Hydroelastic oscillations of a viscous infinitely long liquid column

Journal qf Sound and Vibration (1987) 119(2), 249-265

HYDROELASTIC OSCILLATIONS OF A VISCOUS INFINITELY LONG LIQUID COLUMN H. F. BAUER Institut ftir Raumfahrttechnik,

(Received

Universittit

15 September

der Bundeswehr

Miinchen,

D-8014

1986, and in revised form 9 Feburary

Neubiberg,

Germany

1987)

The coupled frequencies of hydroelastic systems in zero gravity consisting of an elastic shell and a viscous liquid layer with a free surface have been treated. The system exhibits either an annular liquid layer around an elastic cylindrical center shell or a liquid layer inside an elastic infinitely long container. The first case has been evaluated numerically, where the influence of the liquid surface tension parameter, the elasticity parameter of the shell, its axial deflection wave length I and the thickness of the layer have been determined. In contrast to the hydroelastic system with an ideal liquid, the system with viscous liquid exhibits instability of the liquid surface as well as the shell.

1. INTRODUCTION

With the availability of extended manned space flights in an earth-orbiting space laboratory, we are able to perform unique experiments of explanatary nature. There are many experiments that can hardly be performed on earth under the action of gravity. Some of these experiments planned in a micro-gravity environment are concerned with the behavior of liquid, for which the surface tension forces play the dominant role. Since we deal a lot with liquid configurations held together by surface tension, the knowledge of the natural frequencies of such systems are of some importance, since one may have to avoid certain frequency ranges present in a space laboratory either as the so-called g-jitter vibration, the motion disturbances of the crew or the operation of on-board machines, etc. The system investigated here is a viscous liquid around an elastic center-shell and a liquid inside an elastic cylindrical container. These configurations are of practical interest for various reasons. Since the structure in space-systems should usually be very light, the structural parts of the problem have been considered elastic. The system is treated as an infinitely long cylinder with an infinitely long (Figure 1) annular liquid body in contact with the inner or outer surface. To maintain such a configuration the liquid, considered in the following treatment should exhibit wetting behavior. This means that its contact angle to the wall will be of small magnitude, in contrast to a non-wetting liquid such as mercury, which would, because of its large contact angle of about 180”, display a great tendency to globulize on the cylindrical shell rather than remain in such an assumed cylindrical configuration. So it is assumed here that a wetting liquid with a small contact angle is being considered, which will provide an axial wavelength ratio l/a -C T under which condition a cylindrical liquid configuration should be maintained. This is justified since the coupled free vibrations then appearing do not provide an acceleration of a magnitude comparable to that which would be observed by shaking the system. In the investigations which follow the elliptic cross-sectional mode m = 2, n = 1, which exhibits no instability for the liquid by itself, is considered. The axisymmetric mode m = 0 would show instability for the liquid for l/a 3 n, as does an infinite liquid column (for an axial 249 0022-460X/87/230249+

17 %03.00/O

@ 1987

Academic

Press

Limited

250

H. F. BAUER

To

Figure

1. Geometry

and co-ordinates

of the hydroelastic

system.

wavelength of magnitude larger than the circumference of the cross-section of the liquid column), which forms into drops for m = 0. In the axisymmetric vibration case the tendency for globulization of liquid on the shell wall would exist for l/a > 71: For an annular cylindrical liquid system of frictionless liquid around a rigid cylindrical center core and for a frictionless liquid inside a rigid container in zerogravity, the natural frequencies have been determined [l]. For a viscous liquid and a rigid structure results for the damped frequencies of the system may be obtained by following the procedure used in previous theoretical investigations [2] in which only the infinite liquid column has been evaluated numerically. The treatment of the hydroelastic problem in which the structure is considered as an elastic shell has not been performed yet for a viscous liquid. If the liquid surface and the elastic shell are both capable of performing vibrations, one is dealing with a coupled system exhibiting coupled frequencies. These frequencies should be determined. Of further interest is whether the system remains stable or not and what the influence of the various geometric, liquid and structural parameters upon the damped frequencies finally is. Previously it was found [3 3 for the two-dimensional system (z-independent), that the interaction of the structure and the liquid with a viscous liquid may well be unstable, a fact that does not appear for a frictionless liquid. The physical significance of this unexpected result calls for some discussion. Of primary interest here is the fact that this is a case where viscosity provides instability whereas for an ideal liquid there is always stability. Small perturbations of the structure lead to disturbances of the liquid surface and vice versa. In the case of a viscous liquid, for which different phase relations appear, the flow as such is quite different than that of an ideal liquid. A frictionless liquid exhibits slippage in the angular direction while a viscous liquid adheres to the wall and moves with it in the angular direction. Perturbations always result in liquid surface displacements, thus changing the surface tension restoring force, which in the case of a viscous liquid, through the additional disturbance in the angular direction due to the motion of the

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HYDROELASTICITY

elastic shell, is not adequate to damp out the perturbations. These effects, i.e., phase shifts and then additional excitation of the liquid through the angular motion of the elastic shell and vice uersa, phenomena which are not present in the system with a frictionless liquid, seem to be responsible for such instabilities. The investigation reported in what follows is an analysis for the determination of the coupled frequencies of these hydroelastic systems. The main interest is directed towards the behavior of the coupled viscous liquid-shell system, its stability or instability , and the influences of the surface tension parameter Tu/pv’, the thickness of the liquid layer, the density ratio of shell to liquid p/p, the thickness of the shell h, the angular mode number rn and axial mode number n, as well as the elastic parameter Ea*/jj(l - C’)v’. Lord Rayleigh [4] treated the three-dimensional case of a cylindrical air column in an infinite medium and found that the axisymmetric cylindrical configuration becomes unstable and exhibits its most pronounced instability at an axial wave length A, = 12.96~. Lamb [5] performed similar investigations, in which mainly a single liquid column was treated. In addition, the remarkable work of Plateau [6], who performed a large number of useful simulation experiments, should be mentioned. No work, however, except that of the author [3], has previously been performed on the interaction of an elastic structure with a viscous liquid in a zero gravity environment.

2. BASIC EQUATIONS If a liquid and structure in a zero gravity environment is disturbed, it will perform damped oscillations, for which the aim here is to determine the coupled frequencies of the elastic shell and viscous liquid system, which may be able to perform surface oscillations. For small liquid velocities and small elastic deflections the governing equations may be linearized. The motion of the viscous liquid requires the solution of the Stokes equations (see Figure 1) au/at + (I/p)dp/iir

= v{rl*u/ilr*+ (I/r)i?u/ar - (2/r2)av/a(p

aV/ar+(l/p)(l/r)ap/aQ

=

+ ( l/r2)8*u/d(p2

+a’u/a.72),

(1)

V{a’V/ar’+(l/r)aV/ar-(V/r’)+(l/r*)a*V/aQ

+

aw/at+(i/p)ap/az=

- (u/r’)

(2/r’)au/a(p

+ a”v/az’},

(2)

v{a’w/ar’+(l/r)aw/ar+(l/r’)a’w/arp*+a’w/az*}

(3)

together with the continuity equation (4)

aU/ar+(U/r)+(l/r)aV/dQ+aW/aZ=o,

where u, v, w are the components of the velocity of the liquid in the radial, circumferential and axial directions respectively, p is the pressure, p is the density and p is the dynamic viscosity (v = p/p). (A list of symbols is given in Appendix 2.) These equations have to be solved with the appropriate boundary conditions. If the free liquid surface, which is held together by surface tension forces, is at r = a outside an elastic cylindrical shell at r = b, b < a, the free surface conditions are, with the kinematic condition u = a&/at, obtained from Us+ pT, = 0 and 7rc = 7,: = 0 at r = a. These yield, after differentiation of the dynamic condition with respect to time, for the first of these equations, ap/at-2~a2u/arat+(To/a2)[u+a’a+4/az2+a54/acp2]=0

at r=a,

(5)

252

H.F.EAUER

while the other two surface conditions are atr=a.

r(a/ar)(u/r)+(l/r)au/dcp=O,aw/ar+au/az=O

(6)

With the elastic deflections denoted by [( cp,z, t), ~(cp, z, t) and l((p, z, t) in the radial circumferential and axial direction, respectively, the boundary conditions at the shellliquid interface are (adherence to the wail) at/at

=

U,

i)T/dt

and

= v

a{fat =

w

at r = b.

(7)

A cylindrical she11 has to satisfy the shell equations

(9) ag p(1-i;2)b2a21 __=+ “;bazacpaz E at2

b’$+)(l-i$$+)(l+C)b*

These are the Donnell-Mushtari shell equations, which easily may be converged to other circular she11equations by adding some more terms, which are, due to their multiplication by h2/12b2, small. The operator V2= b2(a2/az2)+ (a’/aq’), V is the Poisson ratio, E is the modulus of elasticity, h is the thickness of the shell, 26 is its diameter, p is the shells density and fi = Eh/( 1 - C”). As may be noticed, the motion of the shell is coupled to that of the liquid, which is expressed by the right side of equations (7)-(10). If the liquid is contained within an elastic infinitely long tubular she11 of radius a, the inner free surface equation is -(TJb2)[u+

ap/at -2wa*u/arat

b2a%/az2+a2u/ap2]=0,

(11)

with the condition at the she11 surface being equation (6). In the shell equations (8), (9) and (10) the value b has to be replaced by a, and the sign of the term in the square brackets on the right side of equation (8) has to be changed. 3. METHOD

OF SOLUTION

Applying to equations (l), (2) and (3) the vector divergence operation yields for the pressure the Laplace equation vp

= 0.

(12)

If the functions u, u, w, p, 5, 7, 5 are assumed to depend on 9, z and t through a factor of the form of eicmP-cnrrz”) e”, m, n being integers and s representing the yet unknown complex frequency (s = 6+iG), and if one substitutes v=V-(l/ps)gradp, with p = p(r) ei(mv+“nz/‘)esr, into the Stokes equations obtained: d2tJ/dr2+(l/r)

dU/dr-[{(m2+l)/r2}+(

d’V/dr’+(l/r)dV/dr-[{(m2+l)/r2}+(

(13) the following expressions

are

n21r2/1*)+(s/~)]U-(2im/r’)V=O,

(14)

n27r2/12)+(s/~)]V+(2im/r2)U=0,

(15)

d2W/dr’+(1/r)dW/dr-[(m2/r2)+(n2~2/l’)+(s/v)]W=0.

(16)

ZERO The

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continuity equation (4) is then (17)

dU/dr+(U/r)-t(im/r)V+(in~/l)W=O while equation (12) for the pressure is

(18)

d2P/dr2+(l/r)dP/dr-[(mZ/r2)+(n2~2/12)]P=0. The free surface condition (5) and the shear stresses (6) produce the equations SP-2.S@ [F-:$]+$(l-*‘-7) d

V

rdrr 0 -

im +--=r

(U-t:) P

2im dP

psr [ dr-- r

~0

atrza,

(19)

1

The solution of the pressure equation (18) is given by P(r)=AI,(n~r/l)+BK,(n~r/l), where A and B are integration constants. The axial velocity coefficient W(r) is obtained from equation (16) as W(r) = CI,(J(

n2~*/12)+(~/~)r)+DK,(~(nZ~*/12)+(s/v)r).

(23)

From the continuity equation (17) one obtains (im/r)V=

-(inn/l)

W-[dU/dr+

U/r],

(24)

which, introduced into equation (14), results in

=-~[~I-(~~r)+~K~(~~r)]. The transformation

U = U*/r

(25)

when used in equation (25) gives

-d*U*+ldU* -dr’ r dr

=-2i~[CI,(~~r)+DK,,,(~~r)], of which the inhomogeneous part is in resonance with the homogeneous solution of equation (25) is thus given by U(r) = (EIr)I,(J(

(26) solution. The

n’~2/I’)+(s/v)r)+(F/r)K,(~(n2~2/12)+(s/v)r)

-[in~/IJ(n2a2/1*)+(S/Y)][CI~(J(n2~*/IZ)+(S/~)r) +DKk(J(n’n*/I’)+(s/v)r)].

(27)

Thus the velocity coefficient V(r) is V(r)=(i/m)J(n2c2//2)+(~/v)[EI~J(n’~2/I2)+(s/~)r)+Fi(~(J(n2~2/12)+(~/v)r)] +{mn~/l[(n2~‘/I2)+(s/v)]r}[CI,(J(n2~2/12)+(~/~)r) +DK,(\l(n27r2/12)+(s/v)r)].

(28)

254

H. F. BAUER

The velocities are, according to equation (13), given by u = [ u - (l/ps)

dP/dr]

ei’mcF+nTz”‘ir’,

w=[w_(inn/psj)p]

u = [ V- (im/psr)P]

e”my+nr2”‘+“,

,i(w+n4l)+*r

while the elastic deflections are [=Xe

i(mq+n7rr/l)+sr 9

7j=Ye

ilm.c+nrrz/l)+sl 9

l=Ze

i(mp+n?rz//)+cr

3.1. LIQUID AROUND CENTER SHELL (FIGURE 1) Introducing these forma1 solution results in the boundary conditions (19), (20) and (21) yields, with equations (7) (E= b/a) and the she11 equations, after introducing there the approriate values of u, u, w, nine homogeneous algebraic equations for nine integration constants. The coefficient determinant Il%ll=O

(29)

is the transcendental frequency equation for the determination of the complex frequencies s = 6+i6 for each mode (m, n) (see Appendix 1). (FIGURE 1) 3.2. LIQUID INSIDE AN ELASTIC CONTAINER For a liquid in an infinitely long elastic container the values a and b must be carefully exchanged, while To has to be substituted by - Ti and D/b2 by -B/a*. The elements of the determinant are obtained in a similar fashion, but have been omitted here for reasons of saving space.

4. NUMERICAL EVALUATION AND CONCLUSION For a viscous liquid column around an elastic center shell, the complex frequency determinant (29) with the elements (Al) has been evaluated numerically for the various parameters: namely, the surface tension Tu/pv’, the density ratio p/p, the shell wall thickness h/a, the angular and axial mode numbers m and n respectively, the axial wavelength ratio l/u, the Poisson ratio V, the diameter ratio f= b/u and the elastic parameter E = p( 1 - V2)v2/ Eu’. The results are given as the real (-) and imaginary (- - -) parts of the complex roots S = (@a’/ v) + i( &x2/ v) as functions of the ratio of the shell diameter to the liquid diameter k: This means that if E is in the vicinity of f = l(
and is presented in Figure 2(a) and (b) for n = 1. The natural frequencies o,, of the non-viscous liquid is proportional to the square root of the tension parameter. It decreases with increasing diameter ratio 6 showing for thick liquid layers only a small change in frequency, while for thin layers, i.e., E in the vicinity of unity, a strong decrease occurs. This is more pronounced for higher angular modes tn (Figure 2(a)). In Figure 2(b) the uncoupled natural frequency is presented again for n = 1 and m = 0, 1, 2, 3, 4 for a fixed diameter ratio E= 0.5 and the tension parameter T,a/pv’ = 100 as a function of the aspect

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100

(0)

__________--_------------___ -.m=4 -\

80

‘\ ‘\

i

‘.

I 0.4 _

I 02

0 00

I 06

10

0.8

k = b/a

150.

I I

(b)

:

3

0

2

IT

4

6

I(

Figure 2. Natural frequencies of the inviscid annular liquid column around rigid cylindrical center shell. n = I, T,a/pv’= 100, (a) I/a =2; (b) z=@S. -, Real part; ----, imaginary part.

ratio l/a. It may be noticed that the axisymmetric mode m = 0 exhibits stability up to l/a = 7~ (n = 1), i.e., shows a real natural frequency, while for l/a > P it is unstable. This behavior is shown by the imaginary value of the frequency w,,. Otherwise it can be noticed that the natural frequency decreases rapidly with increasing ratio I/a and remains nearly at a constant value for large aspect ratios. The uncoupled natural frequency for the structure is presented in Figure 3 for the axisymmetric mode m = 0 and n = 1 with the parameters l/a = 2.0, i; = 0.3, h/a = 0.01 and the elasticity parameter p( 1 - fi”) a’/ E = O-01. They are presented as functions of E and with increasing shell diameter & exhibit a strong decay in the radial direction 5, while w,, and oc in the angular and axial directions respectively show only small changes with increasing shell diameter. For a less stiff shell the frequencies would show decreased magnitudes. The coupled natural frequencies of the system treated but for a frictionless liquid may be found in reference [7]. The coupled frequencies of the viscous hydroelastic system consisting of a viscous liquid layer around an elastic shell structure has been numerically evaluated for the mode (m = 2, n = 1). This is a motion of the shell wall of elliptic cross-section and simple sinusoidal axial displacement. All other parameters such as the density ratio p/p = 2 of the density of the shell to the density of the liquid, the wall thickness ratio h/a = 0.01, the length ratio l/a = 2, the Poisson ratio U= f and the tension

256

H. F. RAUER

2551

,

c i

I I

204

‘I I

I

I

153 3

I I

:

t I

00

\

\

02

0.6

0.6

1-l

Figure 3. Uncoupled natural frequencies of the cylindrical shell. m = 0, n = 1, p( 1 - C2)a2/E = 0.01; ii = 0.30, l/a = 2.00, h/a =O.Ol.

parameter T,* = T,,a/pv’ = 100 have been kept constant. The real and imaginary parts of the roots of the characteristic equation (the complex transcendental frequency equation (Al)), namely S = ( sa2/ V) = ( ea2/ V)+ i( (3a2/ v), have been plotted versus the diameter ratio E= b/a for various elasticity parameters E = (p( 1 - P2)b2v2/Ea4) = 0, 10d9, 10M6, lo-“, 10P4. The case E = 0 is that of a rigid structure (E + CO),while E = 10e9 is a slightly less stiff elastic shell structure. The value E = 10V4corresponds to an even less stiff shell. For a rigid structure one just obtains, as presented in Figure 4(a), the roots of the viscous liquid motion in the mode m = 2, n = 1. Note that the root I, corresponds to the natural frequency for the frictionless liquid case of Figure 2 for m = 2, n = 1. Here, however, the damped natural frequency for the viscous liquid case consists of a negative real part and an imaginary part. This means that in the range 0 s E=z O-75 a damped oscillation occurs, which with increasing magnitude of &= b/a (i.e., for decreasing liquid layer thickness) has an initially increasing damping, and an at first slightly decreasing and from E= 0.5 on strongly diminishing oscillation frequency. Above E = 0.75 the oscillation frequency is zero in contrast to the behavior for a frictionless liquid and the surface of the thinner liquid layer for c20.75 performs just a decaying aperiodic motion, if disturbed. Thus such thinner liquid layers are not able to oscillate. In addition one can notice other roots such as IQ in mode m = 2, n = 1, which stem from the viscosity of the liquid and are all negative real and large, meaning a very fast decay (aperiodic). It may be mentioned, that only one root Ia is shown in Figure 4(a). For a small elasticity parameter E = 10e9, i.e., slight elasticity or a slightly less stiff shell compared to the rigid shell, the result does not differ very much of that of the rigid shell. Not much of a coupling effect with the elastic structure is noticed yet. Increasing the elasticity parameter E to the magnitude E = 10m4, which corresponds to a less stiff shell, shows that the coupled liquid root takes a quite different course, since it is strongly influenced by the motion of the shell, especially for small liquid layer thicknesses. Figure 4(b) exhibits the coupled liquid frequency, indicated again by Ia, as a function of the diameter ratio & It can be seen that this liquid root remains complex, i.e., always has both real and imaginary parts. This means that the previously mentioned feature of the liquid not being capable to oscillate any more for E>O*75 and a rigid shell is no longer valid. In addition, the oscillation frequency no longer decays with increasing diameter ratio E as it does for the rigid shell case, but it increases to large values. And

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\

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HYDROELASTICITY

\

i

h

-24-

0.2

0.0

0.6

0.4

O-8

1

0.0

0.2

0.4

0.6

0.0

1.0

0.90

0.92

0.94

096

0.90

1.0

0.9-

0.3 -

-0.9

-

-1.5 0.95

Figure structure)

-120 -

j

1 0.96

4

I 0.97

/.

1 0.98

1

1 0.99

1

-200

l_ k=b/o

0.0

b

5

1 0.2

/

1 0.4

1

1 0.6

I

1 0.0

0

I I 1.0

4. Coupled frequencies of the hydroelastic system (liquid layer around elastic cylindrical shell for E = 0 and lOA presented for varying diameter ratio k = b/a. ti = l/3, I/a = 2, T$ = 100, p/p = 2, h/a =O.OI; m =2, n = 1. (a) E =O (rigid shell); (b) e = 10-f O< EC: 1.0; (c) as (b) but showing additional roots; (d) as (b) but with larger ordinate scale and for 0.90~ k < 1.0; (e) as (b) but with reduced ordinate scale and for 0.95 i k < 1 .O; (f) as (b) but with reduced ordinate scale showing additional roots. -, Re(S); -_ __, Im(S).

258

H. F.

BAUER

as may be seen from Figure 4(e) (where the scale of ff ranges from 0.95 to l-O), the liquid becomes even unstable for very thin liquid layers, i.e., above 63 0.98 (see Is3). The second root of the liquid Ia is also shown in Figure 4(b) and describes as before a fast decaying motion. The roots St are the uncoupled natural frequencies of the shell in the angular direction. The coupled shell root S, in the axial direction shows strong instability and exhibits for nearly all of the diameter ratio range a large positive real value (except for a very thin layer, as may be seen in Figures 4(d) and 4(e)). In Figure 4(b) not all of the course of the root S, is shown. Figure 4(c) shows some roots in addition to these in Figure 4(b), and it can be seen, in particular, that the coupled structural root S, is a strongly decaying root of very small frequency. In the diameter ratio range 0.6 < E < 0.8 the oscillatory part becomes larger again, insofar as it can be indicated in the graph. The root S, is also shown. It may be mentioned here that the continuous search for roots with the computer is hampered considerably by the closeness of other roots. For that reason some of the roots are shown only for a certain range of k In Figures 4(d) and (e) the thin layer range E- l(< 1) is presented in an enlarged form and thus serves for a better identification of the course of the various roots. More roots of the liquid-structure system are finally presented in Figure 4(f), where the coupled liquid roots lo (damped oscillatory) and IQ (decaying) are presented together with the coupled structural roots S, and S,, showing both S, and S, to be unstable roots. With the elasticity parameter E changed to a smaller value, i.e., to a value E = 10U6,being one hundred times smaller than in the previous case, the results are presented in Figure 5.

00

0.2

04

0.6 ii=b/o

Figure structure)

5. Coupled frequencies of the hydroelastic for E = 10e6 presented for varying diameter

syste_m (liquid layer around ratio k = b/a. Other parameters

elastic cylindrical shell and key as Figure 4.

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HYDROELASTICITY

The morphology of the roots in the range 0 G E < 10m6does not change very much, as may be seen. Again the coupled oscillatory liquid root shows similar behavior as for a rigid shell (Figure 4(a)) or a slightly less stiff shell with E = 10-9. The coupled structural root S, in radial direction is shown in Figure 5 only for a short range. In addition, the coupled roots have been investigated for an elasticity parameter E = lo-’ which is ten times larger than that for the results of Figure 5 but 10 times smaller than that for the results of Figure 4. The results are presented in Figure 6(a)-(c) for

k=b/a

240

-

-240

-

-720

-

-200 00

02

o-4

O-6

O-8

1(

I , \ \ \ \

1 0.95

1 0.96

1

I \ \ \

1 0.97

\ \

I I’

/*

I’ \

1

\

\

\

\

\_’ 1 0.98

/’ I 1’ I

1 099

I

. 1.00

k=b/a

Figure 6. Coupled frequencies of the hydroelastic system (liquid layer around elastic cylindrical shell struc_ture) for E = low5 presented for varying diameter ratio E= b/a. Other parameters and key as Figure 4. (a) O< k < 1 .O; (b) as (a) but enlarged ordinate scale; (c) as (b) but 0.95 < E< 1.00 and greatly enlarged ordinate scale.

260

H. F.BAUER

different scales. The various coupled roots of the liquid-structure system are indicated. It is again seen that the coupled structural root SC in the axial direction is unstable. In most of the diameter range E it is again a diverging instability and exhibits only for very thin liquid layers an unstable oscillatory behavior. For comparison and better identification of the various roots of the coupled liquidstructure system the diameter ratio E was kept constant at k= 4, while all the other parameters where kept at the same magnitudes as previously. The roots are presented versus log,, E of the elasticity parameter in the range -9 G log,, E < -4. The scale in the complex S-plane is changed in the following Figures 7(a)-(d). It may be seen that the 120 (bl

I

72 I/

,-

-4-O

-50

-6.0

-7.0

-6.0

-9.0

(d)

-4.0

-4.3

-4.6

-4.9

-5.2

-!

Figure 7. Coupled frequencies of the hydroelastic system (liquid layer around elastic cylindrical shell structure) presented for &= 0.5 as function of the elasticity parameter. Other parameters and key as Figure 4. (a) General plot; (b) as (a) but with enlarged ordinate scales; (c) as (a) but with reduced ordinate scale; (d) as (c) but for -4.O> log,, E > -5.5.

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261

coupled roots S,, in the angular direction have large negative real values, thus indicating a strong decay, while the roots in the axial direction S, are of positive real magnitudes showing therefore strong diverging instability. In conclusion, one can state that (a) for a very elastic shell (E = 10m4) the coupled liquid root becomes unstable for very thin liquid layers k> 0.9765, (b) for a stiffer shell (E = 10m9)the liquid behaves as in the case of a rigid shell, where at a diameter ratio of about E = 0.75 and larger the liquid ceases to oscillate and just performs an aperiodic motion, if disturbed, and (c) for a less stiff (very elastic) shell (E = 10e4) the coupled structural roots exhibit a strong instability, where in the axial direction for E~O.978 a diverging instability appears, while for a larger diameter ratio E> O-978 an oscillatory instability occurs. The physical significance of the results obtained above calls for some discussion and possible explanation. The main point of interest is the observation that the inclusion of viscosity provides instability, in contrast to the case for a frictionless liquid where the roots are all of imaginary magnitude. Small perturbations of the structure lead to perturbation of the liquid and vice versa In the case of a viscous liquid the flow field, due to the adherence condition, is quite different from that of ideal liquid where slipping at the shell appears in both angular and axial directions. Viscous liquid thus also exhibits different phase relations and it happens that for very elastic shells there is no damping out of the perturbation. These effects, i.e., phase shifts and the additional excitation by the adherence of the viscous liquid at the shell wall (not present for an ideal liquid) seem to be responsible for the instabilities. The inclusion of structural damping would, of course, change the results slightly, but, due to its small magnitude, it is not believed that it would qualitatively alter the basic findings. The usual “careless” statement, that the introduction of damping into a system will decrease its vibration amplitudes, or if unstable, would lead the system to stability, yields in many cases wrong conclusions. In fact damping or viscosity may even be the diabolic and roguish cause for instability. This can be observed in a rotating liquid column, where instability occurs for viscous liquid at a much lower angular spin velocity than it does for frictionless liquid. The same is true for a viscoelastic liquid. It is just the fact of the pure presence of viscosity, not even its magnitude, which is solely responsible for such behavior. Similar results have been obtained for a spinning satellite, where damping increases the tumbling until it is spinning about the axis of the largest moment of inertia.

REFERENCES 1. H. F. BAUER 1982 Acta Astronautica 9,547-563. Coupled oscillations of a solidly rotating liquid bridge. 2. H. F. BAUER 1984 Zeitschrif fur Angewandte Mathematik und Mechanik 64, 475-490. Natural damped frequencies of an infinitely long column of immiscible viscous liquids. 3. H. F. BAUER 1985 Forschungsbericht der Universitiit der Bundeswehr Miinchen LRT-WE-9-FB-5. Coupled frequencies of a hydroelastic viscous liquid system. 4. LORD RAYLEIGH 1892 Philosophical Magazine 34, 177-180. On the instability of cylindrical fluid surfaces. 5. H. LAMB 1945 Hydrodynamics. New York: Dover. See pp. 471-473. 6. J. A. F. PLATEAU Smithonian Institute Annual Report 1863, 207-285; 1864, 286-369; 1865, 41 l-435; 1866, 255-289. Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. 7. H. F. BAUER, 1987 Journal of Sound and Vibration 113, 217-232. Coupled frequencies of a hydroelastic system consisting of an elastic shell and frictionless liquid. 8. H. F. BAUER, 1986 Forschungsbericht der Universitiit der Bundeswehr Miinchen LRT- WE-9-FB-7-. Hydroelastic oscillations of a viscous infinitely long liquid column.

262

H.

F. BAUER

APPENDlX

For a liquid around a center shell the elements 6,, of the determinant, and here xz = S+ n’n2a’/l’ are given by the following expressions:

623 ==I[ (I/a)x m

(x)+~[xla(x)-2I,(x)]; (Z/a)x’

-!!!!E824= (l/a)*

K;(x)

+A!.!?-

[xK’,(x)

(l/ah'

-2Kn(x)l;

s,,=imK.,(x)+~{K,(x)(l+~)-~Kb(x)}; 82, = 628 = 629 = 0,

832 = -2in’rr’Kh

E (

(l/a)~[x2-{n2~2/(1/a)2}]; )/

with S = sa’/ u

ZERO

833= I;(x)[x+

VISCOUS

GRAVITY

n2772/x(I/a)2];

XI s35= (I/a)

(x); m

ad2 =

a,, = K;(x)[x+ 83,=

S,,=EL(x);

S,,=-MI:,

263

HYDROELASTICITY

n’~‘/x(~/a)‘]; 83*=

s39=0;

(EE)/(Ua)[x2-$$I;

-nnKk

(Ec)/tll.)

[x’-$$I;

6,3 = -irvrI~(Ex)/(l/a)x, ad4 = -[inr/(

I/a)x]Kk(

h);

b5 = W)I,Nd;

s,,=-[x’-n~~‘/(l/a)~];

846 = W~)K,,,(l;c);

S5, = -imI,

(+yi[x’-$-$];

S52=-imK,

(E~)/+-g$];

s&q= 0;

8,,=[mns~/f(l/a)x~]K,(kx);

653 = /-()r/na);x2 I,(~); 656= (ix/m)Kk(/&);

Sbl = -inTI,

(r~)/w

&,=

a,*=

a,?=

L=

6,,=0;

2im k'2[x2-n2r2/(1/u)-]

6,5=

s,,=o;

-[x2--n27r'/(l/a)'];

[I+$)-p:.(E~)];

k”~x’ _n~~,~r,a)~l

S73=qy,yx

[x2-&$];

K,(L);

a,,=

,

[K,

rnn72 ':"(w+~2(l,a)x2

s7~=E(~~~~xKb(kx)+~x~~~~a)

( Efj

-SK;

[kXIL(Ex)-21,(Ex)];

[~Kin(kx)-2K,,,(6)];

s,,=~I~(&)+~l,~~~)(1+~)-~1:~~~};

a,~=~K~~~)+X{K,,,~~)(l+~)-OK-}; a,,=

659 = 0;

[ x~-$$I:

S,, = -i nr K, (t$)/tlia) 863 = I,(~);

-im(p/p)(h/a)/[p(l-

= 0;

~,,=(ix/m)I~(Ex);

fS5*= -[x2-n27r2/(Z/a)‘];

85, = 0;

&

F)b”v2/Ea4];

($j];

264

H.

S82=Km

( ‘E

F. BAUER

2n2sr2 1 +(l/a)‘[x2-n27r2/(I/a)‘]

{+E)

(lim;j;.$))

6,, = (2,l?)[K,(l;c)

s,,=(2,~*)[1,(l;c)-~xl~(~)];

- I;cK#i)];

j$l-

im

fi2)b2v2, Ea4

,6(1-i;*)b2v2,Ea4;

’

b,=(X+$&) Iin(b); )

n*rr’E’

1 Yj (I-

-!=I s95 = @[/a)

Kin(~);

3m2+

(,,a)2

p(l+

-

C2)b2v2 Ea4

(I;(); m

[ x2--!32)

inn 6,, = W/la)

L(~);

chk’cf,$

jT(l-

(Al)

v )b v /Ea4’

Here T$ = T,,a/pv2, and p(l C2)b’v2/ Ea4 = E is an elasticity parameter. The remaining parameters are the density ratio p/p, thickness ratio h/a, the length ratio I/ a, the Poisson ratio V, the diameter ratio l= b/a and the modal numbers m and n in the angular and axial directions, respectively.

ZERO

GRAVITY

APPENDIX

;:

D E h

Ii,,, K,,, k P

r, cp,z s

VISCOUS

HYDROELASTICITY

2: LIST OF SYMBOLS

radius of liquid, or radius of shell radius of shell, or radius of liquid =Eh/(l-fi2) modulus of elasticity shell wall thickness modified Bessel functions of the first and second = b/n, diameter ratio wavelength in axial direction liquid pressure polar co-ordinates complex frequency, s = d+ i& time surface tension of liquid radial velocity component of liquid angular velocity component of liquid axial velocity component of liquid dynamic viscosity of liquid radial shell deflection angular shell deflection axial shell deflection Poisson ratio = p/p, kinematic viscosity of liquid density of liquid density of shell material free liquid surface elevation normal and shear stresses

kind

26.5