ONSET OF INSTABILITY IN LOW PRANDTL NUMBER LIQUID BRIDGE WITH DEFORMABLE FREE SURFACE
8.1 INTRODUCTION TO PART 8
Liquid bridge with cylindrical nondeformable free surface, considered up to now, is just a simpli- fied model since it is only under zero gravity and when volume of the liquid bridge is V
0= πR
2d the free surface is absolutely cylindrical. Is either V
= V
0or g
= 0, the free surface will become deformed. Moreover, the hydrostatic pressure may be a reason of significant deflections of the liquid-gas interface. The free surface deformation is an additional factor noticeably influencing the onset of oscillatory regime in liquid bridge. One of the first experimental evidences of that the critical temperature difference depends strongly on the liquid volume was obtained by Cao
et al.[11], Hu
et al.[43] and Masud
et al.[72]. It was also discovered experimentally, that wave number m
cr(m
cris wave number at the onset of instability) changes with volume of the liquid bridge. For the case of high Prandtl numbers in terrestrial conditions the stability diagram (∆T
crvs. V olume) consists of two different oscillatory instability branches. It has been experimentally obtained by Shevtsova et al. [125] for the case of 10 cSt silicone oil and aspect ratio Γ = 4/3 that the two branches correspond to different azimuthal wave numbers. The branch on which ∆T
crgrows with increasing volume belongs to m
cr= 1, and the descending branch belongs to the azimuthal wave number m
cr= 2 (see Fig. 8.1).
In some limits the empirical relation m
cr ≈2.0/Γ remains valid for non-cylindrical liquid bridges in zero gravity condition. The radius of a liquid bridge changes with the height but for zero-g condition the free surface shape is symmetrical with respect to mid-plane. For the small Prandtl numbers, P r = 0.01, to take into account the shape effect, Lappa
et al.[62]
suggested the modified empirical relation m
cr ≈2.0/ Γ, ˜ where the radius of the liquid bridge at the mid-plane ˜ Γ = h
(z=d/2)/d is used. It corresponds to the minimal radius of the zero-g configuration. Nienh¨ user
et al.[82] have recently confirmed this formula by linear stability analysis and extended it up to P r = 4.
With respect to the stability of deformable liquid bridges, the theoretical developments are
not as advanced as the experimental results. The first calculations on steady thermocapillary
convection in a floating zone with a deformable free surface were done by Kozhoukharova and
Slavchev [53] for small deformations. A few 2D numerical simulations of thermocapillary flow
in liquid bridges with strong deformation of the free surface have been performed (Shevtsova et
al. [117],[114], Sumner et al. [138]). Tang
et al.[139] have studied numerically the influence of
gravitational vibrations ( g-jitter ) on thermocapillary convection in an axisymmetric half-zone
with deformation of the free surface. For the floating-zone, Lappa [59] made a numerical study
20 40 60 80
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
V /V 0
DTcr
m = 1
m = 2
Figure 8.1: Experimental dependence of the critical temperature difference ∆T
crupon liquid bridge volume, obtained by Mojahed and Shevtsova for Γ = 1.2, P r = 105, Bo
dyn= 2.3.
The branches have different azimuthal wave numbers, m = 1 and m = 2.
of influence of liquid volume and gravity on the flow instability.
8.2 PROBLEM DESCRIPTION
Thermocapillary flow of an incompressible Newtonian liquid confined by parallel circular disks (liquid bridge configuration) is considered. The coaxial rigid disks with an equal radii r = R
0are separated by a distance d, so the aspect ratio is Γ = d/R
0. The lateral free surface is bounded by passive gas and it is assumed to be a function of the vertical co–ordinate, r = h(z). The geometry of the problem is shown in Fig. 8.2. The temperatures T
hotand T
cold(T
hot> T
cold) are prescribed at the upper and lower solid-liquid interfaces respectively, yielding a temperature difference ∆T = T
hot−T
cold.
The surface tension, acting on the free surface, is a linearly decreasing function of the tem- perature
σ(T ) = σ
0−σ
T(T
−T
0),
here σ
T=
−(dσ/dT )
|Tcold. The temperature of the cold disk is used as the reference, T
0= T
cold. The study for the deformed liquid bridge is carrying out for constant viscosity.
The fluid motion is caused by the thermocapillary effect only. The governing Navier–Stokes, energy and continuity equations are written in non–dimensional primitive–variable formulation in cylindrical co-ordinate system.
∂V
∂t +
V· ∇V=
−∇P +
V,(8.1)
∂Θ
∂t +
V· ∇Θ = −Vz+ 1
P r
· Θ,(8.2)
∇ ·V
= 0, (8.3)
where velocity is defined as
V= (V
r, V
ϕ, V
z), Θ
0= (T
−T
cold)/∆T is the dimensionless tem- perature and Θ is the deviation from the linear temperature profile Θ = Θ
0−z. The scales for time, velocity and pressure are V
ch= ν/d, t
ch= d
2/ν and P
ch= ρ
0V
ch2. Two characteristic scales are used for the length, d and R
0for the vertical and the radial coordinate, respectively.
Thus,
∇= [Γ∂
r, (Γ/r)∂
ϕ, ∂
z].
Figure 8.2: Deformed liquid bridge.
At the rigid walls no slip conditions are used
V(r, ϕ, z= 0, t) = 0,
V(r, ϕ, z= 1, t) = 0 and constant temperatures are imposed Θ(r, ϕ, z = 0, t) = 0, Θ(r, ϕ, z = 1, t) = 0.
The stress balance between the viscous fluid and inviscid gas on the free surface, r = h(z), is given by
[P
−P
0+ σ(∇ ·
n) ]ni= ˆ σ
iknk+ ∂σ
∂
τi, (8.4)
where P and P
0, are the pressure inside the liquid and the ambient gas pressure, σ(∇ ·
n) is theLaplace pressure and ˆ σ
ik= µ(∂V
i/∂x
k+ ∂V
k/∂x
i) is the ik-th element of the tensor of viscous stress.
The tangential projections of eq.(8.4) define the driving thermocapillary force
τz·
σ ˆ
·n+ ∂σ/∂
τz= 0, (8.5)
τϕ·
σ ˆ
·n+ ∂σ/∂
τϕ= 0, (8.6)
where
τzand
τϕare the unit tangential vectors to the free surface in r
−z and r
−ϕ planes, respectively. And
nis the unit normal vector directed out of liquid into the ambient gas. The location of the interface is determined by the normal projection.
∆P =
n·σ ˆ
·n−1
C
∇ ·n,(8.7)
The mean curvature is determined as
∇ ·n
= ( 1 R
1+ 1
R
2)
where R
1and R
2are the radii, corresponding to the principal curvatures of the interface. Here
∆P = P
−P
0is the pressure difference. Quite often in literature the ratio of viscous force per unit area to the typical capillary pressure is named as capillary number.
C = ρν
2/σ
0d = Γ (ρν
2/d
2)/(σ
0/R).
The kinematic condition gives
n·V
= dh/dt (8.8)
and the free surface is assumed to be thermally insulated
n· ∇T
= 0 (8.9)
The complete moving boundary problem, described by eqs.(8.1-8.9), except mentioned above parameters Γ and C includes
P r = ν
0k , Re = σ
T∆T d
ρ
0ν
02, Ca = σ
T∆T
σ
0 →M a = Re P r, Ca = C
·Re. (8.10) Prandtl, Reynolds and another Capillary number, Ca, which measures the relative impor- tance of dynamic pressure to the capillary pressure Ca = Γ
−1(σ
T∆T /d)/(σ
0/R).
8.3 BASIC ASSUMPTIONS
The whole problem can be solved by perturbation solution in the asymptotic limit of small capillary number, C
→0 or Ca
→0, which corresponds to the case of small surface tension variation as compared to the mean surface tension. For most of the experimental liquids C
1 and Ca < 1. Usually when studying a liquid bridge it is assumed that Ca and C have the same order of magnitude, i.e. O(Ca)
≈O(C), see [55, 82].
Here the capillary number C is used as a parameter of expansion. In the limit C
→0, the Laplace pressure
∇ ·ncan only be balanced by the jump between the fluid in the liquid bridge and the ambient gas. The pressure jump is asymptotically large,
∼1/C or
∼1/Ca, see [56]. In this limit, C
→0, the following expansions are used to find the leading order contributions to the flow and temperature fields (see [114]):
f (r, z) = f
0(r, z) + C f
1(r, z) + O(C
2), P = C
−1P
st+ P
0+ C P
1+ O(C
2) h = h
0(z) + C h
1(r, z) + O(C
2) here f (r, z, t) reads for the components of vector
Vand Θ.
The solution in the lowest order of expansion, e.g. O(C
−1) determine the free surface shape.
The position of the free surface, r = h(z) is independent on flow. Taking into account that the projections of the unit vectors are
n = (1 + Γ
−2h
2z)
−1/2[1, 0,
−h
z/Γ] , τ
z= (1 + Γ
−2h
2z)
−1/2[h
z/Γ, 0, 1] , τ
ϕ= [0, 1, 0] ,
where the subscript z denotes the derivative, h
z= dh/dz, here h is scaled by R
0.The mean curvature as dimensionless quantity is defined as
1 R
1+ 1
R
2= 1 (1 + h
2z)
1/2
h
zz(1 + h
2z)
−1 h
. (8.11)
The governing equations are simplified to the equations, which define the shape of the static meniscus.
∂
rP ˜
st= 0, ∂
zP ˜
st= 0,
∆ ˜ P
st−Bo
stz = (1 + Γ
−2h
2z)
−1/21/h
−h
zz(1 + Γ
−2h
2z)
−1. (8.12) By definition the static Bond number is the ratio of hydrostatic pressure to the capillary one, Bo ˜
st= (ρgH )/(σ/R
0), where g is the gravity level. Here below the static Bond number is defined as Bo
st= (ρgH
2)/σ = ˜ Bo
stΓ
−1. The constant value of the static pressure ∆ ˜ P
st= ∆P
st+ρgH/σ should be obtained as part of the solution. Remember, that length scales in radial and axial directions are different. To solve the second order eq.(8.12) a conservation of liquid volume and few boundary conditions could be applied (see [114, 56]):
1) Prescribed fluid volume V =
01h
20(z)dz, which is scaled by V
0= πR
2d 2) fixed contact line h
0(0) = h
0(1) = 1
3) fixed contact angle either on the hot wall or at cold wall: Γ
−1h
z(0) =
−cotα
cor Γ
−1h
z(1) = cotα
h.
Here the angles α
cand α
hare counted from rigid disks up to the free surface of the liquid.
Any two of these conditions may be used for the solution of eq.(8.12) and the third one will be used to determine ∆P
stas eigenvalue of the problem.
In the next order of expansion, O(C
0), the flow field problem should be solved with fixed boundary, h(z). At this approximation the shape of the governing equations eq.(8.1-8.3) does not change. Let us write the boundary conditions at the free surface. Equation(8.5) for the tangential balances
τz·
σ ˆ
·n+ ∂σ/∂
τz= 0, becomes 1
(1 + Γ
−2h
2z)
1/2(1
−Γ
−2h
2z)
Γ ∂V
z∂r + ∂V
r∂z
+ 2h
zΓ
Γ ∂V
r∂r
−∂V
z∂z
=
−
Re( h
zΓ
∂Θ
∂r + ∂Θ
∂z + 1). (8.13)
And eq.(8.6)
τϕ·
σ ˆ
·n+ ∂σ/∂
τϕ= 0, becomes 1
(1 + Γ
−2h
2z)
1/2Γ ∂V
ϕ∂r
−Γ V
ϕr + Γ
r
∂V
r∂ϕ
−h
zΓ
∂V
ϕ∂z + Γ r
∂V
z∂ϕ
=
−Re Γ r
∂Θ
∂ϕ . (8.14) As the static shape of the liquid bridge does not change with time, the kinematic condition leads to
n·V
= dh/dt = V
r+ h
zV
z= 0. (8.15)
The free surface is assumed thermally insulated
∂
nΘ(r = 1, ϕ, z, t) = 0. (8.16)
8.4 SOLUTION METHOD
To solve the problem with a fixed boundary, body-fitted curvilinear coordinates are adopted, e.g.
see [114, 117]. The transformation thus converts a nonrectangular grid in the physical domain in the (r, z)–plane to a rectangular one in the computational domain in the (ξ, η) plane (see Fig.
2 in [114]) and greatly facilitates the process of discretization and computer solution:
ξ = r/h(z), η = z
→∂
∂r = 1 h
∂
∂ξ , ∂
∂z = ∂
∂η
−ξ h
h
∂
∂ξ
Here and below h
= dh/dξ and h
= d
2h/d
2ξ. Thus the radial coordinate r is varied from ξ = 0 at the axis, to ξ = 1 at the free surface. The axial coordinate does not change, it varies from η = 0.0 at the cold disk up to η = 1.0 at the hot disk.
In the recent publications the majority of the studies in curved liquid bridge utilize the same coordinate transformation, but there are some divergence in the final form of equations between different papers. Therefore we would like to write down the system of equation in the transformed coordinate system.
Transformation to body-fitted coordinate system (curvilinear coordinates (ξ, η)) of the gov- erning Navier-Stokes, energy and continuity equations equations in non-dimensional primitive- variable will give:
∂V
ξ∂t + Γ hξ
∂
∂ξ (ξV
ξ2) + Γ hξ
∂
∂ϕ (V
ξV
ϕ) + ∂
∂η (V
ξV
η)
−h
h ξ ∂
∂ξ (V
ξV
η) =
−
Γ h
∂p
∂ξ + ∆V
ξ−Γ
2h
2V
rξ
2 −2 Γ
2(hξ)
2∂V
ϕ∂ϕ + Γ h
V
ϕ2ξ , (8.17)
∂V
ϕ∂t + Γ hξ
∂
∂ξ (ξV
ξV
ϕ) + Γ hξ
∂
∂ϕ (V
ϕ2) + ∂
∂η (V
ϕV
η)
−h
h ξ ∂
∂ξ (V
ϕV
η) =
−
Γ hξ
∂p
∂ϕ + ∆V
ϕ−Γ
2h
2V
ϕξ
2+ 2 Γ
2(hξ)
2∂V
ξ∂ϕ
−Γ h
V
ξV
ϕξ , (8.18)
∂V
η∂t + Γ hξ
∂
∂ξ (ξV
ξV
η) + Γ hξ
∂
∂ϕ (V
ϕV
η) + ∂
∂η (V
η2)
−h
h ξ ∂
∂ξ (V
η2) =
−
∂p
∂η + ∆V
η, (8.19)
Γ hξ
∂
∂ξ (ξV
ξ) + Γ hξ
∂
∂ϕ (V
ϕ) + ∂
∂η (V
η)
−h
h ξ ∂
∂ξ (V
η) = 0, (8.20)
∂Θ
∂t + Γ hξ
∂
∂ξ (ξV
ξΘ) + Γ r
∂
∂ϕ (V
ϕΘ) + ∂
∂η (V
ηΘ) + V
η= 1
P r ∆Θ, (8.21)
∆f = Γ
2
1
h
2ξ
∂
∂ξ (ξ ∂f
∂ξ ) + 1 (hξ)
2∂
2f
∂ϕ
2+ ∂
2f
∂η
2+
h
h ξ
2∂
2f
∂ξ
2+
2 h
h
2ξ
−h
h
2ξ
∂f
∂ξ
−2 h
h ξ ∂
2f
∂ξ∂η (8.22)
where V
ξ, V
ϕand V
ηare radial, azimuthal and axial velocities.
Figure 8.3: Dependence of (a) relative volume upon the contact angle near the hot disk and of (b) the pressure jump upon the relative volume.
The boundary conditions at the free surface ξ = 1, eqs.(8.13-8.5), in the body-fitted coordi- nates are
N
2Γ h
∂V
η∂ξ + N
2h
h
∂V
ξ∂ξ + (2
−N
2) ∂V
ξ∂η
−2h
Γ
∂V
η∂η =
−ReN∂Θ
∂η + 1
, (8.23)
N
2∂V
ϕ∂ξ
−V
ϕ+ ∂V
ξ∂ϕ
−h
Γ
2
h ∂V
ϕ∂z + Γ ∂V
z∂ϕ
=
−ReN∂Θ
∂ϕ , (8.24)
V
ξ=
−h
V
η, (8.25) where N is defined as N = (1 + Γ
−2h
2)
1/2.
At the rigid walls no slip, no penetration conditions are used and a constant temperature is imposed:
on the cold disk: V (r, ϕ, z = 0, t) = 0, Θ(r, ϕ, z = 0, t) = 0, (8.26) on the hot disk: V (r, ϕ, z = 1, t) = 0, Θ(r, ϕ, z = 1, t) = 0. (8.27)
8.5 NUMERICAL ASPECTS
The second order boundary value problem, eq.(8.12), was solved by a combined Runge-Kutta and shooting method. The hot-wall contact angle, α
hhas been fixed and the ”shootings” were aimed to arrive at the bottom to the fixed contact point, ξ = 1/Γ. The dependence of relative volume, V , upon this contact angle is shown in Fig. 8.3, as usually experimental results are presented as a function of the volume.
As a first step in code validation for the small Prandtl number liquid bridge we used P r = 0.01 and two aspect ratios 1.0 and 1.2 liquid bridges. Calculations were performed for cylindrical free surface. The results are presented in Tables. 8.1,8.2.
For the unit aspect ratio, the obtained results are in good agreement with all the mentioned works except by Lappa et al. (2001). As for the Γ = 1.2 case, it is only the results by Nienh¨ user
& Kuhlmann that almost coincide with our calculations (discrepancy is only about 0.6%). A
reason for this may be that it is the aspect ratio that was somehow missing in the codes of the
other research groups.
Table 8.1: Critical Reynolds number for unit aspect ratio when P r = 0.01.
Cylindrical LB: Gr = 0, Bo = 0, α = 90, V olume = 1
Γ m
present result Wanschura et al. Levenstamm (1995) Chen et al. (1999) Lappa et al. (2001)3D LSA 3D LSA 3D
1.0 2 1900 1899 1960 1980 2120
Table 8.2: Critical Reynolds number for the aspect ratio Γ = 1.2 when P r = 0.01.
Cylindrical LB: Gr = 0, Bo = 0, α = 90, V olume = 1
Γ m
present result Nienh¨user&Kuhlmann Chen et al. (1999) Lappa et al. (2001)3D (2002) LSA LSA 3D
1.2 2 1760 1770 1550 2500
Table 8.3: Critical Reynolds number for deformed liquid bridge and low Prantdl numbers.
Gr = 0, R
ν= 0
P r ϕ Γ Volume m Re
crNienh¨ user & Kuhlmann
0.02 50 1 0.757 m=2 2554 2540 (m=1)
2380 (m=2)
0.01 62 1.2 0.8 m=2 2306 2200
0.01 70 1 0.882 m=2 1921
0.01 80 1 0.941 m=2 1844
0.01 90 1.2 1 m=2 1758 1770
Figure 8.4: Temperature field disturbances in mid-cross section and on the free surface for P r = 0.017, Γ = 1, Re = 3500, Bo = 0, α
h= 60
◦.
In Table 8.3 some results on the critical Reynolds number for different volumes, P r and aspect ratios are reported. Note, that there are no reliable results to compare with up to now as the deformable liquid bridge problem is rather complicated and it just recently came into focus of attention.
As one can see, there is only in one case of P r = 0.01, Γ = 1.2 and liquid volume 0.8 that a big discrepancy in the critical Reynolds number was obtained (about 5%). For the case of P r = 0.02 and liquid volume 0.757 two different wave numbers are reported by Nienh¨ user &
Kuhlmann. We observed only m = 2 solution (Fig. 8.4) and for this mode the critical Reynolds numbers almost coincide with Re
cr(m = 1), obtained by linear stability analysis performed by Nienh¨ user & Kuhlmann. In comparison to the case of large Prandtl number, the temperature spots are situated close to the free surface, and the spots on the free surface itself do not spread themselves from hot disk down to the cold one. As a proof that at Re = 3500 the observed instability is stationary, i.e., the spots stay in their positions, is given in Fig. 8.5. In this figure the temperature time-series in four different locations (r = 0.9, φ = (0, 0.25π, 0.5π, 0.75π), z = 0.5) are shown.
As it follows from eq.(), the radial velocity on the deformed liquid-gas interface is not zero
as it is in cylindrical case. Figures 8.6 and 8.7 present radial distributions of v
rand axial v
zvelocities on the free surface with z for P r = 0.017, Γ = 1, Re = 2500, Bo = 0, and different
contact angles α
hfor slender liquid bridge. As one can see, the less the contact angle α
h, the
more the absolute values of min(v
r) and max(v
r) are. As for the axial velocity on the free surface,
its maximal absolute value is not too much influenced by the contact angle, but the location of
the point where v
zis minimal shifts towards the hot wall as α
his decreasing (Fig. 8.7). As a
general tendency, one can see that with the decrease of α
hfrom 80
◦down to 60
◦, the v
zplot
2.655 2.668 2.681 2.694 2.708 2.721 dimensionless time
0.0236 0.0289 0.0342 0.0394 0.0447 0.0500
Temperature
