DYNAMIC RESPONSE OF A WETTING
LIQUID ENCLOSED IN A IZOTATING TANK
IN A ZERO-G ENVIRONMENT AND
SUBJECTED TO VARIOI-TS
SU1-KW'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
TO M LOV
Th uhrwse oepeshi hnsadapeito -
TABLE OF CONTENTS
ACK"t4"EMS . . . . . . . . . . . iii
LIST OF TABLES . . * . . . . . . . . v
LIST OF PIGU#SS . . . . . . . . . . . vi
PARTTAL LIST OF SYMOLS . . . . . . . . . vii
ABSTRACT . . . . . . . . . . . . . viii
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
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 $91MCH . . . . . . . . . . 67
LIST OF TABLES
1 THE FIRST AND SECOND EIGENFREQUENCIES FOR BUBBLES
WITH DIFFERENT SHAPE FACTORS IN A CYLINDRICAL
TANK WITH FIXED DIMENSIONS . . . . . . .
LIST OF FIGMES
I A mxridian gection of the defortied cylindrical
coordinate system I 9 ,u ) and the prolste
t phe ro ida 1 c oord in ate sys tam ( x P 0 ) . . . . 10
2 'rhn shape factor 9 'versus the dimensionless number E.
Vor the limiting case E t_- G.5, the bubble becoms
an infinite cylindrical tubi, . . . . . . . is
3 The first and second eigenfrequencies plotted vrrsus
the hubble thape factor t . . . . . . . . 38
4 The typical mode sh-apes in tht meridian plam
corresponding to the first and the second eigen-
frequencies of the oscillations of the bubble
about Its position of stable equilibrim . . . 39
5 A 94wetch for the Cauchy problem . . . . . . 43
6 A Sketcb for the (;Ojjrsat problea . . . . . 44
7 Th* coordinate Aystem eployed at the pole region
of the bubble . * . . . . . . 47
8 Division of the Elow field tnto regions separated by
eharacteristic IL-nes whtre discontinuitios may occur 50
9 TWe pwrturba4tion pras$ate f1*1d for 4 tank-liquid
*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&nk-liquid
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#nk-liquLd sytt4oj
h 0 cosj"*t *%*4#r 4"#4#1 wW*v 'tb*
PARTIAL LIST OF SYMBOLS
t t ime
0 constant angular speed of the rotatitg tank
2L leqgth of the tank
R radium of the cylind-er
C semi-minor-axis of the un4isturb-4od 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
(uvw) dimeasionle** perturbation velocity c*wp*ne nts,
correspff o the directions
P dimens less perturbation pressure
T Coef of t ace te
t he 'U'n ble
Abtato israin'rsetdt 1eGaut oni
in|tilFlile~ fteRqi~kt o h
Dereo otr CPiooh
DYNAIC RSPOSE O A ETTM LIQUD SCLGSD I
A R(TTN A*I ZP- NIOM N
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-
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 self-sustained 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. The-governing
differential equation is hyperbolic, hence the. methodi of analysis is
frm thto h bv etoe litccs.Temteaia rb
le stasore noiter-ifrnta 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
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-
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) in-qide the vessel to take ail elongated spheroLd-like shape,
oitjnted symmetrically about the axis of rotation, Equilibrium con-
figuraticms for v4rioiis constant angular speeds have been stWied by
Rosenthal A concise survey on the literature to this field is
given in a revi,,v article by Habip .
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 t4nk-liquid 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.
#qMpWWwjp blop"s #Ao*r to t%* 1bli<41roh*-
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 integro-differential
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
souineit naclnrcltn fifnt egh h etr
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
(f A + + (2)
r r ?z
The system of differential equations can be made diumnsion-
less ty choosing the semi-minor 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 ca-se 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
C-4 1 Ti + 2Pj + 6 fcje + POL (3)
where PoL is a constant and tha timr-, dependent, Fecond term ol the
right-hand side takes account oE the Perturb4ition forc* effectei. The
eqtiatiorrs of ftotion are then reduced to
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
+ ) 2 -2 P- -a
S + +
Wi t l l*s t p t e i l q a i n o h e tu b t o r s
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
cu4'r/ V- = curl V72
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 .
c) For uo = 2, the partial differential equation is indetermi-
nate in the sense that the C-dependence is arbitrary. Any solution of
the form --
where A (7,e) satisfies the field equation and the boundary conditions
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
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 i-direction, 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 f-direction, must vanish. Hence,
2 2 21
+ - 0 Af (16)
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
In the preient cOSt we have
C( 'Y' (P''g)e (22)
( W 0' (
A t dt ? a t
Neglecting quadratit berms in the above equ*tion th*r# re9ultg
L W (24)
The. perturbation velocity r-cuponent 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
of the interfaim, toZttb*,rQvdth thrMut4aze tepi(ion forew, v-*tt be irl
equilibriua with tb*., rtia roree-s. Now, the- pm;M44-
in the liquid h4,4. ion (3)
Intecvttepesr sacosat *n hntebodr
_22( e+T *- (7
The number E is directly related to the bubble-shaped factor K.
Their relation is shown in Figure 2.
0.0 0-3 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 $. The procedure of calculation follows the method
outlined in Struik .
Al th/nrytase n te motn yaia fet
SOLUTION FOR Tit eLLIPvc CASE.
3.1 Representation of the solutiori of the fteZd equation
Sinct the Laplace equatLon (14) iti separable in the prelate
spheroidal coordinsten the perturbation preosure -way be written #0
(,x, vo) A (.x) M S (0) (33)
After a few stes of standard computation thip 8olutii,n of the fteld
equation ig obtained in the following form-
('X L A ('s) + (34)
in the *bove representation, the functions PNO),
and '(o) are tte Legandre associated functions of the first and the
second (iBd, respectively. The cooeffltients mn and B Mn are constants
to be determined by the boundary condittons.
The dependence on the variable 0 is 9i n by the terms e
wbere the number m is an Integer whie the value$ 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 h|li 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
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
2 11e1P (0v()W(6
Wihti1ceeo eaainO vrals h olwn ytmo
unope riaydfeeta qain sotie
Th i gevle fo/h/a~e maentrlydeemnd8 h
seuneo nees0 ,2 . Teegnauso h aaee
i.e., at the plane end disks of the tank, where tJ
6j~ ~ ~ ~ ~ ;w A. A 9)in7,L40(9
In order to have a system of nontrivial solutions, it is necessary
Hence the eigenvalues are dete mined by
Arl=n TL n = 1,2,3,(1
This mean physically that the normal vt-locity 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
This sequence of eigenfunctions, repretentingthnomlvoc
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
(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 pe-rturbation 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
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
In the present case the representation of the function P is
whtre J is the Bestel function of order m and the b 's are
constants. The corresponding velocity cctftpone'nts u and w are
4) Lt b, An) J" A, 1) + J" (,A,, C'OS P" U) e (45)
(wt 4 MO)
(W'- 4) J, 1-n U) e
(tit 4 MO)
In order to satisEy the prescribed boundary o-ondition it is
sary to put
'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,
tMse eigenfuncticms, evaluated at the exterior boundary, can be-
(Aa a l
at th yimrc
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
approaches,- a cons tant ratio v, gtvv4 by
whev-- cot VO and 00 <
when le, then v = 2.000.
Now the relatWi'- importance of the boundary conditionl4t the cylitidrical
,Oall and the boundary tmdition at the P-nd diaks of the bubble oseilla-
tion problem is am-wed to folloi-, theioam ratio v. Then the new
sequence Qf eigenfunttion.* Lg danumeratr-O from the seq*ne#o(#) and the
sequence (t) in ach a-way that in the first-W-Vffi*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
se~ Ce 0 is
3. h amc(neio)bodr odto
Sic|aiu oe fdfeetvle f i r o ~~ed
th neirbudr odto 3)Er m 2 wl euet
it| ol entdta h emin )i qain(2
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-
alized by using the Hilbert-Schmidt 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
Now the equivalent statement of the homogeneous boundary con-
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
m 6o) A (f')foa m, to)
rl + i m+ M + f ()j
0 m, A,, +
(5) fV 6 m fQ 4412
z (3 K'-4 1)
m' + m 1<2 + K + 6 +( Kl oe'
rn + t i rnl- m t 4 3
2 fA fAz I)
a, ( n' 4 n + 2
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
to a sumation of terms oE TO(f) with Polynbaials *f 0 as the coef-
Eicients. It appears in the form
n) P"3) (pf (M '+ - f P" (p J" (59)
wher,-_ 1,2,3,4, are polynamiiils of Thase Poly-
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
T 2n+1 fz -rn* I + ( n M) (60)
Then, f ina Ilyp the dynamic boundary cmditjon can be vritten in the
(ix.; M, W) Mit + (0(.j M W) B,.k 'D (61)
whor* th* f W cOTAtaOts A"rmtntd
b a, n# Ind contCtning (t) ljr**ly #n t'j#&nAr4beiwr. the
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 pro-ducts of this boundary
condition and the eigenfunctions Pm($) for n 2t m. Together with
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
cofi cins an h IgnaaIe oiscnandi hs q
| os.Sneteoclain r yotiol h eo ihee
nubro|nm r o-eo 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
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 -!(n--m) + 1 X
square matrices. Tht vectors X and Y are two column matrices with
-I(n-m) + I elements. They are
m + 2 (j
Then the exterior boundary condition together with tN interior boundary
coridition can be written in the following concise Eorm-
A(') A2) BO) Be2) X
This matrix equAtion can be solved, employing,#tandard nota-
tion as follokn:
First ve have
X -Im .( 7
(A O X+( 2 2) ,(8
(A()-BB())--11Y)+( 2 2 y 0
Y = s-i, (72)
and hence by equation (67)
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-
J w-4 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
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.
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
T11JR FIRST ANDIC & QU904* F09 RUMM
lwm DIFFERE 'RAP IN A CYLTMD(ICAL
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,
3 G ............
figure 4. The typical mode shapes iT) the meridian plaao,
corr g to t ir igen-
I S tmffu.bb le
SOLMION TO THE HWV"OLIC CASE
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 mathematic-al 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 #1044-W cobdIti0% for iI4llVo*d prob4# in th*
4,"W# of 4***WA j*jW dtk4#W#t fQx NowrJolic Od v#ILPttC' *qua#-
ow "i* OOMPPOW 4% 10.)** to
*owl, *44 And 06
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  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-
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
Along the charactetistic directions, the transformation
dots not determine the gtcond derivativo-s 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-
Throligh the trar#Coieaation (75), the hyperbolic differ-
enti*l equation (74) turns into the nor4i#,I formi
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 0W--dh4"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 -
Figum 5. A sketch for the CaQchy probJem.
The entire system of differential equations and the inittal conditions
can be combined into the foltowing integro-lifterential equation
to +1!% (77)
ax-5T 4 N + r,)' 4(x dt
This equatit;n can aAJB4% tion technique. This j>r*Zm0w:&--
is pr C *in lar4 vhen
(R) The Gourwat problev
We may also, pr**crtbe one boundary condition on the ordinary
(non-characteristic) 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).
Figure 6. A sketch for tht Goursat problem.
Th4A probtawcbm b* forma-10W as an into"I wqu#tion, n*slyp
11 'J' At (79)
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 integro-differential equationi
somewhat complicated. Let us denote the ordinary curve by
The function ((T) is strictly monotonic. Then the integro-differ
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 integro-differential 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
(A) The kin tic botindary Conditions at the walls of the tank:
?i t 1,5 at tIIA cylindrical wall,
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 boun-dary condition at the interface-
SE + B j X-J)
vh* rt 6 is the emponent of 6 in the 11-dire,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
vo4ft d J,#v a t*-VVVFA- 4k t*kjot M&tu**d 0 1--6 tjo ter of tho
in the neighborhood of the equator of the bubble, d ds.
Hence we have the following asymptotic representation of the boundary
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).
Figure 7. The coordinate system employed at the
polot region of the 'bubble.
The formialas applicable tO the Present sitQatton, are sAftmarL2ed below:
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 7-component of the displacem-ent 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", + B-60Y CY(CAS) (84)
The q4anttty 4p) *pproaches onq. ps appro4ches zero.
The combination of the kintmtriti-c- 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
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
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  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
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,&eneralize-d Govrsat problems can be detined in the
regtons It through V succ"sively Their solutions are obtatn-ed 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
+ 'K' 1) 56
+ 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
Th ouini h ein IVII,..,cnesl
tubain ildapra,,e zr a w xtn tesouio o nint
Along the C-d ire ct Lon.
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
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 liquid-vapor 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 is-ensured 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
for a-real 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 tank-liquid
system in low gravity environments.
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
cortt-9ponding to the bubble
shAps factor K R = 0.43
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
ITILVIC _2 0 07 'A (40)1 25
0-1 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.
EXPRES81ON OF TRE PERTURBATION VELOCITY C"4PONM v IN TM
PROLWE "'REROIDAL r-OORDINATR$ (a,0,8)
The expression of v & may be derivp-d from the known expressions
of u and w iu the deformea cylindrical coordinates
(W2- 4) 14 = i W '2
by means of tlia daftneO coor0inate transformation relation, equation (J).
Let us dsfins
Accotding to the coordinate tramsformatiou defiDed tn equation (13),
ji a (A3)
t d t
where the superscript T tn equation (0) denotes the tra"pased
7, L- A Voc
IA Aid (A5)
Ti '-- I Tt
Hence, by using -equations (Al), (A2), (A3), a-nd. (0),
CL 2 + (A6)
-T 1 (d P") 9A,
The txpre**i*o for V,,, follo*;* imftdiately frco equstions 46) anil (46).
FLOW ClLkRT OF THE (",OMPUTI!+G PROGRAM EMPLOYFD
IN CHAPTER ITT
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.
Define complex variable names.
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
Construct a polynomial multiplication segment.
Compute W (P) according to equation (55).
Construct a segment of program for the repeated
application of the recurrence formula (60).
Compute, rn, W
8tor-A the obtained 1K
proper locations of the matrices A(10 x 20) and
R(iC x 20).
READ: The size
of increment for
the walls of jthe
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
FUNCTIONS: The Bessel
function J and the
Construct the sequence of eigen- READ Zeroes of
functions Ii. Compute their values the Bessel func-
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
Store thiR obtairwd v#luet
of j',k Aod %K in PrOPer
loe#tion* of # m4trix
ATX(10 X 20).
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-
Solve the matri-x 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 m-s of the trans-
WRITE: The ranstorre the eige"Vectorg
eigenvactors. b ck to the original basis
by equations (67) and <72).
(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.
 Phillips, 0. M. Centrifugal waves. J. Fluid ch. 7 (1960), 340.
(5) Struik, D. J. Lectures on ClaSsical Differential Geometry. (19SQ),
Addison-Wesley 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.
k---U -1w- _a
Sui-kwong 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.
Sui-kwong Paul Pao is married to the former Juliet Yung-li Zue
He is a member of Phi Kappa Phi.
chaimAn of the d*ndidatV4
approved by al I
the Do-an of the Colle
was was appromd as parl
degree of Doctor of PKIcs