DYNAMIC RESPONSE OF A WETTING
LIQUID ENCLOSED IN A IZOTATING TANK
IN A ZEROG ENVIRONMENT AND
SUBJECTED TO VARIOITS
DISTURBING FORCES
By
SU1KW'0.NG PAU!, PAO
A MSERTATION PRFtNTT`F) TO TTTV GRADIJAT17 COUNCIL, OF
THE UNiVERSITY OF FLORIDA
IN PARTM FTMVILLMENT OF THE REQUIREMENTS FOB THF,
VEGREL OF DOCTOi Of'
UNIVERSITY Or FLORIDA
April, 1967
TO M LOV
AmNWLm4ET
Th uhrwse oepeshi hnsadapeito 
TABLE OF CONTENTS
Page
ACK"t4"EMS . . . . . . . . . . . iii
LIST OF TABLES . . * . . . . . . . . v
LIST OF PIGU#SS . . . . . . . . . . . vi
PARTTAL LIST OF SYMOLS . . . . . . . . . vii
ABSTRACT . . . . . . . . . . . . . viii
Chapter
I INTRODUCTION . . . . . . . . . . I
Il FORMULATION OP THE PRGBIal . . . . . . . 5
III SGLUTION FOR T49 ELLIPTIC CASE . . . . . . 17
IV SOLuTIONTO THE HVPERBOLIC GASF . . . . . 40
V CONCLUSIONS . . . . I . . . . . 57
APPZPMX#S
A EXP4ESSION OF TW PERTURBATION VELOCITY CQ#fPQNM
v cr IN THk FOLNTE SPRU016AL CQORDMTES (o,,J3,0 . 61
P FLOIN CHMT OF THE CCMPtTrING PROWM IPWLOYED
1* cmAPTER III . . . . . I . I . . 1 63
RErgWomm s . . . . . . . I . . . . . 66
1K0Q" M4L $91MCH . . . . . . . . . . 67
iv
,fish.,
MN dam.
LIST OF TABLES
Page
Table
1 THE FIRST AND SECOND EIGENFREQUENCIES FOR BUBBLES
WITH DIFFERENT SHAPE FACTORS IN A CYLINDRICAL
TANK WITH FIXED DIMENSIONS . . . . . . .
i i.........
LIST OF FIGMES
Figure Page
I A mxridian gection of the defortied cylindrical
coordinate system I 9 ,u ) and the prolste
t phe ro ida 1 c oord in ate sys tam ( x P 0 ) . . . . 10
2 'rhn shape factor 9 'versus the dimensionless number E.
Vor the limiting case E t_ G.5, the bubble becoms
an infinite cylindrical tubi, . . . . . . . is
3 The first and second eigenfrequencies plotted vrrsus
the hubble thape factor t . . . . . . . . 38
4 The typical mode shapes in tht meridian plam
corresponding to the first and the second eigen
frequencies of the oscillations of the bubble
about Its position of stable equilibrim . . . 39
5 A 94wetch for the Cauchy problem . . . . . . 43
6 A Sketcb for the (;Ojjrsat problea . . . . . 44
7 Th* coordinate Aystem eployed at the pole region
of the bubble . * . . . . . . 47
8 Division of the Elow field tnto regions separated by
eharacteristic ILnes whtre discontinuitios may occur 50
9 TWe pwrturba4tion pras$ate f1*1d for 4 tankliquid
*Vatoa, rotating ;Ath 6 const,*nt angular 5;po*d,
*d#r the tntluence of a trA*vor** r*'ducod
cravity fLeIA 4 4 # . . . I . . . 53
LO T* p*rttrvtw*.J*v fi#ld for t&nkliquid
r*tating witb j eo*6tftt4ngu1*r *p*44,
un*' tho fnflMwco' of a, t#*4wrtj r4ducid
**,vi t y t it td . . . . . . . . . 5.4
11 Tjo 404mmmd )>ubb4 #*p# for t#nkliquLd sytt4oj
h 0 cosj"*t *%*4#r 4"#4#1 wW*v 'tb*
V04U*04 55
Vi
PARTIAL LIST OF SYMBOLS
t t ime
0 constant angular speed of the rotatitg tank
2L leqgth of the tank
R radium of the cylinder
C semiminoraxis of the un4isturb4od bu"le
2 tz2LYC dimensionless length of th, tank
0 =R 0 A dimensionless radius of tha cylinder
(rOz) cylindrical coordinates
(T, 9, C) dinensi*nlass ylindrical coordinates, 7 = r/C, Z/O
(OJP'G) jimensionlesa spheroidal coordinates
Oa,O,P) perturbation velocity components with respect to the
(rgz) directions
(uvw) dimeasionle** perturbation velocity c*wp*ne nts,
correspff o the directions
P dimens less perturbation pressure
denst liqqid
T Coef of t ace te
t he 'U'n ble
bble iuterfac
Abtato israin'rsetdt 1eGaut oni
intilFlile~ fteRqi~kt o h
Dereo otr CPiooh
DYNAIC RSPOSE O A ETTM LIQUD SCLGSD I
A R(TTN A*I ZP NIOM N
SUBEM TOVROSD5TRIGFP
The large scale vapor bubble takes a stable equilibrium shape
which is similar to an elongated spheroid and located symmetrically
about the axis of rotation. When perturbing forces are prevailing,
the liquid body and the free interface will oscillate about the equilib
rium configuration.
The equations of motion are linearized. Surface tension and
rotational velocity component are both essential in the treatment of
the problem and are taken fully into account.
The selfsustained oscillations are governed by an elliptic
differential equation for the perturbation pressure field. In this
case, the bubble is approximated by a prolate spheroid which is embedded
into a spheroidal coordinate system. Hence associated Legendre func
tions can be employed for the series expansion of the solution. A spe
cial method has been employed in order to account for the homogeneous
boundary condition at the walls of the tank. The resulting eigenvalue
problem for the relative frequency of oscillation is nonlinear. The
eigenfrequencies are obtained through iteration procedure. They are
all real and greater than two. The first two eigenfrequencies can be
computed with great accuracy. The oscillations are stable. The ampli
tude of oscillation is large near the equator of the bubble. For very
slowly rotating systems the eigenfrequencies tend to accumulate near
the critical frequency.
For a forced oscillation, induced by a reduced gravity field
Of constant magnitude and directLon in an inertial frame of reference,
the relative frequency of oscillation is less than two. Thegoverning
differential equation is hyperbolic, hence the. methodi of analysis is
ix
frm thto h bv etoe litccs.Temteaia rb
le stasore noiterifrnta qainswihcnb
inertdn erclyb enso iads ehdo ucesv prx
imtos Ition htased outo naclniia ako
fiit lngh oe nt xit. Hcweer asoutonexst i acyin
drca tako niielnt. Teprtrain xedt niiy
A/ nueia/xml/o hi ocdoclainprbe a are
CHAPTER I
INTRODUCTION
During the past Eew years, a growing interest in the investiga
tion of various problems in the 8rea of the mechanics of contLined
fluids under reduced effective gravity has been observei, stimulated
by the practical needs of spacecraft operations. Among others, the
question of the dynamic response and stability of a large body of fluid
in a tank placed into subgravity, is one of considerable importance.
The matter of concern hns to do with the storage, transportation, and
utilization of liquid propellants in th( operation oE spacecraft power
plants.
In low gravitational environments, forces like surface tension
and the centrifugal force, induced by slow rotation of the fluid, will
Tiave a dominating effect on the large scale equilibrium configuration
and the dynamics of a fluid system. For example, a right circular
cylindrical tank, partially filled with a wetting liquid, spins about
its axis of revolution with constant angular speed and is placed in
a weaV gravitational field. Then the surface tension at the
interface, together with the centrifugal force, causesthe vapor cavity
(bubble) inqide the vessel to take ail elongated spheroLdlike shape,
oitjnted symmetrically about the axis of rotation, Equilibrium con
figuraticms for v4rioiis constant angular speeds have been stWied by
2
Rosenthal A concise survey on the literature to this field is
given in a revi,,v article by Habip [2].
The present study deals with the explGration of the nature of
liquid iftterface. oscillations in a rotatiTig tank and the possibility
of rotational stabilization.
The chosen frat&_ of reference is fixed in the tank and rotates
with it, The perturbation vtlocitieg, the interface wave amplitude,
and the disturbing forces are assuMed to bi2t small. Thus, in the equa
tions of motion, the aquation of contin.Qity, and the boundary condi
tions terms involving quadratic or higher orders of the perturbation
wantities are neglected.
The fluid is assuwd to be invistid and incompressible.
Surface tension and rotational v*locity components are both essential
in the study. They are taken fully into account As far is a 1n;nri7ed
theory will allow.
For this rotating fIqLd oscillation probl,*,m there ore two fre
qvwwcy orange for which the dilt(Irbancev Are of an entirely different
character Let w be tho dLWnsionletsa fr#quenlay of oscillation which
is o"sured rglative to tbo con#tant rot4tion ot the t4nkliquid system.
TW for I#) > 2, tNa type of the governing p*rtial diffarentLRI aqua
ti,&n J* Illiptic, while tor ji < 2, ittm flow field is de$crib*d bY A
tWpwrblolic #quotion. Ths %ethod of #nsly,#is and thw phvtLcal interpre
tAtt0h of t* Vow ph#m"*r4 *ft *ntirely dLrt*;*nt.
1111111 Mli
#qMpWWwjp blop"s #Ao*r to t%* 1bli<41roh*
AL
3
For he llipic ase, the bubble isapproximated by a probate
spheroid which is embedded into a spheroidal coordinate system. Hence,
associated Legendre functions can be employed for the series expansion
of the solution. A special method has been employed in order to account
for the homogeneous boundary condition at the wall of the tank. The
resulting eigenvalue problem for the relative frequency of oscillation
is nonlinear, since the eigenfrequency is included as a parameter in the
formulation of the differential equation as well as the boundary condi
tions. The eigenfrequencies are obtained through a successive approx
imation procedure. This procedure proved to be stable for the numerical
computations performed for a sequence of examples.
The result for this case indicates, that for the circumferential
mode m = 2, the first eigenfrequency greater than two corresponds to
the first mode a[ vibration in the meridian plane. Hence, for higher
modes of oscillation the flow field is certainly elliptic in nature.
For a forced oscillation, induced by a reduced gravity field of
constant magnitude and direction in an inertial frame of reference, the
relative frequency of oscillation is less than two. The governing dif
ferential equation is hyperbolic, hence the method of analysis is dif
ferent. Consequently, the structure of the flow field is different
from that of the above mentioned elliptic cage. The mathematical prob
lem (Cauchy and Goursat problem) is transformed into integrodifferential
equations which can be integrated nuperically by means of Picard's
method of successive approximations. It is found that a steady solution
in a cylindrical tank of finite length does not exist. However, a
al_4i
l/4
souineit naclnrcltn fifnt egh h etr
r V9
6
with as the cooponents of the perturbation velocity vector andwith
rpr FOPF z I as the coMponents of the ext*rnal forae.
The equation of continuity reads
V
(f A + + (2)
r r ?z
The system of differential equations can be made diumnsion
less ty choosing the semiminor axis of the undisturbed bubble radius
C as the length scale and d I at the time scalt. Further, the act
ing external perturbing forces in our case are ajis"aed to originate
from an oscillatory force field which ts tranWverse to the direction
of the tank axis. It iE assumed further that this field has a conStant
direction in the inertia Erame of reforence. Then the pressure cAn be
written at
C4 1 Ti + 2Pj + 6 fcje + POL (3)
where PoL is a constant and tha timr, dependent, Fecond term ol the
righthand side takes account oE the Perturb4ition forc* effectei. The
eqtiatiorrs of ftotion are then reduced to
P
2V (4m)
+ (4b)
?t
1W
it (4c)
whie te euaton f ontinuity becomes
? u + I V + O 0.
2.2 Drivation of te field equation
By means of simple elimination, relationsbewneahvlc
ity component and tepssure may be obtained from teeutoso
motion, namely,
+ ) 2 2 P a
t+2
S + +
(6b
Wi t l l*s t p t e i l q a i n o h e tu b t o r s
su~pis tansfrmedint
+ 9

adi t olw httefrgigeuto so litctpwe
Ic ,ado yeblctpwe i .Tels emo h et
hn si deo 9 aihpsi u=2 iialth toiycmoet
ca hnb xrse xlctyi trso h etrainpesr
an t ervtvs
II=t 1a
cu4'r/ V = curl V72
where
Hence, the velocity field can be characterized as follows:
a) For Lu > 2, the magnitude of the rotational velocity com
ponent is smaller than the magnitude of the irrotational one. Their
ratio tends to zero at the rate 1/W as W tends to infinity.
b) For w < 2, the magnitude of the rotational velocity com
ponent is larger than the magnitude of the irrotational one. Their
ratio tends to infinity of the order O(1/W2 ) as (u tends to zero.
The vorticity effect is so dominating that the velocity field may have
to take up a cellular structure in order to make the flow in the fluid
domain dynamically possible. The hyperbolic type of the field equation
permits such discontinuities in its solution. An example can be found
in Phillips [4].
c) For uo = 2, the partial differential equation is indetermi
nate in the sense that the Cdependence is arbitrary. Any solution of
the form 
where A (7,e) satisfies the field equation and the boundary conditions
M1
cni tin r disbe h ouini o nqe eci h
linaie hoysc lwi ntbe
Fo h ae ( h fiv rnfraino h pc
cori nae Fgre
wil reuetefedeuto (9 toteLpaeeuto
( +(2
t 0(
0!
114W I %%I4 W AWO 4W4 '68e W C
Another coordinate transformation is needed for the analysis
of this problem. The cylindrical coordinates ( ,9,u) are transformed
into the prolate spheroidal coordinates (c,p,9) by means of
S= 1 (13)
e = 9 ,
where a is a scale constant which will be determined later.
In this coordinate system the Laplace equation for the pres
sure field becomes
+O [.p] [ [[P' ) 1+ ? _1}O. (14)
It should be noted that the field equation is separable in the cylin
drical as well as in the prolate spheroidal coordinate system.
2.3 The boundary conditions for the elliptic case
The next step in our analysis is to formulate the boundary
conditions at the rigid tank wall and the free liquid vapor interface.
At the plane end disks of the tank the normal velocity, which
is in the idirection, must vanish. Thus
2 o at = 
0 at t J (15)
At the cylindrical portion of the tank wall the normal velocity com
ponent, which is in the fdirection, must vanish. Hence,
2 2 21
+  0 Af (16)
i II
Th ubeitraesalfrtb pcfe nteshr
cIdlcodnt yt*bfr th bonaycniin ar se up
Th proiae ube hp i sumdt h phri wihha h
samerato ofthesezamajr axs t thesemnino axs asthe exc
shp o hebbbe hesmiioraisi omlze o n iih
diesols oriaestee ytkn noacMttesrth
in detote ffn codiat rator~to tebubl anb
eaedk no h phridlcoriae ytm son fth ordnt
sufcs c=c osat hr sgvnb
K 17
j ,
In the preient cOSt we have
C( 'Y' (P''g)e (22)
hence
dC(
( W 0' (
A t dt ? a t
Neglecting quadratit berms in the above equ*tion th*r# re9ultg
L W (24)
dt
The. perturbation velocity rcuponent in the *,direation,
V may t>e written in terms of the pressure And its derivattws.
It turns out that
(W 4) '3a'! 3? (25)
At C 7(;Y') ao f
where h denotvtit scale fsitor. Then the above retultscan be
summarLzed in the following eqtjationt
t =: _4 (.(7 1) 'a P i 4
A dynatiieal boqndmify condition shall also toa satisfied besldrf
the *bove kiorcpatit:a 1 boundary condition. The pressure at both aides
I
of the interfaim, toZttb*,rQvdth thrMut4aze tepi(ion forew, v*tt be irl
equilibriua with tb*., rtia rorees. Now, the pm;M44
in the liquid h4,4. ion (3)
41
Intecvttepesr sacosat *n hntebodr
codto tteitrosrar
Jl +
_22( e+T * (7
15
The number E is directly related to the bubbleshaped factor K.
Their relation is shown in Figure 2.
2.00
1.50
1.00 E
0.0 03 0.5
Figure 2. The shape factor K versus the dimensionless number B.
For the limiting case E 0.5, the bubble becomes
an infinite cylindrical tube.
Then, finally, the dynamic boundary condition can be written as
with PO as the perturbation pressure to be evaluated at the interface.
The change of curvature J* J is a lengthy expression in terms
of 8, 9, Oe1(P,O), and the first and second derivatives of al (6G
with respect to $. The procedure of calculation follows the method
outlined in Struik [5].
//16
Al th/nrytase n te motn yaia fet
SOLUTION FOR Tit eLLIPvc CASE.
3.1 Representation of the solutiori of the fteZd equation
Sinct the Laplace equatLon (14) iti separable in the prelate
spheroidal coordinsten the perturbation preosure way be written #0
(,x, vo) A (.x) M S (0) (33)
After a few stes of standard computation thip 8olutii,n of the fteld
equation ig obtained in the following form
('X L A ('s) + (34)
M11 'h=M
in the *bove representation, the functions PNO),
n
and '(o) are tte Legandre associated functions of the first and the
Qn
second (iBd, respectively. The cooeffltients mn and B Mn are constants
to be determined by the boundary condittons.
The dependence on the variable 0 is 9i n by the terms e
wbere the number m is an Integer whie the value$ If 2#
Phys1cally, these are tfMAundavental, ieq*d on each of tW
circular crossaActi rotatton. The case
M ate$ that
shw ha ahciclrcrs etinmvs sAwol rnveset
th xs frtain I hs aeth yaoctfet reay merc
Iftepetraio rsur sitertdaln n cneticcru
lar pahisd hli oan hersl snnz .Teei
a ne reultnt orceactng n te flid odybouded y tis ath
Onteohrhni h ceeaio ilidcdb h oain
issm mtitebbl 13fel flaigisd th fli an hnc
th ufc tninofesn eitac oa aymti dsubne
exetlclyfrasalprino h nieitrao a hr
bee noetra oc il cigonti ytm hr ol en
19
it is possible to match a homogeneous boundary condition at this bound
ary by means of a method described in this section.
It may be found also from the computation of the difference of
curvature,' J* J that there is no interaction between the various
modes of e iedue to the surface tension effect. Hence, each mode of
different values of m can be handled separately. From the discussions
of the previous section, only the cases where m ? 2 are considered.
Now the summation with respect to m may be dropped from the represen
tation of the perturbation pressure. Then the number m remaining in
the expressions in question acts as a parameter rather than an index.
In the following a system of eigenfunctions will be obtained in
the cylindrical coordinate system such that:
a) The eigenfunctions are harmonic inside the tank;
b) Each eigenfunction satisfies part of the required
boundary conditions at the rigid wall of the tank;
c) The system of e igenf unct ions is complete over the
entire exterior boundary, i.e., the cylindrical wall
and the plane end disks.
Let it be required that the scalar product formed between P and the
first N eigentfunctions over the entire surface of the exterior bound
ary are all zero, and further let N pass to infinity. Since the sys
tem of eigenfunctions3 is complete, the perturbation pressure satisfies
the homogeneous boundary condition in the mean at the exterior boundary.
The Laplace equation aa written in the cylindrical coordinate
system is
!2
Le1 erpeetda
2 11e1P (0v()W(6
Wihti1ceeo eaainO vrals h olwn ytmo
unope riaydfeeta qain sotie
!u +
CC1 V
Ae (37
0(1 Ai
Th i gevle fo/h/a~e maentrlydeemnd8 h
seuneo nees0 ,2 . Teegnauso h aaee
Ll
21
i.e., at the plane end disks of the tank, where tJ
4
Hence
6j~ ~ ~ ~ ~ ;w A. A 9)in7,L40(9
In order to have a system of nontrivial solutions, it is necessary
that
Hence the eigenvalues are dete mined by
Arl=n TL n = 1,2,3,(1
This mean physically that the normal vtlocity a h nso
the tank vanishes. The cylindrical boundary is left unrestrce n
is free to oscillate. The normal velocity at the freecyidca
surface is given by
cQ
This sequence of eigenfunctions, repretentingthnomlvoc
22
a) The normal velocity at the ends of the tank vanishea;
b) The eitenfunctio"s are orthogonal to each other over the
entire surfac#, of the exterior boundary;
0 The seqwnoe of eigonfuncttoas is complete when its
deeiin is O"tricted to the cylindrical portion of the
exterior boftdary.
(B) The ca" ), 2 < 0
The matn purpo" for con4tructing these sftqqences of eigen
functions is that the normal velocity rspresentation, derived from the
eigenfunrtions of the perturbation preiure at the entire exterior
boundary, form a complwt* *Y;tam 4ance the thoice oE the boundnry
corWition for the iionfonctioni iA mor* or leis free. In the present
C28e th*re is no !straightforward b"ndary condition that can bot Prt
scribed *uch &,q in case (A). Tho Eollowing boundary condition haa b*en
stlectea to *tart with
(43)
Thor roaSoni for choooin# thiA boundary condition ara the following,
Th is A*, tho w#Ak## t boun4#r y c on d i t i*n wh ich a 11 own
choiew 4mon .# varipty ot othar conditionC
b) Thi "Off4petiont thtw obtained do not d*pond on the
0 Oeft otthop*4Wpiption wLth P*#Vcct t* the first
.4A#WMW at *W*mt4pctLqpj *A ot# ived tn tbt cjj* (A),
t* #L*Wbovct4" obt#io#4 in thU 0#*# wil I
oK bowW*ry condittoo *t tb#
C= 10 t OWn
d)
for*#,
to
23
In the present case the representation of the function P is
given by
Cos (44)
whtre J is the Bestel function of order m and the b 's are
M n
constants. The corresponding velocity cctftpone'nts u and w are
2 rn
4) Lt b, An) J" A, 1) + J" (,A,, C'OS P" U) e (45)
n
(wt 4 MO)
(W' 4) J, 1n U) e
nj (46)
and
(tit 4 MO)
n (47)
n=t
In order to satisEy the prescribed boundary oondition it is
sary to put
0 (48)
'Rence the eigenvalues art the zeroes of the function J'() 11
Thus the two kequences of eigenfuncti*ns are completely defind
When thi proper valiies of the coefficients a and b are giveu,
n n
tMse eigenfuncticms, evaluated at the exterior boundary, can be
written a#,t
dim
!2
Seunc1A
atteclndia al
(49
attepln nddss
Seunc
(Aa a l
at th yimrc
a W
25
For sufftctently I*rge numbtrs of n, the nwber ot waves of the fune
tLoa Pal($) *utstdw the hyperboloids and the number of vavei inside
In
approaches, a cons tant ratio v, gtvv4 by
V
IT
whev cot VO and 00 <
iw
Vor examplw:
when le, then v = 2.000.
Now the relatWi' importance of the boundary conditionl4t the cylitidrical
,Oall and the boundary tmdition at the Pnd diaks of the bubble oseilla
tion problem is amwed to folloi, theioam ratio v. Then the new
sequence Qf eigenfunttion.* Lg danumeratrO from the seq*ne#o(#) and the
sequence (t) in ach away that in the firstWVffi*slof the new
sequence th44&ti* of the rC4&rs ft4 sequence (A) to the members
f rom the se shall alvays be the rational rMasper t to V.
For tioned example, the at of the new
RL
se~ Ce 0 is
q27
3. h amc(neio)bodr odto
Sicaiu oe fdfeetvle f i r o ~~ed
th neirbudr odto 3)Er m 2 wl euet
t 53
it ol entdta h emin )i qain(2
26
functions at the cylindrical portion of the boundary and vanishes at
the disk ends, while the normal velocity representation, given by the
eigenfunctions of the sequence (B) forms a complete system of functions
at the end disks and, moreover, is a set of monotone functions, which
vanish very rapidly, as n becomes large, over the half lengths of
the cylindrical portion of the boundary. In the new sequence t n
the members from the sequence (A) and the sequence (B) approach a
constant ratio. It ia eaay to verify that the new sequence of eigen
functions gives a complete system of functions for the normal veloc
ity representation of the exterior boundary.
The eigenfunctions in the sequence I can be orthogon
n,
alized by using the HilbertSchmidt procedure. In the present study,
only the homogeneous boundary condition has to be satisfied. The
sequence ( .n I can be used directly. The results obtained are
equivalent to those obtained by using the orthogonalized sequence of
eigenfunctions.
Now the equivalent statement of the homogeneous boundary con
dition is
z > (_'i ldS,) A n = 1, 2, 3, ,(52)
where Z denotes the entire surface of the tank.
In the above integrals, the function P and the eigenfunc
tions ( n I are expressed analytically in two different coordinate
systems. Hence, thsse integral can be handled conveniently only
...... mm,
28
m 6o) A (f')foa m, to)
Q (56)
A M
rl + i m+ M + f ()j
n (57)
OD
0 m, A,, +
(5) fV 6 m fQ 4412
+ (a,
4
with
z (3 K'4 1)
m' + m 1<2 + K + 6 +( Kl oe'
rn + t i rnl m t 4 3
(3 K
0
a 3
+ 4fA
2 fA fAz I)
f z
z
OL
2fA
(40
a, ( n' 4 n + 2
Jjjjjjj ..........
MIT
a 2 m14 4 3 + 3,xo' (e(,,' +,2) n toto"+ Z), n, I
a 2 a47 n, + jn) 60(, + M 16 '4
A Ili+ + 7n + 12
a. 2( IP
fA
aw""JURT M
7111W TIMMER
30
to a sumation of terms oE TO(f) with Polynbaials *f 0 as the coef
n
Eicients. It appears in the form
(3)
n) P"3) (pf (M '+  f P" (p J" (59)
wher,_ 1,2,3,4, are polynamiiils of Thase Poly
n
nomials are obtaiaed by substituting equations (54), (55), tnd (56)
into equation (53) and multiplying the entire expreasion by a factor
2 2.0 P2 ). The hLghest order ter*A in the polynomials are
8. This boundary condition can be reduced further to a seri** of
P00) with constant coeCtitients by means of the recurrence forraula
n
T 2n+1 fz rn* I + ( n M) (60)
Then, f ina Ilyp the dynamic boundary cmditjon can be vritten in the
form
(ix.; M, W) Mit + (0(.j M W) B,.k 'D (61)
(n)
whor* th* f W cOTAtaOts A"rmtntd
b a, n# Ind contCtning (t) ljr**ly #n t'j#&nAr4beiwr. the
(L)
CO*fAf "Aw#* Of t*K Polly"004 (0), t4o Co"Irttnto
w(n) Cor **00, by
MM d0k or A O*qtW Wopoi*w4lf tor
4JO*. MOO*** 1W fit so *a
A system of linear algebraic equations in terms of A mnand
B wn can be obtained by forming the scalar products of this boundary
condition and the eigenfunctions Pm($) for n 2t m. Together with
n
the exterior boundary condition, there are sufficient algebraic equa
tions to determine the eigenvalues of u), and subsequently the eigen
solutions of A inand B mn, which determine the mode shape of the oscil
lations of the bubble interface.
In concluding this section, there is one remark which should
be made concerning the effects of the rotation of the entire system
and the surface tension of the oscillations of the bubble. In the final
form of the interior boundary condition there is a second summation sipn
which sums over k for nine terms. In an elementary oscillation prob
lem, the second summation sign is not present. Each simple mode of the
harmonic oscillations, as given by the eigenfunctions, is distinct,
such as in the case of the vibrations of a string with fixed end points.
In the present linearized theory of oscillations, the surface tension
effect induces an interaction among nine simple modes of harmonic oscil
lations, while the rotational effect alone will induce an interaction
among three consecutive simple modes only.
3.4 Numerical solution
If the series representation of the perturbation pressure is
truncated to n terms only, the boundary conditions can be reduced to
a system of linear algebraic equations in terms of the unknown constant
In the present case the oscillations are symmpetric with respect
I3
cofi cins an h IgnaaIe oiscnandi hs q
 os.Sneteoclain r yotiol h eo ihee
nubronm r oeo hreaeol nm fkmcn
IIatAm n i Ec ftebudr cniin ilrne
fI )+I lna qa~n fteukoncntns h ytmo
lina eqain is ttriae
Th eutono h ondr odtint t't o le
bri ceutosi gie intefloi .
Th xeirbudr odto 5)yed
33
At this point it is convenient to introduce matrix notation to
handle the system of algebraic equations. All the matricei, G, H, A (1)
(2) (1) (2) 1
A B defined below, are !(nm) + 1 X
square matrices. Tht vectors X and Y are two column matrices with
I
I(nm) + I elements. They are
C, IJ
H ij
A (64)
A ij
rM C(
(m n)
M
m + 2 (j
4M
Then the exterior boundary condition together with tN interior boundary
coridition can be written in the following concise Eorm
A(') A2) BO) Be2) X
0
(65>
This matrix equAtion can be solved, employing,#tandard nota
tion as follokn:
First ve have
HY
!34
wi chyed
X Im .( 7
Furhe
(A O X+( 2 2) ,(8
upnsbtttngfrXIehv
(A()BB())11Y)+( 2 2 y 0
W
or
II II
35
yie Ids
Y = si, (72)
and hence by equation (67)
l
X = G HY.
For any specific example, where dimensions of the tank, the
density of the liquid, the constant speed of rotation, and hence the
static equilibrium shape of the bubble are given, numerical results
can easily be obtained by performing the computations according to this
theory on a digital computer.
In the numerical computation the scale of the affine trans
formation
J w4 in
has to be chosen at the beginning. Then according to the obtained
eigenvalues of a this scale of the affine transformation can be com
puted employing the above given formula for t. The new value of this
scale factor is then used for the computation of a more accurate value
of u.
It is important to point out that a different scale factor
exists for each different eigenvalue of i,. For very large values of ru
the scale factor is close to 1.
3. eutiaddsuno
Nueiaiopttoshv ee aefrtecs s=2
wi c iste lws cicmeeta oiune th scp oftee l 
tic ae h eutsosta h frtegnrqe bv .0
corsod last h is oeo silto ntemrda
pln.Fr n pcfi xml w r t~r ocmpt h irs n
thi eodeznrqece cuatl. W a ocueta l h
naua!rqece ftebbl siltosaegetrta w
an hi rcrepnigprubto r"uefedi eli ic n
naue ic ehv bandte oe on fteefeunis
Sm fte ueialrsls r rsetdi Tbe1 ig
ur ,an iue Te cmuaio Apefre frasquneo
bubl hae wil hetnkdmesin rminth am.Th irs nd
TABLE I
T11JR FIRST ANDIC & QU904* F09 RUMM
lwm DIFFERE 'RAP IN A CYLTMD(ICAL
TAW rivTmom
T4e shape factor The first The Aikocond
K eigenfrequency elgencrwquency
11100 2.296 2.409
1.200 2.401 2.558
1.400 2.516 2.780
1.600 2.620 2.977
1.800 2.754 1.,*35
2.000 2.941 3.605
2.200 3 166 4.007
Th t 2.000,
3II2 0
3.0
1
3.1
1l
2.5
Fis1ienrqec
39
3 G ............
2.0
0
figure 4. The typical mode shapes iT) the meridian plaao,
corr g to t ir igen
I S tmffu.bb le
f qu
;;BE. IMMEMMUM
CHAPTSR TV
SOLMION TO THE HWV"OLIC CASE
4.1 Introduction
In this chapter the
fluid systev with respect to the perturbation Jue to a constant
reduced gravity field transverk to the tank axis will be ttudled.
From the discussion on page 18, we know that for the tiwda of oscil
lation M t 1, the motion has to W a4companied by an external
force field. The governing equation is hyperbolic. Hence the
mottho43 of analysis and the mathematical formul,ition of the problem
are dlfferevt from the ellipti4 cas< studied above.
For the physical conditi*ns given abova, the frequency of
the perturbation n relative to the rotating syst4m is one. How
ever, the analysis for this p#rticulior case, as pe"U*d in the fol
lowing section* of this chapt*r, is valid atjo f*r nay frequgncy
in ihs, pnge 0 < i[ < 2.
Tb% mAtbematicnI formul,*tion her* is w Ltten In cyiin
driewl coovdWSMS and t4w coordin#t*#" The 14ktttr
iki I I b* in 04A ch*ptiir.
AW tow #1044W cobdIti0% for iI4llVo*d prob4# in th*
4,"W# of 4***WA j*jW dtk4#W#t fQx NowrJolic Od v#ILPttC' *qua#
4
ow "i* OOMPPOW 4% 10.)** to
AL A
*owl, *44 And 06
41
interface, 8s well as a dynatutc condition at the interface. Such
a system of boundary conditions is too stringent For the solution to
exist. We shall see in section 4.4 that the boundary condition at
the ends of the tank has to be relaxed. Bence we are considering
a cylinder of infinite length instead of one with finite length.
Since the differential equation governing the pressure.
field is hyperbolic, there are real characteristic surfaces in the
flow domain. Across these surfaces the normal derivatives may be
discontinuous. Hence the velocity field will suffer a finite jurmp
at the same location. The work by Oser [6] provides a good example.
4.2 Formulatio ofte problem
The field equation in cylindrical coordinates (,,)for
the case m = I and ar I is given by
?P  3 J = 0
The first order derivative term in this equation can be eliminated
by means oL the transformation
f ( ,( = 2 (q.EC) *(73)
Then the partial differential equation for takes the form
This transformation has been carried out for the sake of conven
42
For the above hyperbolic equation (74), there ar* two
f"ilies of real characteristic curves through each point of the
plane. Both of thejw families are straight lines.
These characteristics may be chosen as ojr coordin4te
system, The characteristic traasformation in quettion is
A I f = 7 437 ;(
/ 3 , = fj C 7 (75)
where y = const. and T = const. are the characteristic lines
fr
Along the charactetistic directions, the transformation
dots not determine the gtcond derivativos uniquely. Hance across
a characteristic line, the normal derivative may suffer a jump.
However, the 6inctivn t is quppoUd to be 4oDtinuoun or a hydro
dynamical problem.
Throligh the trar#Coieaation (75), the hyperbolic differ
enti*l equation (74) turns into the nor4i#,I formi
(76)
For thi# #quation two typo of probAWW can bV posed.
(A) 74 Owhy Probl"
TO jn*t*j crodjt"a 1'" p"jc*JW on # cut** PQ wbich
nmop" t to 0Wdh4"ttef*#LC d ft*et toni 4 thu t4 V*ItAt
or 00 IMMOMB 4#* *no it#, Aar#* 4"1"**" ft thts cwvw #rv
0 M4 Aftifto y 46k#40"4 4 n t* tf i *n#TA 
4T
WNW
43
Figum 5. A sketch for the CaQchy probJem.
The entire system of differential equations and the inittal conditions
can be combined into the foltowing integrolifterential equation
to +1!% (77)
ax5T 4 N + r,)' 4(x dt
R
This equatit;n can aAJB4% tion technique. This j>r*Zm0w:&
4
is pr C *in lar4 vhen
9 A
44
(R) The Gourwat problev
We may also, pr**crtbe one boundary condition on the ordinary
(noncharacteristic) curve and one on a charactari!stic liae. Tt the
value of # is pretcribed on these curves, the problem is (,,alled the
Coursat problem. A solution exi!!ts also and is uniquely determtned in
the triaagular do"JTI CrrQ (Vigure 6).
1 T,
T
= V0
P
Ne
41
Figure 6. A sketch for tht Goursat problem.
Th4A probtawcbm b* forma10W as an into"I wqu#tion, n*slyp
11 'J' At (79)
4(I+V
This equation can be solved by means of Picard's method of success
approximations as in (A). When the boundary condition on the ordinr
curve is replaced by a mixed type condition
the existence and uniqueness of the solution remain unaltered. Th
problem is well posed. However, the integrodifferential equationi
somewhat complicated. Let us denote the ordinary curve by
The function ((T) is strictly monotonic. Then the integrodiffer
ential equation in question can be written in the form
=I (T) =s 4 +T)
The solution of the Goursat problem fralinear equation exists i
the large, as in case (A) .
Numerical computations have benperformed according to th
above given integrodifferential equtios.
The boundary conditions prescribd for this problem hnavcien
derived in Chapter 11. They are give by equations (15), (16), (2)
and (32). They are listed blow as rertten in the cylindrical
46
(A) The kin tic botindary Conditions at the walls of the tank:
?i t 1,5 at tIIA cylindrical wall,
(81)
at the plaRe disk ends.
(B) The kineiftatic boundary condition at the interface:
Vn S = 0 (82)
The normal velocity v n and the iiormal displacement 5 of the liquid
vapor interface are actually 90 0 out of Phase when 9 and t are
taken into account.
(C) The dynAmLc boundary condition at the interface
SE + B j XJ)
vh* rt 6 is the emponent of 6 in the 11dire,tion. This boundary
,condition, esWciatly it# right hand side, is very complicated. We
*hAAll Mplaci it by two asymptotic reprt*eRtations. T'his approximation
Producog Oo%* quantitative wrrorat while it demonstrate, however, the
qualitatiV* natur** of thg problem *uch more c1farly,
For a mall displacAmwmt 6 of the bubblo interface, the ch4ogo
of curvaturi say twk approxjoawtod by
4Z6
J, dsl
vo4ft d J,#v a t*VVVFA 4k t*kjot M&tu**d 0 16 tjo ter of tho
JAW inv*444*
1%
47
in the neighborhood of the equator of the bubble, d ds.
Hence we have the following asymptotic representation of the boundary
condition:
4 P. ) + (83)
In the neighborhood of the poles of the bubble the curvature
of the bubble is pronounced. The geometry in this region is best
described through the spheroidal coordinates defined in the previous
chapters (Figure 7).
0 O
di
Figure 7. The coordinate system employed at the
polot region of the 'bubble.
49
The formialas applicable tO the Present sitQatton, are sAftmarL2ed below:
KI T
FK T
Cf
K
d 5 + J
Near the pole the value of cp it small. The curvature may te written as:
j 26 i's I
dS I  614 2
The 7component of the displacement 6, 611 is also very small in
tKis rtgion. tt can be neglected. Hence we have Another asymptotic
repreaentation for the dynamic boundary condition:
i's LIf 12
SEP", + B60Y CY(CAS) (84)
The q4anttty 4p) *pproaches onq. ps appro4ches zero.
as
The combination of the kintmtritic condition (82) and the dyn&Tn
ical condition 03) at thw intarf,3ce providta ut only with on# boundary
condition for t* field equAtion (76).
(D) TO condition of A#W#try at tho *quotorial pione
U A*#iMA that tt4 flw Ijkld i* *Wwtric witb rOptct to th*
ul 4##vo t1W condition
49
There are two more conditions for the determination of the flow
field. These conditiont will be introduced in the next section in
connection with the construction of the solutLon to this problem.
4.4 Construction of the solution
For a hyperbolic equation discontinuities of the normal deriv
atives are admissible across a characteristic linr. We shall construct
a solution to this problem with all the possible discontinuities in
mind. Figure 8 shows the division of the flow field into regions
separated by characteristics where discontinuities may occur. In this
figure the point G is located such that the bubble is tangent to the
characteristic line.
As indicated in 4.3, we have to postulate two additional condi
tions to construct the flow field.
a) In the tmmedLate neighborhcod of the equatorial plane, the
flow field is symetric. The perturbation velocity normal to this
plane vanighes. The bubble is tangent to a cylinder. Hence the per
turbation pressure distribution given in Phillips [4] is exactly the
same an the perturbation pressure distribution at the equatorial plane.
In our notation there follows
j _L ' ( 3
0 2 SE 12 _! (85)
b) The perturbation velocity At the axtt of rotatim is finite.
Frcm t4e expreit'ion of the perturbation velocity (6), we arrive at
the conditiqn
A
t*fMWm1tIto1ta4#*11t4b
We can now procttd with the construction of the solution.
The condition 4.30)) tojether with condition 4.4(a) form a set
of initial conditions for the determination of the flow in the region I.
Consequently, the,&eneralized Govrsat problems can be detined in the
regtons It through V succ"sively Their solutions are obtatned by
means of equation (78) or equation (80). 1" the region VI, the solu
tion is obtained as follows.
The displacement of the bubble 6 at cp m 'D vAnished for the
geometrical compatibility requirement. in a neighborhood of the pole
such that 7 < I B 1, both the inertia and the coatrifugal field
forc*5 are smaller than the surfacie tmnsion and the reduced forces by
at least two border. of magnitude. Hence from equation (84) we have
jq) 'I = 8 5
dS
or,
+ 'K' 1) 56
hence
+ in (f + sin +
where the constant a 0 Is determined by tUs condition that the 4is
placement at G is continuous. By 4*ans of equations (82) pnd (84)
the solution in is determined .
proceed to *fiP;Asdtttion in further UWovtl
we fi ined he tondition
at t ti to
Wudayi !reoew wtimdaeyta th souine eds o
inint. Pyial tien thtased stt souinde nt
exs o ako iielnt. Teslto xssfracln
dica ako niielnt. I t[ aeteprubto rp
agte toifntmeitl. Sc nuepce hnmnna on
inasmlrpolmivsiaedbejmnadBradI]
Th ouini h ein IVII,..,cnesl
htdtrie!yasqec fgnrlzdCustpolm.Tepr
tubain ildapra,,e zr a w xtn tesouio o nint
Along the Cd ire ct Lon.
53
00 w
& 0
bc
00 ao0
0 do
Wa
rC&
4 0
no a
C,4 w
IN
461 on
'K Ln
54
mm
551
PPI"
56
tension provides the rmtc"ary adjustment to absorb th, pressure field
perturbatLons. In this rWgion, the liquid particle perforvo maLnly
an oscillatory motion in the neighborhood of its own e
position.
CPAPTER V
CONCLUSIONS
In the previous chapters we have determined the dynamic
response of the rotating fluid system for the entire range of fre
quencies. In conclusion, we shall Kscuss hatt iome of the phys
ical implications oE the results obtained ab*ve.
The slow rotation of the system with a constant angular speed
has a profound effect on the dynamic responss. For a rotating system,
all the small oscillations of the liquidvapor interface, or rather,
of the entire liquid body, are stable. I small transverse disturb
ance to the system will induce one or 7everal modes of otcillattun
about the stable tqUilibrAm configuration.
OR the other hand, unber the Qfluenoe of a tranwtersa constant
force fiell, the stability of the configuration of the system in the
Anighborhood of the eqwor of the bubble isensured by the tentrif
ugal pressure field and the Wertia force produced by a small pertur
bation to the tonyant rotating base flow. The surface tension effeat
is nealtAbie in thia region. For a system without rotation, a AS
turbance tontAining such a force component is Rable to poite
instability.
for areal system viscosity effetts are always
present* with a uce, 60=me
secondary circulation. The Asturbance vill be dissipatV by means of
the vincous; mechanism. The surfato tension effect alone is a too
d0onsistal menhanism, It can reAtore a d0turbed systim to an equi
libriam conUguration in a much longhr period of time.
Finally, we shall present here an example to show the actual
magnitude of various quantities characterizing a rotating tankliquid
system in low gravity environments.
Given:
The radius of the tank R = 100 cm
Si_miminor axis of the
undisturbed bubble C = 40 cm
Bubble shape factor R = 2.00
The dimonlionles* nlxftbar E
cortt9ponding to the bubble
shAps factor K R = 0.43
3
DenmLty of the liquid = 0.070 9/cm
Surtace tension T = 2.00 dyne/cm
The liquid Qo*en in this gxample is liquid hydrogen. From the nbove
given d4t*, *a may wompKe the corr*4pondiag constant *ygular *pled of
rototion p and th* allaw*bhe **gnitode of the transvorse Peduced
gravity 101d.
rod
ITILVIC _2 0 07 'A (40)1 25
9E
Assume that
Then
01 O0 x Q x C /C
For this small acceleration the system can cover a distance of 100
feet in fifteen minutes. This would allow the system to perform a
s low maneuver like changing the orientat ion of the tank, etc.
APPNDII
APP3NDIX A
EXPRES81ON OF TRE PERTURBATION VELOCITY C"4PONM v IN TM
CY
PROLWE "'REROIDAL rOORDINATR$ (a,0,8)
I
The expression of v & may be derivpd from the known expressions
of u and w iu the deformea cylindrical coordinates
LP +
(W2 4) 14 = i W '2
(Al)
by means of tlia daftneO coor0inate transformation relation, equation (J).
Let us dsfins
lop
71 91
Accotding to the coordinate tramsformatiou defiDed tn equation (13),
we find
(A2)
1 21
Ok,
J t
ji a (A3)
t d t
and
9
U
(A4)
where the superscript T tn equation (0) denotes the tra"pased
matrix, and
d 0
7, L A Voc
IA Aid (A5)
Ti ' I Tt
Hence, by using equations (Al), (A2), (A3), and. (0),
we obtain
CL 2 + (A6)
T 1 (d P") 9A,
The txpre**i*o for V,,, follo*;* imftdiately frco equstions 46) anil (46).
Aoki
APPENDIX B
FLOW ClLkRT OF THE (",OMPUTI!+G PROGRAM EMPLOYFD
IN CHAPTER ITT
G D
READ: Tank dimensions.
Bubble nhape factor.
Assumed scale of transformftion FUNCTION:
The relation of
the shape factor
K versus tha
J imens ion less
Dimension statement. number E.
Equivalence statement.
Define complex variable names.
t 
Aritlimetic statement for COSH(X
Compute o(,, according to formula (17).
Determine constants of the coordinate
transformation from (je,,U) to
Compute all constants related to equations (55),
(56), (57), 2nd (58). Store results by means of
Lbscript variables.
Construct a polynomial multiplication segment.
Compute W (P) according to equation (55).
n
Construct a segment of program for the repeated
application of the recurrence formula (60).
Compute, rn, W
8torA the obtained 1K
proper locations of the matrices A(10 x 20) and
R(iC x 20).
64
READ: The size
of increment for
the numerical
integration along
the walls of jthe
'ze
tank.
FUNCTIONS:
The associated
Legendre functions. COMPUTE: The values of
(c' rn F M()
) F, (P) and
at points on the tank walls.
FUNCTION: The About 25 to 50 points are
hypergeometric taken f or a quarter of the
function. tank along a meridian section
line.
FUNCTION The
Gamma function.
FUNCTIONS: The Bessel
function J and the
modified Bessel
function I.
tjon
Construct the sequence of eigen READ Zeroes of
functions Ii. Compute their values the Bessel func
g
4t the aasigned points on the tank tion J' The
walls. Store result in matrix form. denumeration of
the sequence of
e Piigenfunct ions.
Computation of the numerical
LnttegratiouS aCuOrding to
the foroulaa
P""(') FMK
Store thiR obtairwd v#luet
of j',k Aod %K in PrOPer
loe#tion* of # m4trix
ATX(10 X 20).
3
65
2
READ: The range
of variation of n.
Say, 4 to 7.
T rai n s r r i be, n x n blocks
from the matrices A, B, and
ATK to form the new matrices
(11) (2) (1) (2)
A A B B G,
and H in equation (65).

READ: A scale
factor for over
flow control.
1Pr
Solve the matrix equation (65) SUBROWINg
by the steps (66) through (71). Matrix inverson.
SUBROUTINE: For finding
the eigen"alues and eigen
vectors for a real matrix.
WRITE: The The eigenvectors given
eigenfrequencies./ in te ms of the trans
formed basis.
WRITE: The ranstorre the eige"Vectorg
eigenvactors. b ck to the original basis
by equations (67) and <72).
END
REFERENCES
(11 Rosenthal, D. K. The shape and stability of a bubbl,&_ at the axis
of a rotating liquid, 1. Fluid Mech. 12 (1962), 358.
(21 Habip, L. M. On the mechanics of liquids in subgravLty.
Astronautica Acta 11 (1965), 401.
[31 Morgan, G. W. A 3tudy of motions in a rotating liquid.
Proc. Roy. Soc. A, 206 (1951), 108.
[4] Phillips, 0. M. Centrifugal waves. J. Fluid ch. 7 (1960), 340.
(5) Struik, D. J. Lectures on ClaSsical Differential Geometry. (19SQ),
AddisonWesley Pres5, Cambridge, Mass.
[61 Oser, Ii. Erzwungsne Schwingungen in rotierenden Flissigkeiten.
Arch, NzIt. Mech. Anal. 1 (1957), 81.
[71 Garabedian, P. R. Partial Differential E Vations. (1964),
John Wiley & Sons, New York.
[81 Bepj,min, T. B., and Barnard, B. J. S. A study of the motion of
a cavity in a rotuting liquid. J. Fluid Mech. 19 (1964), 193.
66
kU 1w _a
BI10GRAPHICAL SKETCH
Suikwong Paul Pao was born on October 23, 1940, in Canton,
China. He received the degree of Bachelor of Science in Engineering
from the National Taiwan University in July, 1961. He enrolled in
the Graduate School of the University of Florida in February, 1962.
He worked as a graduate assistant in the Department of Engineering
Mechanics until April, 1963, when he received the degree of Master
of Science in Engineering. He worked as a research assistant in the
same department in the following summer, and as a structural engineer
in an engineering firm in New York, N.Y., between September, 1963,
and December, 1964. He resumed his graduate study in the Department
of Engineering Science and Mechanics of the University of Florida in
January, 1965, and was granted the College of Engineering Fellowship.
Suikwong Paul Pao is married to the former Juliet Yungli Zue
He is a member of Phi Kappa Phi.
This dimsertation
chaimAn of the d*ndidatV4
approved by al I
the Doan of the Colle
was was appromd as parl
degree of Doctor of PKIcs
,*pril, 1967
Am.