A cylindrical liquid bridge supported between two circular-shaped disks in isorotation is considered. The effect of a lateral gravitational field on the stability of the liquid bridge is investigated. A numerical method is used to find stable and unstable shapes and to determine the stability limit for different values of lateral gravity

Lapuerta González, María Victoria

Laverón Simavilla, Ana

Perales Perales, José Manuel

Rodríguez Otero, Jacobo

Archivo Digital UPM

Effect of rotation on the equilibrium shapes and stability of liquid bridges in a lateral gravitational field

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Proceedings of Third International Symposium on Physical Science in Space 2008 
J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 
Effect of rotation on the equilibrium shapes and stability of liquid 
bridges in a lateral gravitational field 
A. Laveron-Simavilla1 , V. Lapuerta1, J. Rodriguez1 and J. M. Perales1 
1
 E.T.S.I.Aeronauticos (Universidad Politecnica de Madrid), Spain, ana. laveron@upm. es, 
mariavictoria.lapuerta(a>upm.es 
Abstract 
A cylindrical liquid bridge supported between two circular-shaped disks in isorotation is considered. The 
effect of a lateral gravitational field on the stability of the liquid bridge is investigated. A numerical 
method is used to find stable and unstable shapes and to determine the stability limit for different values 
of lateral gravity. 
1. Introduction 
The behavior of liquid bridges has been widely 
studied, both theoretically and experimentally, due 
to the use of this configuration in a crystal growth 
technique known as the floating zone technique. In 
this technique, rotation of the supports is used to 
achieve an uniform temperature field. 
In this paper a cylindrical liquid bridge 
supported between two circular-shaped disks in 
isorotation is considered. The effect of a lateral 
gravitational field on the stability of the liquid 
bridge is studied. Numerical and analytical studies 
for the case with rotation and no lateral gravity15'2) 
have been made. 
The stability limits and the equilibrium shapes of 
the configuration are calculated using an extension 
of an already implemented numerical method35, 4). 
The numerical method is used to find stable and 
unstable shapes and to determine the stability limits 
for different values of the lateral gravity. 
2. Problem description 
The fluid configuration consists of a liquid 
bridge as sketched in Figure 1. The liquid column is 
held by surface tension forces between two disks of 
radius i^ , placed a distance L apart. Both disks are 
parallel and coaxial. The volume of the bridge is that 
corresponding to a cylindrical one, V = KR\L . The 
liquid bridge is in an lateral gravitational field, and 
is rotating at an angular speed Q around the axis 
of the disks. 
The equation governing the steady shape of the 
liquid bridge is obtained by balancing the different 
forces at the interface 
aM(R) + P + -fa2R2+p\g!\Rcos0 = O (1) 
where R = R(Z,0) is the equation of the 
gas-liquid interface, a is the surface tension, 
M(R) is twice the mean curvature of the interface, 
P is the pressure difference at the origin, 9 is the 
azimuthal angle, p is the liquid density, and g; is the 
lateral gravity. 
Equation (1) has to be integrated with the 
boundary conditions 
R(±L/2,9) = R0 
R(Z,6) = R(Z,0 + 2TT) 
(2) 
(3) 
-. L/2 In 
- \ dZ JR\Z,0)d6 = xLRl 
/  2  
'o (4) 
• " - L / 2 0 
Eq. (2) indicates that the liquid column remains 
anchored to the disk edges, Eq. (3) comes from the 
azimuthal periodicity, and Eq. (4) expresses the 
conservation of the volume of the liquid bridge. 
dislcs a d s 
1 
Q 
• H^\ 
-y 
Figure 1. Geometry and coordinate system for the liquid 
bridge problem. 
Let us introduce the following dimensionless 
variables and parameters 
A = L/(2R0), W = fa2R3Q/a, 
0 ' (5) P = PRJ a, F(z, 6) = R(z, 6) I R{ 
Bj= p\gt\R02 /a, z = Z/R0, 
where A is the liquid bridge slendemess, W the 
Weber number, P the dimensionless reference 
pressure, and Bi is the lateral Bond number. The 
formulation of the problem then becomes: 
M(F) + P + -WF2 + B,F cosO = 0 (6) 
with 
J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 : — 219 — 43 
M(F) = F(l + F:
2)(F-F) + FF7z(F2+F2)-2Fe(Fg + FFF) 
F2(l + Fz2) + i £ ) 
(7) 
The dimensionless boundary conditions for Eq. 
(6) are: 
F(±A,0) = 1 
F(Z,0) = F(Z,0 + 2TT) 
and 
A lit 
jdz JF2(z,0)d0 = 47rA 
(8) 
(9) 
(10) 
- A 0 
3. Numerical method 
An algorithm, based on a continuation method5^ 
capable of stepping over bifurcation points and 
turning points was developed using a 
finite-difference method3^ and used to obtain the 
bifurcation diagrams and equilibrium shapes of 
liquid bridges subjected to lateral acceleration and 
other effects. The stable or unstable character of 
each of the shapes is determined to locate the 
stability limit. In this paper the system of 
equations (6)-(10) is solved by using an extension of 
that algorithm adapted to liquid bridges rotating 
around an eccentric axis to study the effect of 
combined eccentricity and angular rotation. 
The method is based on linearizing Eqs. 
(7)-(ll) around a known solution (F0(z,6),P0) by 
seeking solutions of the form 
F(z,0) = Fo(z,0) + f(z,0) + o(\f\), 
P = P0+p + o(\p\), 
where | / / F 0 | « 1 and \p/PQ\ « 1 • The leading 
order terms obtained from Eq. (7) result in an 
equation forflz, 6) 
J
 ( i i) 
+Ef^ + Gfss+HfzS+P0+p + -W(F02 + 2Fj) + 
+B, (FQ+f )cosd = 0 
where 
A, B, C, D, E, G, H, 0, Q, S and f are 
known functions of F0 (z, 0) and PQ and thus of 
the particular point on the interface. The leading 
order terms obtained for the boundary conditions are 
f(±A,0) = O (12) 
f(z,0) = f(z,0 + 27r) (13) 
^Adz^Fo(z,d)2d0 + 2^dz^[Fo(z,0)f(z,d)]d0 = 47tA 
(14) 
If (F0(z,0),PO) is an exact solution of the 
problem, Eqs. (11)-(14) can be simplified, but all 
terms have been retained here because 
A + \B-
(FQ(z,0),PQ) will only be an approximation to 
the solution in the iterative scheme used here. 
In order to develop a center finite difference 
scheme the domain has been characterized by a 
mesh, defined as the intersection of the free surface 
with the following planes: 
A 
J J 
0,1,..., J 
0=*?-i, 
I + l 
i = 0,l,...,I 
(15) 
(16) 
By doing so, the system (11)-(14) yields a 
linearized system of finite-difference equations 
which can be written as follows: 
+«w"-yj;-/A'+#1,)+i'=n. I=O,...,I, J=O,...,J 
(17) 
f°-fr=<> 
/ , ' = ! - < , . 
fi=i-K 0' 
z = 0 , 
z = 0 , 
(18) 
(19) 
(20) 
where the coefficients <*u>Pv>ru, Sv, <p,j, frj, y , , a y 
(21) 
and 
A are functions of FQ . and PQ. 
The iterative scheme is started from a known 
solution(p
 s po 5 A, W = 0, B, - 0), which is introduced 
into equations (17)-(21). The value of the Weber 
number is then increased by a small amount/^ = w)> 
and the linear terms (f,p) are calculated. The new 
values 
{Fo=Fo+f,Po=Po+p,A,W = Wl,Bl=0) 
are then introduced into equations (17)-(21) to 
obtain new corrections(f,p), and this procedure is 
repeated until | | / | 2 + ^ 2 <10"6 and the final 
equilibrium shape is obtained 
{F,=F0+f,P^P0+P, A, W = W1,Bl=0). 
The iteration then repeats with an increased value of 
the Weber number (w = W2) giving a new 
equilibrium shape 
(F2=Fo+f,P2=Po+p,A,W = W2,Bl = 0). 
All equilibrium shapes obtained in this manner are 
stored to be used as initial conditions for an iterative 
scheme identical to the one just described, but with 
increasing values of the eccentricity, e. Together, 
these procedures allow the bifurcation diagram to be 
completed. 
If no further modifications are made, the 
algorithm becomes unstable when crossing a critical 
point. To stabilize the algorithm a supplementary 
44 J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 — 220 — 
equation defining the arc-length parameter needs to 
be included. The details of the numerical method 
used to locate bifurcation and limit points are 
identical to those outlined elsewhere3^ and will not 
be repeated here. We only remark that an analysis 
of the errors of the numerical method was made3), 
which indicates that a minimum number of mesh 
points is needed to find the complete bifurcation 
diagram and that this minimum increases as the 
slenderness of the liquid bridge decreases. The size 
of the mesh used to obtain the results presented in 
this paper is 7xJ=19x28. 
4. Results 
Perales et al. ^ looked for a solution of Eqs. (l)-(4) 
with fl/=0 of the form: F=l+ef+o(e), P=l-W/2+ep 
+o(e), e<<7 (small departures from a cylindrical 
liquid bridge), finding the non-zero solutions: 
(i) Non-axisymmetric shapes (C-mode) 
Wn = — , f = cos#cos — z 
0
 UAJ UA 
(ii) Axisymmetric shapes (amphora mode) 
Wn=\—\ - 1 , f = sm\—i 
p = 0 
A VA 
,p = 0 
The functions W0(A) represent in the A- W plane the 
curves where the transition between stable and 
unstable equilibrium shapes appears. These two 
curves have been plotted in Figure 2. The point B, 
for which expressions (24) and (25) are equal 
(A = v3/z72 ), marks the transition between the 
C-mode and the amphora mode instabilities. 
0.9 
0.8 
0.7 
0.6 
fe 0.5 
0.4 
0.3 
0.2 
0.1 
\ A \ 
\ 
v C mode \ 
\ \ 
-
f 
" 
\ 
V
"
V
"-'^? 
\ - --- . ._ 
\ Amphora 
\*^mode 
Figure 2. Stability diagram for zero lateral gravity. 
The bifurcation diagrams are sketched in Figure 3 
and Figure 4 for J3/=0. Note that the bifurcations 
are both subcritical and hence that both equilibrium 
shapes (the C-mode shape and the amphora mode 
shape) are unstable. 
If the rotation speed is slowly increased from zero, 
one has stable cylindrical shapes until W reaches W0 
at which point an instability occurs. Thus, for Bf=Q 
liquid bridges with A<V3;r/2 lose stability to a 
non-axisymmetric mode and liquid bridges with 
A > -Jin 12 lose stability to an axisymmetric mode. 
6 
" x^— 0<Bl<Blc 
^J\ / w 
B,=0 
Figure 3. Bifurcation diagram for A < yfeft 12 
B,=0 
Figure 4. Bifurcation diagram for A > >/3;r / 2 . 
With the numerical method we have recovered the 
stability threshold forj^ =0 , and we have analyzed 
the dependency of this threshold on the lateral 
gravity and the angular speed. Figure 5 shows the 
variation of the stability threshold for increasing 
lateral gravity. The line fori?, = 0corresponds to the 
threshold represented in Figure 2. As we can see in 
the figure, for small values of the lateral gravity, it 
has a somewhat stronger effect on short liquid 
bridges (A<A/3;T/2), although for increasing values 
of the lateral gravity the stability region is strongly 
reduced even for long liquid bridges ( A > -\/3;z: / 2). 
0.7 
0.6 
0.5 
S ! 0.4 
0.3 
0.2 
0.1 
; \ . 
0.25 
0.1 
e=o 
0.01^^« 
-
-
-
^ ^ 
^~~"~^»\ 
1 ^ ^ ^ 
1 ^ ^ 
4: ^ "Kx 
Figure 5. Stability threshold for different values of Bh The 
dashed line marks the transition between a subcritical 
pitchfork bifurcation (B) and a turning point (TP). 
The analysis shows a complex behaviour of the 
instability boundary for liquid bridges. For a given 
liquid bridge with A<>/3^/2 , increasing the 
J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 — 221 — 45 
lateral gravity leads to: 
(i) A subcritical pitchfork bifurcation (for 
BrO). 
(ii) A : turning point (for 0<5/<5/c). The 
bifurcation diagram has been sketched in 
Figure 3. 
(iii) A subcritical pitchfork bifurcation (for Bk < 
Bi). 
The region in the angular rotation corresponding 
to the turning point instability decreases as A 
increases, disappearing for A = V3;r/2, the point 
where the transition between C-mode and amphora 
mode occurs for B/=0. 
For A > V3;r / 2 the effect of lateral gravity does 
not change the character of the initial instability, as 
happened for the C-mode. Instead, a subcritical 
pitchfork bifurcation persists, as in the Bi =0 case. 
The bifurcation diagram has been sketched in Figure 
4. 
Fig. 5 shows the variation of the stability 
threshold for increasing lateral gravity. The dashed 
line marks the change of the primary instability from 
subcritical pitchfork bifurcation (region B in the 
figure) to toning point (region TP in the figure). 
(i) 
Figure 6. Effect of the lateral gravity and the slenderness 
on the stability threshold. 
Figure 7 and Figure 8 show the bifurcation 
diagrams obtained by representing the area of the 
section at z=# and z=A/2 respectively, as a function 
of W for A=2.6 and two values of lateral gravity 
marked in Figure 5: 5/=0.1, which corresponds to 
the turning point and Bj=0.25, which corresponds to 
the subcritical pitchfork bifurcation. The stable part 
of the branches is represented with a solid line, and 
the unstable part with a dashed line. 
Figure 9 and Figure 10 show the equilibrium 
shapes for the cases indicated in Figure 7 and Figure 
8, respectively. In Figure 10, equilibrium shapes for 
both the main branch and the bifurcated one are 
represented. As we can see in both figures, the 
stable shapes are non-axisymmetric modes (that is, 
C-modes). The behaviour of the equilibrium shapes 
for the two possibilities with A < 73^-/2 is as 
follows: 
For both the pitchfork bifurcation point 
with Bf=0 and the turning point with 
0<Bi<B!c the associated eigenfunction is 
non-axisymmetric (that is, a C-mode) so 
that the breaking process can be expected to 
lead to a non-axisymmetric configuration. 
For the subcritical pitchfork bifurcation 
point with B!c< B\ the associated 
eigenfunction is axisymmetric (that is, an 
amphora-mode) so that the breaking 
process can be expected to lead to a 
combined amphora and C-mode 
configuration. 
(ii) 
Figure 7. Bifurcation diagram for A = 2.6 and B, = 0.1 • 
3.8 
••)./ 
3.b 
3.4 
3.3 
3.2 
31 
3 
2.9 
»^. 
- VIb 
I 
•— 
s 
--N 
N 
x s 
"Vvb 
\ 
\ 
II III 
1 
* 
' I V 
Vim 
-
-
Vm 
w 
Figure 8. Bifurcation diagram for A = 2.6 and^ = 0.25 • 
Figure 11 shows the bifurcation diagram obtained 
by representing the area of the section at 2=-A/2 as a 
function of W for A=2.9 and B/=0.l (marked in 
Figure 5), which corresponds to the subcritical 
pitchfork bifurcation. The stable part of the branches 
is represented with a solid line, and the unstable part 
with a dashed line. 
Figure 12 shows the equilibrium shapes for the 
cases indicated in Figure 11. 
46 J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 
— 222 — 
Ill 
Figure 9. Evolution of the equilibrium shapes for A = 2.6 
along the main solution branch f o r ^ = 0 . 1 . Shapes I to 
IV are stable while V and VI are unstable.. 
IV Vb VIb 
bifurcated brach 
Vm 
main brach 
Vim 
Figure 10. Evolution of the equilibrium shapes of a 
cylindrical liquid bridge with A=2.6 along the main branch 
and the bifurcated one for_g = 0.25 • Shapes I to IV are 
stable and the rest are unstable. 
The results obtained here for liquid bridges in 
isorotation in a lateral gravity field can be compared 
with those obtained for liquid bridges rotating 
around an eccentric axis6) '7), as both effects 
correspond to lateral forces. The comparison shows 
that, although the stability behaviour is similar, 
lateral gravity has a stronger effect for long liquid 
bridges ( A < -^bnll )• For example, liquid bridges in 
isorotation with A > 2.7 cannot support lateral 
gravity w i t h ^ > 0.25, that is, they are unstable for 
any value of B > 0.25, while for non-lateral gravity 
the same liquid bridges can support eccentric 
rotation for any value of the eccentricity, although 
the limit value of the rotation rate decreases as the 
distance between the axis of the disk and the axis of 
rotation increases. The explanation for this is that the 
lateral force induced by eccentric rotation is 
proportional to the rotation rate, while the force of 
the lateral gravitational field is independent of the 
rotation rate. 
Figure 11. Bifurcation diagram for A = 2.9 and 
5 ; = 0 . 1 . 
IV V VI 
Figure 12. Evolution of the equilibrium shapes of a 
cylindrical liquid bridge with A=2.9 along the bifurcated 
branch for B, = 0.1 • Shapes I to IV are stable and the rest 
are unstable 
5. Conclusions 
The stability limits of liquid bridges in a lateral 
gravitational field rotating around an eccentric axis 
have been calculated numerically. The numerical 
method is used to find stable and unstable shapes 
and to determine the dependence of the stability 
threshold on the slendemess and the strength of the 
lateral gravity. 
• Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 223 47 
The effect of lateral gravity is to drastically reduce 
the stability region. Gravity affects short liquid 
bridges more, although for increasing values of the 
gravitational field the stability region is also strongly 
reduced for long liquid bridges. 
Acknowledments 
This work has been sponsored by the Ministerio 
de Education y Ciencia (MEC) and is part of a more 
general endeavor for the study of fluid physics in 
processing under microgravity (Proyect No. MEC 
ESP2007-65221). The authors want to thank Jeff 
Porter for his help. 
References 
1) Perales, J. M., Sanz, A. and Rivas D., Appl. 
Microgravity Tech. II, 4, 193, 1990. 
2) Vega, J. M. and Perales, J. M., ESA SP-191, ESA, 
Paris, 247, 1983. 
3) Laveron-Simavilla; A., Perales, J. M., Phys. 
Fluids, 7, 1204, 1995. 
4) Laveron-Simavilla; A., Checa, E., Phys. Fluids, 
9, 817,1997. 
5) Keller, H. B., "Lectures on Numerical Methods in 
Bifurcation Problems", Springer-Verlag, Berlin, 
1987. 
6) Laveron-Simavilla, A., Lapuerta, V., 
Rodriguez, J. and Gonzalez, M. A., Fluid 
Dynamics and Materials Processing, III, 1, 2008. 
7) Lapuerta, V., Laveron-Simavilla, A., Rodriguez, J. 
and Gonzalez, M. A., Advance in Space Research, 
(in press). 
48 J. Jpn. Soc. Microgravity Appl. Vol. 25 No. 3 2008 
— 224 


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Effect of rotation on the equilibrium shapes and stability of liquid bridges in a lateral gravitational field

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