Response of a rotating finite annular liquid layer to various axial excitations in zero gravity

Journal

ofSoundand

Vibration (1991) 149 (2), 219-234

RESPONSE OF A ROTATING FINITE ANNULAR LAYER TO VARIOUS AXIAL EXCITATIONS ZERO GRAVITY

LIQUID IN

H. F. BAUER lnstitut ftir Raumfahrttechnik, Universitiit der Bundeswehr, Werner-Heisenberg- Weg 39, D-8014 Neubiberg, Germany (Received 27 April 1990, and in revised form 20 September 1990) A solidly rotating finite annular liquid layer consisting of non-viscous liquid is subjected to different axial harmonic excitations in a zero-gravity environment. The response of the free liquid surface elevation and the velocity in the radial, angular and axial directions has been determined in the elliptic forcing frequency range 0 > 2fi0 and in the hyperbolic range R < 20, for different diameter ratios.

1. INTRODUCTION of prolonged orbital space flights, where a nearly zero-gravity environment can be maintained, furthers the desire to perform unique experiments, the results of which may at a later stage be used as the basis of manufacturing processes which may not be performed under normal-gravity conditions on Earth. An area of some importance is the vibrational behavior of liquids under micro-gravity. For this reason there will be experiments on vibrating liquid bridges in orbiting space laboratories. In these experiments the vibrational behavior of liquids under a lack of gravity, held together just by surface tension, will be investigated, in such a way as to determine detrimental frequency ranges caused by g-jitter, crew motion, and operation of on-board machines and other vibrationinducing instruments. Knowledge of the natural frequencies of a rotating annular liquid layer and its response to harmonic excitation is therefore of importance before the actual flight experiments, in order to be able to interpret the experimental results properly, and to plan the experiments in such a way as to obtain the most efficient results from them. For non-rotating liquid columns linearized theory has been used previously for frictionless, viscous and visco-elastic liquids [l-6], to determine natural frequencies, the damped natural frequencies and stability. This was also done for rotating systems, in which the liquid column was in motion with constant angular speed about its axis of symmetry [7]. In addition, the stability of viscous liquid columns was investigated [8-131. The possibility

For a rotating axially excited liquid column [14] and a rotating annular liquid layer around a rigid central core [ 151 the response of the free surface and velocity distribution have been determined by the author previously. For other axial excitation modes no results are available. What follows, therefore, is

an account of an investigation of the harmonic response of a constantly rotating annular liquid layer due to various axial excitations. 2. BASIC EQUATIONS

An incompressible liquid layer of thickness (a - 6) and length h lies around a rotating rigid center core cylinder of radius b. The layer consists of non-viscous liquid of density 219 0022-460X/91/170219+16$03.00/0

@ 1991 Academic

Press Limited

220

H. F. BAUER

p and at r = a exhibits a free liquid surface. The rotational speed about the axis of symmetry is R, (a list of symbols is given in the Appendix). The system is harmonically excited axially by moving its top or bottom wall, or both, in a prescribed way. The layer is held together by surface tension a, which acts as the restoring force when the free liquid surface is displaced (see Figure 1). The system is in a zero-gravity environment

(b)

z=h

-

Figure 1. Geometry and co-ordinate excited.

f h

system. (a) One-sided excitation; (b) counter-excited;

(c) “phase”-

and exhibits in equilibrium a free surface at r = a. In axial excitation the system will respond with axisymmetric motion (independent of the azimuthal co-ordinate cp). With the velocity v = ue, + ue, + wk relative to the moving system, the equation of motion is the Euler equation ~v/t3t+2[~,xv]+[~,,x[f10xr]]+(1/p)gradp=0,

(1)

where p is the mass density of the liquid, p is the pressure distribution and fi, = f&k is the rotational speed of the liquid column. In writing these equations it has already been assumed that the velocities and oscillation amplitudes are small. The second term in equation (1) represents the Coriolis acceleration, while the third term is the centrifugal acceleration. The vector r = re, + zk is the position vector in the moving co-ordinate system. The continuity equation div v = 0 is for axisymmetric flow (independent of cp) given in component notation by du/ar+(u/r)+dw/dz=O.

Upon introducing

the acceleration

(2)

potential

p = {(P-Po)IP}-SR~(r*-a’)

(3)

the Euler equation (1) may be expressed as av/at+2[0,xv]+grad

Application of the vector operation (a/at) (curl v) = 20,(av/az).

?P =O.

(4)

“curl” reveals that the rotation is not constant, but

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After application of the vector operation “divergence”, differentiating with respect to time twice and applying the continuity equation, equation (4) yields the expression (a2/at2)(A?P) +4R;a2’P/az2

= 0,

(5)

where h=a2/ar2+(1/r) ajar+a’/az’. The boundary conditions at the upper and lower walls are given by w=o

w = ifl$ einr

and

at z=O

atz=h

(6)

for the one-sided excitation case, at z=ih/2

w = TiL& eiRf for the counter-excitation

case, and

w = ifi,f, ei(Rlf+*) at z = h for the phase-different

(7)

and

w = ifl,T, eiRZ’ at z=O

(8)

excitations. At the central core wall it is u=o

at r = b,

(9)

while with c = 5( z, t) as the axisymmetric free surface displacement the free liquid surface at r = a + l( z, t) exhibits the kinematic condition ag/at=u

at r=u

(10)

and the dynamic condition pW+p~R~~+(a/a~)[1;+a~a~~/a~~]=a/a

at r=a.

(11)

In this equation a is the liquid surface tension between the liquid and gas regions. These equations (5) with (6) (or (7) or (8)), (9), (10) and (12) constitute the set of equations that have to be solved for the response of the liquid column with rotational speed 0, due to an axial excitation.

3. METHOD

OF SOLUTION

We shall treat three cases of axial excitation, which are in practice of some importance. 3.1.

ONE-SIDED

EXCITATION

In this case equation (5) has to be solved with the boundary conditions (6), (10) and (12). The velocity distribution and the acceleration potential 9 are assumed to be of the forms

(134 (13b) (13c) 1JI(r, z, t) = e

k21n~]+&~2(f)2+“~,P~(rjcos(~)}.

(13d)

Here k = b/a, the ratio of the central core to free surface diameters. The velocity w satisfies the two boundary conditions (6).

222

H.

F. BAUER

From equation (4) one obtains iflU,-2L&V,=

inU, -2R,V,,

-P,,la,

2R,U,+i0V,=O

and

02=2P2/h,

Expanding

= -dP,/dr,

(144

2L&U, +iRV, = 0,

(14b)

iL!W,, = (nm/h)P,,.

(14c)

z’/ h* into a Fourier cosine series in the range 0 c z s h, i.e., z2 1 -_=-+h2 3

4 ao (-1)” ~2,,?,~cos

(15)

one finally obtains, from equation (5), d”P,/dr2+(l/r)

dP,,/dr-[(n2~‘/h2)(1-46!~/~‘)]P,

=0

(16a)

and F, = -(qa2/h2)(1

-4&/02)

= -(a2/2h)(R2-40:).

(16b)

These have to be solved with the boundary conditions (12) and (9). According to equations

(14a-c), U,=

i&G, a(O*-40;)’

v,= -

2&F,

%a v, = 7

a(R*-40:)’

(17a)

and ia V,(r)=(a2-4@)

ndP dr ’

20, V,(r) = -

(R2_4R3

dP, dr’

W.Cr)=-EP,.

(17b)

The free surface condition (12) yields, with equation (15) at r=a

inP,+a~:u”+~[l-~]U”=2i~~~~-1)fl

(18)

and (19)

According to the magnitude of the forcing frequency R with respect to twice the spin frequency L&,, one has to distinguish two basic regions for the forcing frequency: i.e., the elliptic region 0 > 2L$, with the solution P,(r) =AI,(nmxr/h)+D&(nmrr/h),

with a* = 1-40i/fl*

(20a)

> 0, and the hyperbolic range R < 24, with the solution P,(r) = BJ,(n?rpr/h)+

CY,(m$r/h)

(20b)

with p2 = (4&/0’) - I> 0. In these equations I0 and K,, are the modified Bessel functions, while Jo and Y0 are the Bessel functions. The constants A, D or B, C are obtained from equation (18) and dP,/dr=O at r = b.

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They are as follows: A, 2(hla)(--I)” -= 7r2n2 &$I

(214 D, = A,I,(n~crak/h)/K,(nn~ak/h)

@lb)

for the elliptic case fl> 2Ro, and &I 2(hla)(-I)” -= rr*n* FOa

x[Y,(T)J,(Tk)-J1(T)Y,(Tk)]},

(22a)

C, = -B,J,(nrrSak/h)/Y,(n~~ak/h)

(22b)

for the hyperbolic case 0 < 20,. The values U, , V,, PI, 4, & are obtained from equations (5) and (14) and are U, = -iOa/2h,

Introducing

V,=fl,a/h,

P,=-(a2/2h)(02-4.(3:),

these results into the velocity distribution

(13) yields

224

H. F. BAUER

w( r, 2, t) = Zo eiR’iO

i-5

j,

(A,/%~) y

~[~~(~)~~(~k)+~(~)I,(~k)]sin(~)]

(23~)

for the elliptic case 0 > 26!,,, and u(r,2,1)=ioein(in(-~(s-~)+~~~~~~”~,~~

[~,(~),,(~~)-J*(~)~,(~k)l

cos(y)},

(23d)

Y(r,~,f)=igeinlin{-~~(d-~)+~.,l~~~”~~~~ XIY1(~)J,(~k)-J,(~)Y,(~k)]cos(~)}, w( r, 2, t) = To einfiO

{

“+A

(23e)

cm ~“[YO(!$o)

_JO(~~~,~!.+~j]

1,. (?)I

(23f)

for the hyperbolic case 0 < 2&. It may be noticed that with zo* 0 and 0 9 o,, where w, are the natural frequencies of the axisymmetric oscillations, the free liquid oscillation case is obtained. In this case the denominators of equations (21) and (22) vanish and represent the natural frequency equations. They are, for R > 2Ro, (l_~)[Io(~)~,(~~)+~(~)~,(~~)]+~~~

X[ 1_~(~_1)1[~~(~)~,(~~) and for L? < 2no,

-I++(~~)]

=o, (24a)

(l_~)[Jo(~)Y,(~)_Yo(~)J,(~)]+~~~

~[~-~(~-‘)][Y,(~)l,(~)-J,(~)Y,(~)]=o. (24b) The response curves show singularities at the resonances R = 0,. ‘I’he free surface elevation is obtained from the kinematic condition by integrating u with respect to time. It is g( z, t) = 4 eiR’ -$(l-kl)f0/+2;~~,$~ { X[I,(~)K,(~)-K,(~)I,(~)]cos(~)}

(25a)

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GRAVITY

and

-;(1-k2)+n2‘4a;n;l~~

((2, t) = 2, eiR’

~[~,(~),,(““““-~,(~)~,(~)]co (25b)

for R <2L$). The case of a non-rotating liquid layer 0, = 0 yields the results for a one-sided axially excited liquid layer [ 161. In this case only the elliptic range is valid, since f2 > 2Ro = 0. The velocity distribution and free surface displacement may be derived from equations (23) and (25). It is

{_$(i-kL)+zp,+.r eiR’

U(r, 2, t) = ilEo x

[I,(n~rlhHG( nrruk/h)-K,(nrrr/h)I,(nrruk/h)J

cos (n~z/h)

[I,(n~u/fi)K,(n~ak/h)+I,(n~ak/h)~(n~u/h)](~2,-R2)

I

with v = 0 and w(r,z, t)=iRFOeiR’ x

z 2R2 Oc (-1)” h-lr n=1 c n

1

Mn~rlhK,(

n~ak/h)+Ko(n~r/h)Il(n~uk/h)]

sin (n~z/h)

[I,(nlru/h)K,(n~uk/h)+I,(n~uk/h)Ko(n~u/h)](w2,-R2) The free surface displacement l( z, t) =

is given by

.foeio' -;(I-k’)+$ x

I *

2 ao (-1)” C n=, n

L(n4h)K

, ( nrruk/h)-K,(n~u/h)I,(nlrak/h)] [Io(n~u/h)K,(n~uk/h)+~(n~u/h)I,(nrruk/h)](w2,-R2)

cos (n~z/h)

where

3.2. COUNTER-EXCITATION If the liquid layer is excited counter-wise, equation (5) has to be solved with equations (7), (10) and (12). One assumes velocity and acceleration potential distributions of the forms

~~~{4~--3~,u2”(r)cos(~)], v(r, t) = ein’ {&VO[~--$--+nf, V2.(r)cos(y)},

u(r,z,t)=e

z,

w( r, z, f) = ein’

2n7rz + .r, C W2n(r) sin (-2if%(i) ( )I h

,

(26a) Wb) (26~)

With a similar procedure

as above one obtains

A Zn h/+1)“-’

-= a&

2n2n2

D2” = Al,, Il(2n~uka/h)/K,(2~nak~/h)

(27b)

for R > 2&,, and B2, -= &a

(-1)“~‘h/a 2n2r2

(274

DZn = -B2,J,(2n~~ak/h)/Y1(2n?rpak/h) for R <24,,

(274

the velocity distribution

u( r, z, t) = iGO ein’

(2W

(28b)

(28~)

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for 0 > 2&, and, for R < 2R,,, u( r,

2,

t) = iR& e

x[Y,(F)J~(~)

-J1(y)Y,(y)]

D(,r,z, t) = ia;‘, ein’{~;(;__K_)+

n(;f”;n:,

cos(y)},

“E, z

Y@

+,(y)“,(@$@)-‘,(~).,(~)]cos(~)},

(29b)

J, (y)]

The free surface displacement

(29a)

sin (?)I.

(29~)

is given by

for R > 2fi0, and

(3Ob)

for R <20,. 3.3.

PHASE-SHIFTED

EXCITATION

For phase-shifted excitation the boundary condition (8) has to be used. The response may be obtained by superimposing the two one-sided solutions, satisfying w = 0 at z = 0 and w = ifliz, ei(n,‘+g) at z = h and w = i0,f2 einz’ at z = 0 and w = 0 at z = h, respectively. This may be obtained from the results presented in section 3.1. 4. NUMERICAL EVALUATIONS AND CONCLUSIONS Some of the previous analytical results have been evaluated numerically and will be presented here. In Figure 2 we present the natural frequency ratio won/2J& for the axisymmetric mode rn = 0 for the diameter ratio k = b/a = O-5 and the surface tension parameters a/pa3# = 1 (or Weber numbers We * pa3@J c). The elliptic frequency range won> 2a. and the hyperbolic range won ~24, are indicated. Note that the natural frequency decreases with increasing liquid height ratio h/a. For a decreased Weber number or for a/pa30i= 10 the natural frequencies have been presented in reference [17]. First of all, a strong increase of the frequency with decreasing height ratio h/a has been noted around ma/h = 1, In addition, the natural frequency exhibits, for an increased

228

H.

6

Elliptrc

t

F. BAUER

range: w >2fio

,,Hyperbollc

range: w < 2 520

-6 IStable -10

0

1

I

I

1

2

3

1 4

5

nra/h

Figure 2. Axisymmetric natural frequencies (M = 0, b/a = 0.5, u,/pa3Ri Elllptlc I I

4

= 1).

region: D >2slo

2.26

7,36

,

q ._m

I,”

Ellqtlc

.

region:9>2.& 6.85

0

1

2

3

4

5

6

7

8

Figure 3. Response of free surface displacement for counter-excitation and diameter ratio k = 0.5 (a) and 0.8 (b); h/a

=2-O,

cr/pa’Rg=

1. z/h; -,

0.5; - - -, 0.25; -. - ’-, 0.

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GRAVITY

surface tension parameter, i.e., decreased Weber number We, an increased natural frequency of the rotating liquid bridge. For a further increase of the surface tension parameter to a/pa’Ri = 100 the increase of the axisymmetric natural frequency is larger and, shortly after mm/h = 1, shows a more pronounced increase, as may be seen in reference [17]. It should be mentioned that the rotating liquid column is stable only for the amphora-type mode m =0 (axisymmetric mode), for which pL!$z3/cr< ( n2r2a2/ h2) - 1, which means that for ma/h > ~&8%% one has stability. For We = 10 -’ stability may be obtained at ma/h > 1.005, for We = 10-l at ma/h > 1.049, and for We = I at nra/ h > v’?, as may be seen from the results of the various figures in reference [8], where the boundaries are indicated by straight lines at the abscissas. 4.1.

COUNTER-EXCITATION

For a counter-excited rotating liquid column the response of the free liquid surface is presented in Figure 3(a) for the diameter ratio k = 0.5 and in Figure 3(b) for k = 0.8. The aspect ratio was chosen to be h/a = 2 and the Weber number pa3Ri/a = 1. The response amplitude (l/&J is presented for the radial location r = 0.9~ and at the axial locations z/h=0 (-a-.-), z/h=0*25 (---) and z/h=0*5 (-). According to the relation of the forcing frequency 0 to twice the speed of rotation L?,, one has to distinguish the elliptic (0 > 2&) and hyperbolic (0 < 2&J regions. The latter one exhibits an infinite amount of infinities of very small width and would be damped out in a treatment of a viscous liquid column. The response in the elliptic forcing frequency range plays an important role for engineering purposes. They exhibit prominent resonances, of which the frequencies are indicated on the top of the figures. The response amplitude ][/%I at various axial locations is shown in Figure 3(a). It may be noticed that the response at the location z = h/4 does not exhibit a peak at the first resonance. This is due to the proportionality to cos (2nrz/h), which has for z = h/4 the value cos (m/2). There is, for this reason, a nodal plane at z = h/4. The same is true for the response of the radial and circumferential velocity distribution (Figures 4 and 5), which are presented for the diameter ratio k = 0.5. The radial velocity is shown at I = O-9 a for h/a = 2 and at the

Elliptic

region: J2>2Rc 7-36 I

Figure 4. Response of radial velocity parameters and key as Figure 3.

for counter-excitation

at r = 0.9 a and diameter

ratio

k = 0.5. Other

230

H. Elliptic

F. BAUER region:

JJ >20, ‘7‘36

0

2

4

6

0

Figure 5. Response of circumferential velocity for counter-excitation at r = a and diameter ratio k = 0.5.Other parameters and key as Figure 3.

). For the diameter ratio axial locations z = 0 (- . -), z = h/4 (- - -) and at z = O-5 h (k = 0.5 the first resonance frequency is at w,/2Dno--2.26, while the second resonance frequency has the magnitude wz/2Ro=7*36 (see Figure 4). For k = O-8 one has 0,/2& = 1.80 and oz/212,, = 6.85 (see reference [ 171). It may be noted that an infinity of resonances are located in the hyperbolic region 0 < 2&. They are of no vibrational consequence and are not presented in these figures. In addition, those “inertial” resonance peaks are heavily damped in an actual (viscous) liquid. In Figure 5 are shown the response of the circumferential velocity ](B/z,)iR eiR’] at r = a and for u/pa30i = 1 at the same axial locations z = 0, h/4 and hf 2 for k = O-5. The response of the axial velocity is presented in Figure 6 for k = 0.5 at r = a. Elliptic

region:

.Q>2fio

Figure 6. Response of axial velocity for counter-excitorion at r = (I and diameter ratio k = 0.5.Other parameters and key as Figure 3.

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The response at the axial location z = h/2 is unity as shown by the straight solid line. At z = 0 there is a nodal plane. It may be noticed that the second resonance peak is not present since it is proportional to sin (2nrz/ h), i.e., for z = h/4 and n = 2 it vanishes.

4.2.

ONE-SIDED

EXCITATION

The numerical results for the response to a one-sided excitation (see section 3.1) are shown in Figures 7-10. They are presented for the surface tension parameter o/pa30* = 1, the aspect ratio h/a = 2, the diameter ratios k = O-5 and 0.8, and at the axial locations z/h = 0, l/4, l/2, 314 and 1-O. In addition to the course of the response in the elliptic region R/2&,> 1, some of the response peaks in the hyperbolic forcing frequency range R/2L&< 1 are also shown.

Hyperbolic region C

4

Elliptic

region:

a > 2Ro 4.61

3 -

3

2

1

~~~ c 4

! 1

0 0

Hyperbolic region

1

2

1 I I

Elliptic

3

region.

4

5

6

56 > 29

n/2no

Figure 7. Response of free surface for one-sided excitation and diameter ratios k = 0.5 (a) and 0.8 (b); other parameters as Figure 3. z/h; -, 1; - - -, 0.75; - - . -, 0.5; . 1,0.25; - . -, 0.

232

H. F. Hyperbolic region

! 1 / I / I I I

049

4

~ (

3-

0

1

Figure 8. Response of radial velocity parameters and key as Figure 7.

Hyperbolic region 4

k9

Elltpt~c

BAUER

regron a>

2Ro

2-26 I

2

4.61

/

3

4

5

6

for one-sided excitation at r = 0.9 a and diameter

, I I

Elliptic

i

2.26

region:

ratio k = 0.5. Other

.12>2ao 4.6’ I

3 g.G ._ 1: o)

2

\ za

t

a I_ 0

Figure 9. Response of circumferential Other parameters and key as Figure 7.

5

velocity

for one-sided excitarion

at I = a and diameter

ratio

k = 0.5.

In Figures 7(a) and 7(b) is shown the response of the free surface displacement ](C/Z,) eiR’] as a function of the forcing frequency o/2&. Since the free surface elevation of the rotating liquid column above r = a is proportional to cos (nrrz/h), it is noticed that at some axial locations nodal planes appear and show no resonance peaks. Resonances shown for the diameter ratio k = O-5 are w/2R0 = O-49, 2.26 and 4.61, while for k = 0.8 they appear at o/2&, = O-285, 1.80 and 4.04 [ 171. The response of the liquid layer for the radial velocity ](u/$)i0 eiR’J is presented for r = O-9 a in Figure 8 for k = 0.5, while the response for the circumferential velocity II is presented in Figure 9.

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Finally, the response for the axial velocity w may be seen in Figure 10. Hyperbolic region

) , I

Elliptic

region- .G > 2Ro 4.61

Figure 10. Response of axial velocity for parameters and key as Figure 7.

one-sided

excitation

at r = a and diameter ratio

k = 0.5.

Other

REFERENCES 1879 Proceedings of the Royal Society London 29, 71-97. On the capillary of jets. 2. LORD RAYLEIGH 1945 Theory of Sound. New York: Dover Publications. 3. H. LAMB 1932 Hydrodynamics. Cambridge University Press. 4. H. F. BAUER 1983 Forschung Ingenieur Wesen 49, 117-126. Natural damped frequencies of an 1. LORD RAYLEIGH

phenomenon

infinitely long column of immiscible viscous liquids. 5. H. F. BAUER 1984 Zeitschrif fir angemandte Mathematik und Mechanik 64,475-490.

6. 7.

8. 9. 10. 11.

Natural damped frequencies of an infinitely long column of immiscible viscous liquids. H. F. BAUER 1986 Acta Asrronautica 13, 9-22. Free surface- and interface oscillations of an infinitely long visco-elastic liquid column. H. F. BAUER 1982 Acta Astronautica 9, 547-563. Coupled oscillations of a solidly rotating liquid bridge. S. TOMOTIKA 1935 Proceedings ofthe Royal Society London A-150, 322-337. On the instability of a cylindrical thread of viscous liquid surrounded by another viscous fluid. L. M. HOCKING and D. H. MICHAEL 1959 Mathematika 6, 25-32. The stability of a column of rotating liquid. L. M. HOCKING 1960 Mathematika 7, l-9. The stability of a rotating column of liquid. J. GILLIS 1961 Proceedings of the Cambridge Philosophical Society 57, 152-159. Stability of a column of rotating viscous liquid.

12. J. GILLIS and K. S. SHUH 1962 The Physics ofFluids

5, 1149-1155. Stability

of a rotating

liquid

column. 1962 Quarterly ofApplied Mathematics 19, 301-308. The stability of a rotating viscous jet. 14. H. F. BAUER 1989 Acta Mechanics 77, 153-170. Response of a spinning liquid column to axial excitation. 15. H. F. BAUER 1989 Forschung Ingenieur Wesen 55(4), 120-127. Response of a finite rotating

13. J. GILLIS and B. KAUFMANN

annular

liquid layer to axial excitation.

234

H. F. BAUER

16. H. F. BAUER 1990 Forschung ZngenieurWesen 56, 14-21.Response of a liquid column to one-sided axial excitation in zero-gravity. 17. H. F. BAUER 1990 Forschungsbericht der Universitiit der Bundeswehr Miinchen LRT- WE-9-FB-8. Response of a rotating finite annular liquid layer to various axial excitations in zero-gravity.

APPENDIX: it h IO, 11

Jo. JI

k KotK, yo, y, P r, cp,z t u, v, w zo a2 -2 ;2 P2 Yzn-I

P u YI 00 D 0

LIST OF SYMBOLS

radius of liquid column radius of rigid central.core length of liquid column modified Bessel functions of the first kind Bessel functions of the first kind = b/a, diameter ratio of the system modified Bessel functions of the second kind Bessel functions of the second kind liquid pressure cylindrical polar co-ordinates time velocity distribution in the radial, circumferential and axial directions respectively axial excitation amplitude = 1 -405/fi2 > 0, elliptic case (0 > 2Ro) for forced oscillations = 1 -40$02 > 0, elliptic case (0 > 2Ro) for free oscillations = (4@fi2) - I> 0, hyperbolic range (a < 2Ro) for forced oscillations = (4J2$w2) - 1> 0, hyperbolic range (w < 2R,) for free oscillations =(2n-l)m/h

liquid density liquid surface tension acceleration potential rotational speed axial forcing frequency natural frequency of rotating system