- Email: [email protected]
Note on buble motion in a rotating liquid under residual gravity
NOTE ON BUBBLE MOTION IN A ROTATING LIQUID UNDER RESIDUAL GRAVITY*? JULIUS Institut
fur Mechanik,
and KLAUS
SIEKMANN
DITTRICH
Hochschulstrasse I, Technische D-6100 Darmstadt, Germany
Hochschule
Darmstadt,
(Received 25 April 1977)
Abstract-The motion of a small gas bubble, presumed to retain its geometrical shape and contained in a rotating liquid, has been investigated. The fluid system liquid-gas is subject to the influence of a reduced gravitational field. It is demonstrated that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.
1. INTRODUCTION
Many space processes require the removal of gas bubbles distributed at random in the molten material. This objective can be achieved by spinning the melt about a definite axis through the center of the liquid body. In order to study such processes in some detail, the motion of a shape-preserving gas bubble in a revolving liquid has been investigated. The liquid-gas system is placed into a low gravity environment and it is shown that due to residual gravity the bubble executes a circular motion within the liquid globe, as seen from an observer moving with the bulk of the liquid. This case occurs if the direction of the axis of rotation and the direction of gravity are orthogonal. 2. ANALYSIS
According to Siekmann equations of second order M#
and Johann
[l],
the non-linear
system of ordinary
= [email protected](Rt+cp)+k,. MR$
= 2mh
- 2M&j
differential
(1)
f mg . sin(Qt + q) + k,,
(2)
q(O) = Cp, Q(O) = 0
(3)
subject to the initial conditions R(0) = i?, permits
the steady-state
l?(O) = 0,
solution R = R, = const,
provided
cp = q+, --fit,
cppo = const
(4)
that k,=
-Kd,
k,=
-KR@,
K=const(>O).
Pa, b)
An analysis of the perturbed motion in the first approximation shows, that the above solution (4) is asymptotically stable. The system under consideration describes the motion of a small and undeformable gas bubble, immersed in a rotating homogeneous incompressible and viscous liquid, where the fluid is assumed to consist of a system of particles [2]. The liquid mass spins with constant angular velocity fi about a fixed horizontal axis (Fig. l), the rotating polar reference frame (R, cp) moves with the bulk of the liquid. In the foregoing equations we denote the virtual mass of the bubble by M( = m, +~m), the mass of the bubble (in the computations assumed to be negligible) by m,, the mass of the displaced liquid by m, the virtual mass coefficient by K, the acceleration due to gravity by g, *Dedicated tpresented
to Professor Dr. Kurt Magnus in part at the GAMM-DCAMM
on the occasion of his sixty-fifth birthday. Congress, Copenhagen, May 31-June 3, 1977. 409
JULIUS SIEKMANN and KLAUS DITTRICH
410
Fig. 1. Coordinate
system notation.
and the time by t. Moreover, we denote differentiation with respect to time by a dot. The direction of gravity is parallel to the vertical and perpendicular to the spin axis, its magnitude is not necessarily restricted to its terrestrial value. Equations (5a, b) state that the frictional force acting on the bubble is proportional to the velocity of the bubble. For a spherical bubbk of radius a the constant is assumed to be K = 12. n. p. a. Moreover, the bubble Reynolds number Re = !!$. (A2 +R24,2)1/2
(6)
where p, b and L are, respectively, the density of the liquid, the viscosity of the liquid, and a characteristic length (radius of a spherical or cylindrical bubble), is supposed to be fairly large (> 1). However, it can be shown that the solution (4) holds also in case of a general power law k, = - Kl+,
k, = - K(Rqip,
K=const(>O),
ct=f,
where CIis a rational number, i.e. p and q are integers. Straightforward, calculations yield the result that R, and ‘pOare determined by
4#0,
(7)
but elementary
(8) and tango
= (-l)“&*a-2R;-1.
For a = 1, the formulae (9) of [l] are readily recovered, while for a quadratic = 2), valid for larger velocities (Re >>l), one obtains, after some manipulations, R,
=
+$(“+Km)R 1-1+r(M+4 ,n14
m
drag law (a
(Frl”’ - l]‘;:
(10)
and K
tan ‘pO= P’R, M+m Both cases (a = 1,2) are of practical
importance.
(11)
Note on bubble motion
in a rotating
In what follows, it will be demonstrated general force law
liquid under residual
that the solution
gravity
411
(4) exists even for the more
k, = -K,~d-Kz~l?2’“sgnfi,
(12a)
k, = -K,~Rt~5-K,~(Rqi)~‘~sgn(i), where the positive
constants
(12b)
K, and K, are given by K, = 4,rap2:3.a’:”
K, = 12Z$,
(13)
In these constants o is the surface tension coefficient at the liquid-gas interface, and a and h are, respectively, the radius and the height ofa cylindrical bubble. The system (l)-(3) with the resistance term (12a, b) (instead of (5) or (7), respectively) governs the very slow motion of a bubble between two liquid-filled closely spaced, plan-parallel, rotating plates. Such a device, sometimes called a zero-g centrifuge [3], allows to simulate reduced gravity conditions in an earthbound workshop. Concerning the drag term (12), it should be pointed out that the relation D = K,.
U+K,.
U213,
(14)
derived by Eck [4] for the translational motion of a small bubble with velocity U in a Hele-Shaw cell, is presupposed to hold for slow rotational motion, i.e. for flows with very small (<< 1) modified bubble Reynolds numbers h 2. 2a
0
Re* = !3d”+Rz92)1~2.
(15)
In order to check the validity of the proposed force law (12) for the problem under study, extensive computations [S] have been carried out and compared with test results. Good agreement was found between numerical and experimental data. Substitution of equations (4) and (12) into the basic equations (1,2) yields cos400
(“+m)aL209 .R
=
(16)
PI
KtQ sin ‘p. = ---Ro----mg
K2a2’3
. ~2’3
(17)
0 .
mg
whence (VUJ)~= ([(M+m)R2]2+Kj~z}R~-2K1 follows. Defining
X2Q5’3.R;‘3+K;fi14’3
.R;‘3
now the quantities A = [(M+m)Q2]2+K:0Z B = -2K CZK~.~‘/~
1
(19a)
> 0,
(19b)
.K 2 fi5’3 < 0,
2
>O
(19c)
D = (ins)’ > 0, equation
(18)
(19d)
(18) becomes D --Ri= A
Rp A
(BR;‘3 + C).
With that R, can be determined numerically or graphically. In the graphical the abscissa of the intersection point of the (quadratic) parabola H,(R,)
= ; - R;
(20) solution,
R, is
(21)
412
JULIUS SIEKMANN and KLAUS DITTRICH
1 6000
4000-r" 3000-f ZOOO--
IO
20
30
40
50
60
70
-lOOO-Fig. 2. Plot of functions
. WEG EmER
BLASEIM
H, and
H, vsR,,
ROT~ERENDEN
MEDIUM
t MM=0.500 OMEGA
=
KAPPA
=
RPUNKT SCHRITT
2.00
I PRO
I.0 0 = 0.0 H = 0.040
MARKIERUNGSPUNKTE
MM
MM
SEK
PRO SEK SEK ALL E 5.0
SEK
Fig.3. Bubble path in a rotating
BLASENWRCHMESSER
D = 20.0
RHO FLUESSIGK.
=0.86E
-03
EROBESCHL.
=2000.
MUE
=0.35E-02
FLUESSlGK.
liquid under residual gravity
(absolute
MM
reference
MM
G FROMMxx PRO
%x2
G PRO MMxS
frame).
Note on bubble motion
in a rotating
WEG EINER BLASE
IM
liquid under residual
ROTIERENDEN
413
gravity
MEDIUM
RELATIVBEWEGUNG
I MM =0.500 OMEGA
= 2.00
KAPPA
MM
I PRO SEK BLASENDURCHMESSER D ~20.0 MM
I
=
I.0
RPUNKT =
0.0
MM PRO SEK
RHO FLUESSIGK. =O.B6E-03 G PRO MMma:
SCHRITT=
0.040
SEK
= 2ooO. MM PRO Sxx2 EROEESCHL. MUE FLUESSIGK. z 0.35E-02 G FRO MM XS
MARKIEt?UNGSPUNKTE
ALLE
5.0 SEK
Fig. 4. Bubble path in a rotating
liquid under residual
gravity
(rotating
reference frame).
with the curve H,(R,)
=
$7. (BR;'3 +C).
(22)
Finally, ‘pOresults from tan ‘pO= -
K,0+KK,Q2’3R,1’3
(iV+m)@
.
(23)
If we neglect the influence of surface tension (i.e., K, = 0), it turns out that R, is given as the intersection of the parabola (21) with the abscissa. This solution (with K, = K) corresponds, however, to the solution (I&, cpe) of [l]. 3. NUMERICAL
EXAMPLE
Let us select a cylindrical bubble (K = 1) of radius a = 1Omm and height h = 2 mm, immersed in a rotating liquid (S2 = 2s’) of density p = 0.86 x 10m3gmme3, viscosity c and interfacial tension c = 25 g se2. The residual gravity g is = 0.35 x 10-2gmm-‘s-1, assumed to be 2000mrn~-~. From these data the different constants are evaluated as (omitting the dimensions) A4 = m = plra’h = 0.5404; K, = 6.5973; K2 = 8.4704; A = 192.79; B = -354.8279; C = 180.7911; D = 1167929.5. From Fig. 2 we read R0 = 63.1 mm, while equation (23) yields tan vpo= -3.83, i.e., i& = -75.38”. Taking into consideration the position of the reference axis, we find 40~= 360” + @e = 284.6”. In order to examine this result numerically, a computational experiment was carried out. With the initial values R(0) = 130 mm, R(O) = 0, ~(0) = 31t/2, e(O) = 0, the system (l-3) was
414
JULIUS SIEKMANN and KLAUS DITTRICH
1
R(t)
1
IMM
ZEIT
T (SEK)
5050MM
I SEK = 5015 MM 2GRAD=0.02 MM
Fig. 5. Plots of radius and polar angle vs time.
integrated by means of the Runge-Kutta-Nystrom method. Figures 3, 4, 5 show, respectively, plots of the absolute and relative bubble path as well as radius (R) and polar angle (cp)as functions of time (t). In the absolute reference frame the bubble migrates toward a stable point (R,, tp,,), while in the rotating reference frame the bubble travels finally on a concentric circle about the center of rotation. The ticks on the trajectories in Figs. 3 and 4 mark equal time intervals (5 seconds). Computation as well as inspection of the figures furnish the values R, = 63.1 mm and ‘pO= 283.9”. The transient motion is clearly exhibited in Fig. 5. Acknowledgement-The work reported is this note was supported in part by the Bundesministerium fur Forschung und Technologie, represented by the Deutsche Forschungsund Versuchsanstalt fur Luftund Raumfahrt, Bereich Projekttragerschaften. REFERENCES 1. J. Siekmann 2. 3. 4. 5.
and W. Johann, On bubble motion in a rotating liquid under simulated low and zero gravity, Zngenieur-Arch 45,307 (1976). J. Siekmann and K. Dittrich, Uber die Bewegung von Gasblasen in einem rotierenden Medium, Ingenieur-Archiu 44,131 (1975). J. Siekmann, W. Eck, W. Johann, Experimentelle Untersuchungen iiber das Verhalten von Gasblasen in einem Null-g-Simulator, Z. Ffugwiss. 22,83 (1974). W. Eck, Blasenbewegungen zwischen parallelen, ebenen Wanden. Dtssertatton. Techmsche Hochschule Darmstadt (1975). J. Siekmann and K. Dittrich, Computer study of bubble motion in a rotating liquid, Computer Methods in Applied Mechanics und Eugiwering 10, 291-301 (1977).
Note on bubble motion
in a rotating
liquid under residual
gravity
R&I& On &die le mouvement d'une petite bulle de gaz suooosge aarder sa forme ahome'triaue et contenue dans un liquide ei rotation. Le systeme fluide liquide-gaz est soumis a l'influence d'un champ de qravite reduit. On demontre que sous certaines conditions (axe de rotation et direction de gravitQ perpendiculaires entre eux) la bulle, vue par un observateur se d6plasant avec l'ensemble du liquide) parcourt un cercle autour de l'axe de rotation.
Zusammenfassunq: Die Bewegung einer kleinen undeformierbaren Gasblase in einer rotierenden Flussigkeit wird unter dem EinfluB eines Restschwerefeldes untersucht. Das Svstem Flussiqkeit-Gas befinde sich in einer sog. Nul-g-Zentrifuge. Es-wird gezeigt, daB sich die Blase innerhalb der Flussiqkeit auf einer Krisbahn bewegt, deren Zentrum mit dem DurchstoBpunkt Drehachse-Zentrifuge zusammenfallt, betrachtet vom Standpunkt eines mitbewegten Beobachters. Dies gilt fur den Fall, daB Drehachse und Richtung der Schwerebeschleunigung aufeinander senkrecht stehen.
415
fur Mechanik,
and KLAUS
SIEKMANN
DITTRICH
Hochschulstrasse I, Technische D-6100 Darmstadt, Germany
Hochschule
Darmstadt,
(Received 25 April 1977)
Abstract-The motion of a small gas bubble, presumed to retain its geometrical shape and contained in a rotating liquid, has been investigated. The fluid system liquid-gas is subject to the influence of a reduced gravitational field. It is demonstrated that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.
1. INTRODUCTION
Many space processes require the removal of gas bubbles distributed at random in the molten material. This objective can be achieved by spinning the melt about a definite axis through the center of the liquid body. In order to study such processes in some detail, the motion of a shape-preserving gas bubble in a revolving liquid has been investigated. The liquid-gas system is placed into a low gravity environment and it is shown that due to residual gravity the bubble executes a circular motion within the liquid globe, as seen from an observer moving with the bulk of the liquid. This case occurs if the direction of the axis of rotation and the direction of gravity are orthogonal. 2. ANALYSIS
According to Siekmann equations of second order M#
and Johann
[l],
the non-linear
system of ordinary
= [email protected](Rt+cp)+k,. MR$
= 2mh
- 2M&j
differential
(1)
f mg . sin(Qt + q) + k,,
(2)
q(O) = Cp, Q(O) = 0
(3)
subject to the initial conditions R(0) = i?, permits
the steady-state
l?(O) = 0,
solution R = R, = const,
provided
cp = q+, --fit,
cppo = const
(4)
that k,=
-Kd,
k,=
-KR@,
K=const(>O).
Pa, b)
An analysis of the perturbed motion in the first approximation shows, that the above solution (4) is asymptotically stable. The system under consideration describes the motion of a small and undeformable gas bubble, immersed in a rotating homogeneous incompressible and viscous liquid, where the fluid is assumed to consist of a system of particles [2]. The liquid mass spins with constant angular velocity fi about a fixed horizontal axis (Fig. l), the rotating polar reference frame (R, cp) moves with the bulk of the liquid. In the foregoing equations we denote the virtual mass of the bubble by M( = m, +~m), the mass of the bubble (in the computations assumed to be negligible) by m,, the mass of the displaced liquid by m, the virtual mass coefficient by K, the acceleration due to gravity by g, *Dedicated tpresented
to Professor Dr. Kurt Magnus in part at the GAMM-DCAMM
on the occasion of his sixty-fifth birthday. Congress, Copenhagen, May 31-June 3, 1977. 409
JULIUS SIEKMANN and KLAUS DITTRICH
410
Fig. 1. Coordinate
system notation.
and the time by t. Moreover, we denote differentiation with respect to time by a dot. The direction of gravity is parallel to the vertical and perpendicular to the spin axis, its magnitude is not necessarily restricted to its terrestrial value. Equations (5a, b) state that the frictional force acting on the bubble is proportional to the velocity of the bubble. For a spherical bubbk of radius a the constant is assumed to be K = 12. n. p. a. Moreover, the bubble Reynolds number Re = !!$. (A2 +R24,2)1/2
(6)
where p, b and L are, respectively, the density of the liquid, the viscosity of the liquid, and a characteristic length (radius of a spherical or cylindrical bubble), is supposed to be fairly large (> 1). However, it can be shown that the solution (4) holds also in case of a general power law k, = - Kl+,
k, = - K(Rqip,
K=const(>O),
ct=f,
where CIis a rational number, i.e. p and q are integers. Straightforward, calculations yield the result that R, and ‘pOare determined by
4#0,
(7)
but elementary
(8) and tango
= (-l)“&*a-2R;-1.
For a = 1, the formulae (9) of [l] are readily recovered, while for a quadratic = 2), valid for larger velocities (Re >>l), one obtains, after some manipulations, R,
=
+$(“+Km)R 1-1+r(M+4 ,n14
m
drag law (a
(Frl”’ - l]‘;:
(10)
and K
tan ‘pO= P’R, M+m Both cases (a = 1,2) are of practical
importance.
(11)
Note on bubble motion
in a rotating
In what follows, it will be demonstrated general force law
liquid under residual
that the solution
gravity
411
(4) exists even for the more
k, = -K,~d-Kz~l?2’“sgnfi,
(12a)
k, = -K,~Rt~5-K,~(Rqi)~‘~sgn(i), where the positive
constants
(12b)
K, and K, are given by K, = 4,rap2:3.a’:”
K, = 12Z$,
(13)
In these constants o is the surface tension coefficient at the liquid-gas interface, and a and h are, respectively, the radius and the height ofa cylindrical bubble. The system (l)-(3) with the resistance term (12a, b) (instead of (5) or (7), respectively) governs the very slow motion of a bubble between two liquid-filled closely spaced, plan-parallel, rotating plates. Such a device, sometimes called a zero-g centrifuge [3], allows to simulate reduced gravity conditions in an earthbound workshop. Concerning the drag term (12), it should be pointed out that the relation D = K,.
U+K,.
U213,
(14)
derived by Eck [4] for the translational motion of a small bubble with velocity U in a Hele-Shaw cell, is presupposed to hold for slow rotational motion, i.e. for flows with very small (<< 1) modified bubble Reynolds numbers h 2. 2a
0
Re* = !3d”+Rz92)1~2.
(15)
In order to check the validity of the proposed force law (12) for the problem under study, extensive computations [S] have been carried out and compared with test results. Good agreement was found between numerical and experimental data. Substitution of equations (4) and (12) into the basic equations (1,2) yields cos400
(“+m)aL209 .R
=
(16)
PI
KtQ sin ‘p. = ---Ro----mg
K2a2’3
. ~2’3
(17)
0 .
mg
whence (VUJ)~= ([(M+m)R2]2+Kj~z}R~-2K1 follows. Defining
X2Q5’3.R;‘3+K;fi14’3
.R;‘3
now the quantities A = [(M+m)Q2]2+K:0Z B = -2K CZK~.~‘/~
1
(19a)
> 0,
(19b)
.K 2 fi5’3 < 0,
2
>O
(19c)
D = (ins)’ > 0, equation
(18)
(19d)
(18) becomes D --Ri= A
Rp A
(BR;‘3 + C).
With that R, can be determined numerically or graphically. In the graphical the abscissa of the intersection point of the (quadratic) parabola H,(R,)
= ; - R;
(20) solution,
R, is
(21)
412
JULIUS SIEKMANN and KLAUS DITTRICH
1 6000
4000-r" 3000-f ZOOO--
IO
20
30
40
50
60
70
-lOOO-Fig. 2. Plot of functions
. WEG EmER
BLASEIM
H, and
H, vsR,,
ROT~ERENDEN
MEDIUM
t MM=0.500 OMEGA
=
KAPPA
=
RPUNKT SCHRITT
2.00
I PRO
I.0 0 = 0.0 H = 0.040
MARKIERUNGSPUNKTE
MM
MM
SEK
PRO SEK SEK ALL E 5.0
SEK
Fig.3. Bubble path in a rotating
BLASENWRCHMESSER
D = 20.0
RHO FLUESSIGK.
=0.86E
-03
EROBESCHL.
=2000.
MUE
=0.35E-02
FLUESSlGK.
liquid under residual gravity
(absolute
MM
reference
MM
G FROMMxx PRO
%x2
G PRO MMxS
frame).
Note on bubble motion
in a rotating
WEG EINER BLASE
IM
liquid under residual
ROTIERENDEN
413
gravity
MEDIUM
RELATIVBEWEGUNG
I MM =0.500 OMEGA
= 2.00
KAPPA
MM
I PRO SEK BLASENDURCHMESSER D ~20.0 MM
I
=
I.0
RPUNKT =
0.0
MM PRO SEK
RHO FLUESSIGK. =O.B6E-03 G PRO MMma:
SCHRITT=
0.040
SEK
= 2ooO. MM PRO Sxx2 EROEESCHL. MUE FLUESSIGK. z 0.35E-02 G FRO MM XS
MARKIEt?UNGSPUNKTE
ALLE
5.0 SEK
Fig. 4. Bubble path in a rotating
liquid under residual
gravity
(rotating
reference frame).
with the curve H,(R,)
=
$7. (BR;'3 +C).
(22)
Finally, ‘pOresults from tan ‘pO= -
K,0+KK,Q2’3R,1’3
(iV+m)@
.
(23)
If we neglect the influence of surface tension (i.e., K, = 0), it turns out that R, is given as the intersection of the parabola (21) with the abscissa. This solution (with K, = K) corresponds, however, to the solution (I&, cpe) of [l]. 3. NUMERICAL
EXAMPLE
Let us select a cylindrical bubble (K = 1) of radius a = 1Omm and height h = 2 mm, immersed in a rotating liquid (S2 = 2s’) of density p = 0.86 x 10m3gmme3, viscosity c and interfacial tension c = 25 g se2. The residual gravity g is = 0.35 x 10-2gmm-‘s-1, assumed to be 2000mrn~-~. From these data the different constants are evaluated as (omitting the dimensions) A4 = m = plra’h = 0.5404; K, = 6.5973; K2 = 8.4704; A = 192.79; B = -354.8279; C = 180.7911; D = 1167929.5. From Fig. 2 we read R0 = 63.1 mm, while equation (23) yields tan vpo= -3.83, i.e., i& = -75.38”. Taking into consideration the position of the reference axis, we find 40~= 360” + @e = 284.6”. In order to examine this result numerically, a computational experiment was carried out. With the initial values R(0) = 130 mm, R(O) = 0, ~(0) = 31t/2, e(O) = 0, the system (l-3) was
414
JULIUS SIEKMANN and KLAUS DITTRICH
1
R(t)
1
IMM
ZEIT
T (SEK)
5050MM
I SEK = 5015 MM 2GRAD=0.02 MM
Fig. 5. Plots of radius and polar angle vs time.
integrated by means of the Runge-Kutta-Nystrom method. Figures 3, 4, 5 show, respectively, plots of the absolute and relative bubble path as well as radius (R) and polar angle (cp)as functions of time (t). In the absolute reference frame the bubble migrates toward a stable point (R,, tp,,), while in the rotating reference frame the bubble travels finally on a concentric circle about the center of rotation. The ticks on the trajectories in Figs. 3 and 4 mark equal time intervals (5 seconds). Computation as well as inspection of the figures furnish the values R, = 63.1 mm and ‘pO= 283.9”. The transient motion is clearly exhibited in Fig. 5. Acknowledgement-The work reported is this note was supported in part by the Bundesministerium fur Forschung und Technologie, represented by the Deutsche Forschungsund Versuchsanstalt fur Luftund Raumfahrt, Bereich Projekttragerschaften. REFERENCES 1. J. Siekmann 2. 3. 4. 5.
and W. Johann, On bubble motion in a rotating liquid under simulated low and zero gravity, Zngenieur-Arch 45,307 (1976). J. Siekmann and K. Dittrich, Uber die Bewegung von Gasblasen in einem rotierenden Medium, Ingenieur-Archiu 44,131 (1975). J. Siekmann, W. Eck, W. Johann, Experimentelle Untersuchungen iiber das Verhalten von Gasblasen in einem Null-g-Simulator, Z. Ffugwiss. 22,83 (1974). W. Eck, Blasenbewegungen zwischen parallelen, ebenen Wanden. Dtssertatton. Techmsche Hochschule Darmstadt (1975). J. Siekmann and K. Dittrich, Computer study of bubble motion in a rotating liquid, Computer Methods in Applied Mechanics und Eugiwering 10, 291-301 (1977).
Note on bubble motion
in a rotating
liquid under residual
gravity
R&I& On &die le mouvement d'une petite bulle de gaz suooosge aarder sa forme ahome'triaue et contenue dans un liquide ei rotation. Le systeme fluide liquide-gaz est soumis a l'influence d'un champ de qravite reduit. On demontre que sous certaines conditions (axe de rotation et direction de gravitQ perpendiculaires entre eux) la bulle, vue par un observateur se d6plasant avec l'ensemble du liquide) parcourt un cercle autour de l'axe de rotation.
Zusammenfassunq: Die Bewegung einer kleinen undeformierbaren Gasblase in einer rotierenden Flussigkeit wird unter dem EinfluB eines Restschwerefeldes untersucht. Das Svstem Flussiqkeit-Gas befinde sich in einer sog. Nul-g-Zentrifuge. Es-wird gezeigt, daB sich die Blase innerhalb der Flussiqkeit auf einer Krisbahn bewegt, deren Zentrum mit dem DurchstoBpunkt Drehachse-Zentrifuge zusammenfallt, betrachtet vom Standpunkt eines mitbewegten Beobachters. Dies gilt fur den Fall, daB Drehachse und Richtung der Schwerebeschleunigung aufeinander senkrecht stehen.
415