnitude of the shear stress is close to its maximum
value, and the bulge is moving rather quickly, there is considerable distortion of
the interface between the thinnest (around z = 0.6) and thickest parts of the
film since the contribution of the shear stress to the surface axial velocity is higher
where the film is thicker. This enhances the stabilizing effect of the axial curvature
component, and induces capillary wave.s on the interface. A bulge for the 0 = 0
case with the same rmin as the profiles shown in Fig. 3 has a lower maximum axial
curvature. Even at times when sin Ot = 0, corresponding to times when the shear
stress attains its minimum value, there is not sufficient time for the instability to
grow and induce closure.
The effect of the tidal volume parameter, A, on closure is also shown in Fig.
4, where closure time is plotted versus frequency. Increasing the interfacial shear

45. 37
(a)
/.·/ - - "'
_
:::--:: ---
"
r------ '-
------
/ -----
1 ....................
/ //~~~---~~-
// ~
- r
"1------ , r
, ---~~ 1
/j
I - ~~
I ' / I
I - / I
I I
I I
I 1
r
I I
I
I
I
0.5
-1 -0.5 0 0.5
z
(b)
-
.
rmin
• -
._
0.5
~----- -----
~--
0 20 40 60 80
Figure 1: (a) The development of the capillary instability. The shape of the
air liquid interface at various times. (b) The minimum core radius versus time.
Symbols correspond to the curves shown on (a).

46. 38
0.5
100
n
- - - 0
- - - - - 0.714
- -- - - 1.43
2.145
200 300
Figure 2: The effect of oscillatory interfacial shear stress on the minimum core
radius evolution, A= 1, € =0.13.
stress either by increasing A or f! delays the onset of closure, and at some critical
frequency which is dependent on A, the closure time tends to infinity suggesting
that A and f! can have a stabilizing effect in preventing closure from occurring.
Fig. 5 demonstrates the effect of the unperturbed film thickness, €, on closure
time. Since the transverse component of curvature is enhanced by making the film
thicker, the closure time decreases with increasing €. Also, higher frequencies, and
shear amplitudes are required to stabilize the film. Note that c appears in the
definitions for the time scale and 0.
Previous studies (4] have shown that the effect of wall compliance is destabi-
lizing since as the collar grows, the fluid pressure inside it decreases with respect
to the external constant pressure, causing the wall to be pulled in phase with the
collar. Therefore, by mass conservation, the collar has to grow at a faster rate for
the compliant case. In Fig. 6, the effect of the wall compliance parameter, r is
shown by plotting Tmin as a function of time for a fixed frequency 0 = 2.145 and
A = 1. Note that at this frequency and amplitude, the tube does not close if it
is relatively stiff. The effect of frequency and shear amplitude on closure time,
as shown in Fig. 7, is similar to that displayed in Fig. 4, except that the critical
frequency above which closure occurs happens at higher values. For example, the
critical frequency doubles from 15 to 30 for a shear amplitude of 0.1
4. Conclusions
We have demonstrated by examining the stability of a coupled system of evo-
lution equations for the air-fluid and wall-fluid interfaces that nonlinear saturation
of interfacial instabilities of oscillatory flows is possible if the frequency and the

47. 0.9
r
0.8
0.7
0.9
r
0.8
0.7
(a)
7 /"/!/
"ji
{i
{i
{i
{i
{i
{i
{i
{i
{i
{I
ti
{i
fi
. J.i
0
(b)
I
I
I
0
2
z
2
z
3
3
n
_._0
- -.6.-- 1
-·-T"-
...
--+--
--I! -
4
2
3
4
5
n
_._5
- - .6.- - 6
- -T"-
...
--+--
--·-
i'
'.!
4
7
8
9
10
39
Figure 3: Profiles of the air-liquid interface for !1 = 2.145, t = 0.13 and A= 1 at
t = 200.659 +mr/(5!1) for (a) n = 0, · · ·, 5; (b) n = 5, · · · ,10.

48. 40
A
800
- 0
.1)
- - - - - 0.25
- - - - - ~5
600
/)
·-·-_.,___-~
400
200
0
0 5 10 15
Figure 4: The effect of shear amplitude and frequency on the the closure time.
500
- - 0.13
- .. - - - 0.15
400 . - - - 0.17
100
0.5
n
1.5 2.5
Figure 5: The effect of the unperturbed film thickness, € on closure time, for A= 1.

49. 0.7
0.6
r
0
0.1
0.2
0.5 +-~_!__,-~--..,------,-------~-,
0 100 200 300
41
Figure 6: The effect of wall compliance on the minimum core radius, for € = 0.13,
!1 =2.145, A= 1, T1 =0, ¢> =0.1.
600 A
0.1
0.25
0.5
1
400
tc
200
--------_____-/
0
0 5 10 n 15 20 25 30
Figure 7: The effect of shear amplitude and frequency on closure time for r = 0.1,
€ =0.13, 1i =0, ¢> =0.1.

50. 42
amplitude of the oscillatory shear stress are sufficiently large. Given an initial layer
with sufficient fluid to form a plug if there is no shear, there is a critical frequency
or amplitude above which closure does not occur. The critical frequency increases
if either the unperturbed film thickness or the compliance of the tube wall are
increased. This could occur if the surface tension, a, is raised, since r t, 0 .(., At,
but from (6), lrsl.!., as at. A reduction in the magnitude of the interfacial shear
stress also occurs if the unperturbed film thickness parameter ~: increases, which
can happen in patients with above average amounts of intra-airway fluid [16].
Acknowledgements
This work was supported by NSF grant CTS9412523 and NASA grant NAG3-
2196.
5. References
[1] Macklem, P. T., D. F. Proctor, & J. C. Hogg Respir. Physiol. 8 (1970), 191.
[2] Mansell, A., C. Bryan, & H. Levison J. Appl. Physiol. 33 {1972), 711.
[3] Johnson, M., Kamm, R., Ho, L.W., Shapiro, A. & Pedley, T.J. J. Fluid
Mech. 233 (1991), 141.
[4] Halpern, D. & Grotberg, J.B. J. Fluid Mech. 244 (1992), 615.
[5) Patel, C. A. & Klein, J. M. Archives of Pediatrics & Adolescent Medicine
149(3) (1995), 317.
[6) Paulson, T. E., Spear, R. M., Silva, P. D., Peterson, B. M. Journal of Pedi-
atrics 4 (1996), 566.
[7] Macintyre, N.R. Critical Care Medicine 26(12) (1998), 1955.
[8) Edo H., Kimura H., Niijima M., Sakabe H., Shibuya M., Kanamaru A.,
Homma I. & Kuriyama T. J. Appl. Physiol. 84(5) (1998), 1487.
[9) West, J.B., Elliott, A.R., Guy, H.J.B., Prisk, G.K. (1998) JAMA 279(4)
(1998), 275.
[10] Halpern, D. & Frenkel, A. L. Submitted to J. Fluid Mech. (1999).
[11) Coward, A.V., Papageorgiou, D.T. & Smyrlis, Y.S. Z. angew Math. Phys.
46 (1995), 1.
[12] Halpern, D., Moriarty, J. A. & Grotberg, J. B. IUTAM Symposium on Non-
linear Singularities in Deformation and Flow, Kluwer Academic, Dordrecht,
The Netherlands. (ed. D. Durban & J. R. A. Pearson) (1999), 243.
[13] Halpern, D. & Grotberg, J .B. J. Biomech. Eng. 115 (1993), 271.
[14] Gauglitz, P.A. & Radke, C.J. Chem. Eng. Sci. 43 (1988), 1457.

51. 43
[15] Atabek, H.B. & Lew, H.S. Biophys. J. 6 (1966), 481.
(16] Notter, R.H. & Wang, Z. Rev. Chem. Eng. 77 (1997), 1206.
[17] Heldt, G.P., Merritt, T.A., Golembeski, D., Gilliard, N., Bloor, C. & Spragg,
R. Pediatric Research 31 (3) (1992), 270.

52. EXPERIMENTAL AND MODELING STUDIES OF WAVE OCCLUSION AND
EVOLUTION ON FREE FALLING VISCOUS FILMS IN A VERTICAL PIPE
E.K. DAO, L.T. NGUYEN AND V. BALAK.OTAIAH
Department ofChemical Engineering
University ofHouston
Houston, 77204, Texas
Experiments were conducted to study wave occlusion on falling films in a vertical
pipe. Aqueous solutions of glycerin with viscosity in the range 22 to 1250 cp were
used. Pipe diameters were varied in the range 0.635 to 1.5875 em. For each case, the
critical liquid flow rate at which the free falling wavy annular flow transfonns into
slug flow due to wave occlusion was determined. The experimental results were
correlated in tenns ofthe liquid Reynolds number at occlusion (ReLS = 4q /7tvtD) and
two dimensionless groups influencing the transition, the Kapitza number, Ka =
cr/[(v~.&)113J.Ld, and the Bond number, Bo = pi.&D2 I cr. The annular-slug transition
correlation was found to be ReLS = 0.062 Ka 413 Bo 213 and is valid in the range 0.1 <
ReLS < 200, 0.2 < Ka< 55 and 7< Bo< 50.
In the second part, a new simplified model is developed for describing the
characteristics of free falling wavy liquid films. The model consists of a set of two
partial differential equations (in x and t) for the local film thickness and volumetric
flow rate. It is shown that the new model is a substantial improvement over all
existing simplified models of wavy films such as the Long Wave equation, the
Shkadov model, and the Kapitza boundary layer model. The present model is capable
of predicting large amplitude solitary waves and yields the experimentally observed
relationship between the wave amplitude and the Reynolds and Kapitza numbers.
Local bifurcation analysis of the model for small Re gives the following analytical
relations for the velocity and maximum amplitude of the solitary waves: (h.,..,. -1) =
0.096Re513Ka-1 = (3-Ce)/6. Experimental studies of free falling viscous films were
conducted using water-glycerin solutions for Reynolds numbers in the range of2 to 20
and Kapitza numbers in the range of6 to 22. Comparison ofthe experimental data on
wave amplitudes with analytical correlations shows excellent agreement.
1. Introduction
Free falling wavy liquid films occur in many chemical and industrial processes
such as condensers and reboilers, falling film exchangers (liquid coolers and
condensers, evaporators, absorbers, freezers), gas-liquid reactors and thin film coating
operations. To understand such processes better and design them economically and
45
H.-C. Chang (ed.), /UTAM Symposium on Nonlinear Waves in Multi-Phase Flow, 45- 55.
@ 2000 Kluwer Academic Publishers.

53. 46
effectively, numerous investigations have been carried out over the past several
decades on wavy films. Experimentally it is observed that the interface of a liquid
film falling down a vertical pipe is a complex wavy surface. Most design procedures
assume a smooth interface for heat and mass transfer. As a result, these design
methods have been approxiniate and overly conservative. The wavy interface can
increase the rate of heat and mass transfer by two hundred to three hundred percent.
However, when wave occlusion occurs, the gas-liquid interfacial area reduces and
affects the operating conditions ofthese processes. For liquids with high viscosity, the
slug-annular transition is very sensitive to changes in the flow rate. To keep a process
operating in annular flow regime, the transition boundary between wavy annular flow
and slug flow needs to be understood.
Understanding of the mechanism of the annular-slug transition is also important in
developing flow pattern maps for two-phase gas-liquid flows. For free falling liquid
in a vertical pipe, there are three major flow regimes; namely, bubble, slug and
annular flow. The liquid surface in annular flow is always covered with a complex
wave pattern. Two different types ofwaves have been observed. The first type, called
"ripple" waves, occur at all liquid flow rates. They are of high frequency and small
amplitude and appear to rapidly lose their identity on passing downstream. These
waves are not axially syrmnetric. At higher liquid flow rates, "disturbance" or "roll"
waves are observed. These waves have far greater amplitude than the ripples. They
are axially symmetric (for low Reynolds and Kapitza numbers) and coherent and the
passage of these waves downstream can be readily followed. As the liquid flow rate
increases, the amplitude of the roll waves increases. When the wave amplitudes are
large enough, wave occlusion will occur as shown in Figure 1. Most of the previous
work on two-phase gas-liquid flows in vertical pipes dealt with determining the flow
pattern maps for air-water systems. Systematic studies of wave occlusion on falling
films in vertical pipes are currently lacking.
,
Flal fila
(ideU)
Figure 1: Schematic diagram illustrating wave occlusion and annular-slug flow
transition in vertical pipes.
The first part of this work is focussed on wave occlusion on falling films in a
vertical pipe. The main objective is to determine experimentally the critical flow rate

54. 47
at which wave occlusion occurs as a function ofthe liquid physical properties and pipe
diameter.
2. Experiments on Wave Occlusion in a Vertical Pipe
The experiments were conducted using liquid glycerin and water solutions with
various concentrations and temperatures close to the ambient value. The solution
viscosity varied from 22 to 1250 cp, densities from 1.17 to 1.26 glcm3 and surface
tensions from 64 to 68 dynes /em. Four different pipe diameters in the range 0.635 em
(0.25 in.) to 1.5875 em (0.625 in.) were used. One of the two flow loops used in the
experiments is shown schematically in Figure 2. Liquid glycerin solution was
introduced on the inside wall of the tube as a smooth film and fell by gravity. After a
short distance the liquid layer formed a wavy film. As this wavy film traveled down
the pipe, the amplitude of the wave increased and reached an asymptotic value.
Beyond a certain flow rate, this asymptotic value exceeded the pipe radius and the
wave bridges and formed a liquid slug traveling down the pipe. The details of
experimental equipment, set up, procedures and results can be found in [1].
Gear Pump
To Feed Section
2.5 m
w.n probes connect
to diU acqUISition 0. t Z7 m
devices
Liquid
Reservoir
Figure 2: Schematic diagram of the flow loop used in the experiments for solutions
with viscosity from 22 to 90 cp.
In our present system, the significant variables influencing the critical flow rate or
superficial velocity (uLS) are PL. J.lL, cr, D and g. Here, PL and IlL are the liquid density
and viscosity, cr is the surface tension of the liquid, D is diameter of the tube and g is
the gravitational acceleration. Using D, PL and IlL as the core variables, we identify
three independent dimensionless groups: the liquid Reynolds number based on the
superficial velocity, ReLS = PLu~/llL• the Suratrnan number, Su =crDptfllL2, and the
Galileo number, Ga = gD3/vL2• This dimensional analysis shows that the

55. 48
dimensionless critical flow rate (or the liquid Reynolds nwnber) at which wave
occlusion occurs can be correlated by the other two dimensionless groups which are
the Suratman number and Galileo number, ReLS = f(Su, Ga). Since we varied mainly
the viscosity and the pipe diameter, these two variables can be isolated from the
Suratman number and the Galileo nwnber. The resulting two dimensionless groups
are the Kapitza number, Ka = Su/Ga1
13 = cr/((viE)113Jld and the Bond number, Bo =
Ga/Su = PIEii/cr. Hence, the critical liquid Reynolds number at which wave
occlusion occurs can also be expressed as a function of the Kapitza and Bond
numbers, ReLS=f(Ka, Bo).
Re1.s =0.062 !Ka'Ro),.,
to' Rr,_, =0.062 (Su (;a'"''"' •,
Slu.:t'low /
Ocrlusion/
./ :·
to'
/ .
./ /
tO' ' v
Annular t'low
Figure 3: Dependence of the critical flow rate (Reynolds number) at occlusion on the
Kapitza and Bond (or Suratman and Galileo) numbers for all experiments.
The raw experimental results and values of various dimensionless groups can be
found in [l]. The results show that the critical liquid Reynolds number for occlusion
decreases when the viscosity of the glycerin solution increases. The critical Reynolds
number was plotted versus the Kapitza number to determine the dependence of the
flow rate at occlusion on the liquid viscosity. This plot gave a straightline. To
examine the dependence of critical flow rate on the tube diameter, experiments were
conducted using the same liquid glycerin solution with different tube diameters. A
plot ofcritical ReLS versus Bo for the two data sets also gave a straightline. To get the
final correlation between ReLS and the Kapitza and Bond numbers, we plotted the
critical Reynolds number versus the product Ka2Bo in Figure 3. The twenty different
experiments with various fluids and tube diameters covered approximately five orders
of magnitude of the product Ka2Bo. The data shown on a logarithmic scale in Figure
3 are well correlated by a straight line with a slope of2/3. Since Su Ga1
13 = Ka2Bo, a
plot ofReLS versus Su Ga113 also gives a straight line with identical slope.
Thus, our fmal correlation between the critical liquid Reynolds number at which
occlusion occurs and the dimensionless groups representing the physical properties of
the liquid and the pipe diameter was found to be:

56. 49
(1a)
or
(1b)
where
crz'Jplot9D4'lgZ19
Ka413 Bom = Su213Ga 219 = (2)
1-116/9
This correlation is valid in the range of our experiments 0.1 < ReLS < 200,
0.2<Ka<55, 7 < Bo <50, 0.6 < Su < 2070 and 12 < Ga < 57000. We now compare this
correlation with related results in pipes and capillaries.
Paper [2] shows that the slug-annular transition for gas-liquid two-phase flows in
pipes in microgravity is correlated by
ReLS = 0.215 X 10"3 Su213 ReGs (3)
Comparing Eq. (3) with Eq. (lb), we see that the exponent on the Suratman
nwnber is the same for both cases. In nonnal gravity, the driving force for the liquid
film is the gravity force, which is represented by the Galileo number. In microgravity,
the driving force for the liquid film is the pressure gradient in the gas phase, which is
represented by the gas Reynolds number. Thus, Eqs. (lb) and (3) show that one
important parameter that determines the slug-annular transition in both normal and
microgravity conditions is the Suratman number.
For an ideal free falling flat film, assuming axisymmetric flow with zero pressure
gradient along the vertical axis, we have the relation:
(4)
where
and ho is the flat film thickness. Since f(R) = 1, Eq. (4) shows that if there are no
waves on the film then the critical Reynolds number at occlusion is equal to the
Galileo number (ReLS = Ga/32). Comparison of Eqs. (4) and (lb) shows that the
presence of waves reduces the exponent on Ga from 1 to 2/9. This comparison also
shows that the critical flow rate has a 2/3 power dependence on the surface tension.
The numerical values indicate that the critical Reynolds nwnber at occlusion is
approximately two orders ofmagnitude smaller than the Galileo number.
The experimental results were combined with Eqs. (4) and (5) to determine the
dimensionless flat film thickness (ho/R) at occlusion as well as the liquid fraction
(AJA) in the pipe. These results show that the liquid fraction at the annular-slug
transition varies from 0.4 to 0.7. This number should be compared with the empirical
transition criterion (AJA = 0.35) suggested in (3] for co-current gas-liquid downflow
in vertical pipes. The experimental results reported here clearly show that the liquid

57. 50
fraction at occlusion is higher than 0.35, at least in the laminar regime covered by our
experiments (Rets < 200).
Figure 4 shows a plot of the dimensionless film thickness at occlusion as a
function of the Capillary number (Ca =J.ltUu/cr =Reu;/Su) for experiments conducted .
in a half-inch diameter tube. Within the range of the data, the plot gives a straight line
on a log-log scale with the following correlation:
~ = 0.73 Ca318 •
R
(6)
This relation may be compared with the empirical correlation in [4] and [5] for the
thickness of the annular film in a capillary tube, which is initially filled with liquid,
and gas (or gas bubble) is blown through the tube. The correlation for the flow of
large bubbles through horizontal capillary tubes in (4] and [5]
~ =0.5 Ca 112 ,
R
(7)
where the Capillary nwnber is defmed based on the gas velocity. Comparison of Eqs.
(6) and (7) shows that while the exponent on Ca is smaller in Eq. (6) the nwnerical
value ofthe scaled film thickness at occlusion is about twice that predicted by Eq. (7).
10" 1
11,/R =o.n Ca _,..
10" 1
Ca
lo"
Figure 4: Dependence ofscaled average film thickness at occlusion on Ca nwnber.
A fmal comparison we make is with the experimental data in paper [6] on the
formation and breakup of liquid bridges in capillaries. Their experimental data show
that liquid bridges are formed in capillaries when hoiR exceeds 0.09. For the case of
falling films the smallest scaled film thickness at occlusion was found to be 0.23.
Thus, the scaled film thickness for the formation of liquid bridges (slugs) on falling
films in vertical pipes is at least twice that of the value in capillaries.

58. 51
3. A New Simplified Model for Wave Evolution on Free Falling Liquid Films
In this section. we consider a film of viscous {luid flowing freely down a vertical
plane. The adjacent gas phase at the free surface is assumed quiescent, and its density
and viscosity are significantly smaller than that of the liquid. Let h(x, t) be the film
thickness measured from the wall. The motion of falling film flows can be described
by the two-dimensionless Navier-Stokes (NS) equations and the following boundary
conditions:
Jau +uau +vau)=-/JP+E_+~(a2u +&2 a2
u). (Sa)
_at ax 0y ax Re Re O_y2 ax2
&2(0v +UOv+VOv)=- ap +~(&a2v +&3 a2
v)•
at ax 0y 0y Re ey2 & 2
8u+ilv=O
Ox Oy '
at y =h(x, t),
at y =h(x, t),
(8b)
(8c)
(8d)
(8e)
(8f)
(8g)
where We is the Weber nwnber, Re = 4uj~Ju is the Reynolds number, uN is the
Nusselt average velocity, and hN is the Nusselt film thickness. Here, we scale the
coordinate x by the wavelength A., yby hN' u by uN' vby &uN' pressure by pu/ and
time by i.JuN (where &=hjA.). Next, we apply the integral method to derive the
simplified evolution equations for falling films. We assume the self-similar velocity
profile ofthe form:
u(x,y,t) =A(x,)_Y__.!_(_Y_)z}. (9)
l h(x,t) 2 h(x,t)
While it may be argued that the parabolic velocity profile assumption is arbitrary, we
show in [6] that the resulting evolution equations retain all qualitative features of the
NS equations and agree well with experimental results. We substitute Eq. (9) into
Eqs. (8) and integrate them across the film. For Re~O(l) and We is either 0(&.1) or
0(&- by retaining terms up to order &2• except in the tangential shear stress where we

59. 52
keep only leading order tenns (i.e. ou!Oy=O aty=h}, we obtain the following new two-
equation integral boundary layer model (it is noted that, while this model is not
consistent, it is the best two equation integral boundary layer type model that can be
obtained using the self-similar velocity profile assumption):
oq + oh =0, (lOa)
ax or
{oq+~.i.(£)} =E.(h_.!L)+{J![h~(g_)]-...i..02q}+Weh 03h . (lOb)
ot 5 ax h Re h2 Re ax2 h Re ox2 ax3
Here, h is the local film thickness and q is the volumetric flow rate.
4. Comparison of Linear Stability Results
In this section. we attempt to establish (or verify) the range of applicability of our
two-equation model as well as three simplified literature models by comparing their
neutral stability results to the Orr-Sommerfeld (OS) equation obtained from
linearization of the full 20 Navier-Stokes equations. Although the linear theory alone
can not explain the complex dynamics of wave evolution, it can detect any
inadequacies of the simplified models near wave inception. In Fig. 5a and 5b, we
compare the neutral stability results generated from the LW, Shkadov, Kapitza, and
2EQ model to the OS results, as functions of Reynolds numbers in the range from 0 to
200 and Ka of 10 and 3371, respectively.
"
~
~ 0.4
..
.J:.
a
=
z
..
..
..
3::: 0.1
...
·!:
·;:
u
STABLE
(a) Ka=3371
..................-······--/
.......····....
·-·-·· LW
•
.••/ -K3Pit7a
-···-···- Shkadov
V>ISTABLE
so 100
Re
-os
····-··- 2EQ
ISO 100
Figure 5: Comparisons ofneutral stability results at (a) Ka=3371 and (b) Ka=lO.
For high Ka fluids such as water, all the models are equivalent in predicting the
linear stability results for small Re and wave numbers, Clc:$0.05. This is as expected
since all the models were derived for small flow rates and inftnitesirnal disturbances.

60. 53
The strong surface tension of water suppresses waves on the film surface. The wave
amplitudes remain small if Re is small. All the curves are within ±5% of the OS
results for Re~lO. Thus, all models seem to be valid in this region. As Re increases
above 50, the LW, K.apitza, and the Shkadov model start to deviate from the OS
equation while the 2EQ still remains close. For low K.a fluids, the unstable region
increases. All the literature models distinctively branch off at a much smaller
Reynolds number while the 2EQ model still follows the OS equation closely. Thus,
the linear analysis shows that the new model is better than all simplified literature
models in predicting the neutral stability curves, particularly at low to intermediate
Reynolds numbers and low Kapitza numbers.
5. Analysis of Wavy Film Behavior in Traveling Wave Coordinate
Since our main interest focuses on studying the asymptotic behavior of the wavy film,
we consider a particular type of waveform that travels at a constant velocity without
changing its size and shape. By varying the velocity within some bounded region, we
can explore the entire spectrum of asymptotic behavior of these wavy films. The
coordinate system moving at a steady velocity with respect to a stationary frame of
reference (laboratory) is defmed as z=x-Cet, where x and tare the dimensionless axial
coordinate and time normalized by hN and hjuN. Introducing the traveling wave
transformation into Eqs. ( l0) produces a three-dimensional system of autonomous
ordinary differential equations (ODEs) whose behavior is parameterized by Ce
(11)
T T
where we defme the vector of dynamic variables 0 =(h1, hu h3] =[h, h., h..] and
f = {_g_(h(- (1- Ce +Ce h1))+ _g_h3[~hfCe-h1(1- Ce +Ce h.))
Re Re 6
+h
2(h1
2Ce2 - 2.4Ceh1(1- Ce +Ce h1)+1.2(1-Ce +Ce h1) 2 )
-!:hi{h1Ce-(1-Ce+Ce h1))}/{-We h{ }
.
The asymptotic form of the wavy film can be obtained by integrating directly the
above set of ODEs. The numerical procedure requires only an initial condition, which
can be taken as a small perturbation from the smooth film solution. A complete
comparison ofnumerical computation and experimental data is presented in [8].
When the Reynolds number is small (i.e. near zero), we can perform a weakly
nonlinear analysis to explore the flow dynamics near criticality. The technique is
quite involved and requires the use of the center manifold theory, the normal form
theory, and the Melnikov homoclinic analysis, etc. We report here only the following

61. 54
important correlations relating the amplitude and velocity of solitary waves and flow
properties obtained from these analyses and refer interested readers to [8] for a
complete discussion:
( ) 1( ) 7 1.4 513 -1
hmax -1 =- 3-Ce =--=-=0.096Re Ka .
6 5We We
(12a)
Figure 6 presents a comparison of experimental data and analytical results predicted
by Eq. (12a) for the film thickness. Experimental data are very scarce in the literature
for free falling viscous flows near criticality. We used three water-glycerin
corresponding to Ka=22, 13 and 5.9. For these low Ka fluids, waves could grow to
such large amplitudes even at low Reynolds nwnbers. We varied the Reynolds
nwnbers in the range from 2 to 20, both sinusoidal and solitary waves were clearly
observed in these experiments. The experimental data are evenly distributed above a
straight line predicted by Eq. (12a). However, Fig. 6 confirms the functional
dependency of the film thickness on the Weber number. As stated earlier, this is due
to the dropping of some terms in the tangential stress boundary condition. If we kept
all terms up to order &
2, the coefficient in Eq. (12a) will change from 0.096 to 0.132:
(12b)
Eq. (12b) agrees well with experimental data. Although Eqs. (12) were derived for
flows near criticality only, Fig. 6 shows convincing evidences that such correlation is
also useful in predicting the maximum film thickness for low to intermediate
Reynolds number and especially for low Kapitza fluids. The analytical result, Eq.
(12a), may be used to determine the critical flow rate at occlusion for a vertically
falling film between two parallel plates. For plates separated by a distance 2h and a
flat film thickness ofhN, we have
h = }!_ Ga = g(2h)3
max hN ' v2 '
(13)
Combining Eqs. (12a) and (13), we get
(
Ga )X' s/
- - -1 = 0.096Rel5/J Ka-I.
6ReLS
(14)
If the second term on the l.h.s. is negligible compared to the first, Eq. (14) simplifies
to
(15)
This result, though valid only for occlusion between flat plates, is similar to the
experimental result, Eq. (Ia), for occlusion in vertical pipes.

62. I 0.1
..
"
e
-=
.....,
0.01
0.01
we·'
6. Kapilza's Alalbol
0 Kapilza's Wata
• Ka•22
e Ka-13
T Ka-S.9
--Eq. (l2a)
Eq. (12b)
0.1
Figure 6: A comparison ofanalytical results vs. experimental data for film thickness.
55
Acknowledgement: This work was supported by a grant from the NASA Glenn
Research Center (NAG3-1840) and the Graduate Student Researchers Program
(GSRP) fellowship to EricK. Dao and Luan T. Nguyen.
6. References
[I) Dao E. K. and Balakotaiah V: Experimental studies of wave occlusion on falling films in a vertical
pipe, A/ChE J., submitted for publication (1999).
[2) Jayawardena S. S., Balakotaiah V. and Wine L.C.: Flow pattern transition maps for microgravity two-
phase flows, AIChE J., 43, (1997), 1637-1640.
[3] Bamea D., Shoham 0. and Taite! Y.: Flow pattern transitions for vertical downward two-phase flow,
Chem. EngngSci., 37,(1982), 741-743.
[4) Fairbrother F. and Stubbs A. E: Studies in Electro-endosmosis. Part VI. The "bubble-tube" method
ofmeasurement,J. Chem. Soc., I, (1935), 527-529.
[5) Goldsmith H. L. and MasonS. G.: The Flow of Suspensions through Tubes II. Single large bubbles,
J. Colloid Sci., IS, (1963), 237-261.
[6) Gauglitz P. A. and Radke C. J.: An extended evolution equation for liquid film breakup in cylindrical
capillaries, Chem. Engng. Sci., 43, (1988), 1457-1465.
[7) Nguyen, L.T. and Balakotaiah, V.: Modeling and experimental studies of wave evolution on free
falling viscous films, Phys. Fluids, submitted for publication (1999).
[8) Nguyen, L.T.: Experimental and modeling studies of wavy films in annular gas-liquid flows under
nonnal and microgravity conditions, Ph.D. Dissertation. University ofHouston (1999).

63. NONLINEAR WAVES AND (DISSIPATIVE) SOLITONS
IN THIN LIQUID LAYERS SUBJECTED TO
SURFACTANT GRADIENTS
MANUEL G. VELARDE and ALEX YE. REDNIKOV
Instituto Pluridisciplinar, Universidad Complutense de Madrid,
Paseo Juan XXIII, n.l, 28040-Madrid, Spain
Thermal gradients along or across an interface, or the open surface of a liquid
layer, create surface tension gradients, that lead to flow or convective instability
(Marangoni effect). We provide an asymptotic nonlinear theory of long (surface)
waves excited and sustained, past an instability threshold, by such Marangoni
effect.
1. Introduction
When there is an open surface or an interface exists between two liquids, the
interfacial tension, a, accounts for the jump in normal stresses proportional to the
surface curvature across the interface, hence this Laplace force affects its shape
and stability. Gravity competes with it in accommodating equipotential levels with
curvature. Their balance defines, for instance, the stable equilibrium of spherical
drops or bubbles. If the surface tension varies with temperature or composition,
and, eventually, with position along an interface, its change takes care of the jump
in the tangential stresses. Hence its gradient acts like a shear stress applied by the
interface on the adjoining bulk liquid (Marangoni stress), and thereby generates
flow or alters an existing one (Marangoni effect). The variation of surface tension
along an interface may be due to the existence of a thermal gradient along the in-
terface or perpendicular to it. In the former case we have instantaneous convection
while in the latter flow occurs past an instability threshold (1-3]. The Marangoni
effect is the engine transforming physicochemical energy into flow whose form and
time-dependence for standard liquids rests on the sign of the thermal gradient and
the ratios of viscosities and diffusivities of adjacent fluids (4-6]. Surface waves, ei-
ther capillary-gravity or dilational (longitudinal) waves are known to be excitable
by the Marangoni effect (5,7,8].
57
H.·C. Clwng (ed.), IUTAM Symposium on Nonlinear Waves in MuJti·PhLlse Flow, 51-fJ?.
© 2000 Kluwer Academic Publishers.