Part 7

EFFECT OF AMBIENT CONDITIONS NEAR THE INTERFACE ON FLOW INSTABILITY

7.1 INTRODUCTION TO PART 7

Before it was assumed absence of heat flux through the liquid-surface interface. However, due to natural reason, e.g., evaporation of the liquid [50], or because of intensionaly implied heat flux, e.g., cooling of the free surface by keeping the ambient temperature less then T

, the heat flux through the interface can not be neglected and thus theoretically may play a significant role in the mechanism of hydrothermal instability. Moreover, by isolating the liquid bridge by puting it under a cover one can change temperature distribution near the free surface due to convective motion in the ambient gas. It might be an additional factor influencing the critical parameters. Recent experiments by Kamotani et al. [49] revealed that the air motion around the liquid column, mainly caused by buoyancy due to the heating and cooling arrangement of the experiment, has a strong effect on the onset of oscillations. Velocity of the ambient gas they measured was very small

1cm/s, but according to the analysis performed by the authors such a weak airflow significantly stabilizes the liquid flow.

The question of the influence of the thermal conditions near the free surface of the liquid bridge on the onset of instability remains obscure. In the case of medium and high Prandtl numbers, the intensive heat loss through the free surface increases the temperature gradient along the liquid-gas interface in the vicinity of the hot corner. Consequently, near the cold end, where the driving force in case of large P r is more concentrated, the temperature gradient along the free surface lessens. If it is so, then the critical Reynolds number will be higher in case of the heat loss.

Wanschura

analysis for the full zone in the case of radial heating. It was shown that the critical Reynolds

number (M a

) reaches a minimum near Bi

18 and the type of instability depends on the Biot

number, when P r = 4, Γ = 1. Recently Schwabe

(1998) have experimentally investigated

heat transport at the free surface in a hollow floating zone. They obtained that ∆T

and

the frequency, both increase significantly when the free surface is cooled. The linear stability

analysis by Neitzel

(1993) reveals an increase of the ∆T

, when the Biot number is taken

into consideration.

7.2 EXPERIMENTAL SET-UP

Experiments were carried out by Dr. Mojahed and Dr. Shevtsova to understand the influence of the thermal conditions around the liquid bridge on the threshold of instability and on the flow patterns.

Figure 7.1: Experimental set-up used for study of thermocapillary convection in liquid bridge by Shevtsova and Mojahed. It corresponds to the series of experiments in shielded liquid bridge.

The scheme of the set-up they use is shown in Fig. 7.1. The detailed description of the experimental set-up one may find in earlier report by Shevtsova

(1999). Here one should mention a few important points for the present study. The test liquid is 10cSt silicone oil, P r = 108. The reasons for using such a high Prandtl number silicone oil are the reproducibility of the surface tension and their well-defined dependency upon temperature. The upper rod was fixed in such a way, that three–dimensional movements are possible. A heating element (Resistor Minco R

100 Ohms) was mounted around the upper rod to heat the fluid from above. The lower rod was kept at a constant temperature using thermoregulated water–cooling system. The rods meant to hold the liquid zone were made from Aluminum alloy (λ = 164W/mK ). The rods had the same diameter 2R = 6mm and a liquid zone had a length d = 3.6mm, as a result the aspect ratio was equal to Γ = d/R = 1.2. In tiny liquid bridges, the role of Marangoni convection is increased in comparison with a buoyancy effect.

To establish a floating zone the liquid was injected by a dedicated push syringe into a gap between rods. The push syringe allowed us to measure the volume of liquid with high accuracy.

This is important, as the onset of time dependence is sensitive to the fluid volume. To prevent

Table 7.1: Physical properties of the silicone oil 10 cSt

ν β σ k σ

= dσ/dT ρ

m

/s 1/K N/m m

/s N/mK kg/m

10

1.08

10

20.1

10

0.95

10

6.8

10

934.

liquid creeping over the edge of the lower rod, the lateral surface was coated with the anti-wetting FC723.

The temperature oscillations due to time-dependent convection were measured by insert- ing five shielded thermocouples (D = 0.25mm) inside the liquid at the same radial and axial positions and different azimuthal angles. These thermocouples were embedded into the liquid through the upper rod, to prevent the disturbance of the free surface. Unlike to general opinion, we did not observe a strong influence of the amount of thermocouples on the critical ∆T

. A few experiments were repeated with 2 and 5 thermocouples and the comparison of the results was carefully made. It should be mentioned that the free surface, which is responsible for the driving force, was not disturbed. The choice of 5 hermocouples was made to determine without ambiguity the critical wave number and the type of hydrothermal wave. One ought has an even amount of thermocouples, but 3 were not enough to determine clearly m = 2 mode. All temperature signals given by thermocouples were amplified and band-pass filtered before being recorded by a computer. The signals were recorded with a time interval of 0.1 s.

The experiments were carried out under two different thermal conditions in the ambient gas.

In the first case the measurements were done in the open air. This series of experiments will be referred to as non-shielded liquid bridge experiments. For another series of experiments the liquid bridge was placed in a cylindrical co-axial pipe of larger internal diameter 2R = 12mm (see Fig. 7.1). The double walls of the pipe were filled by flowing water, the temperature of which could be easily controlled. The thickness of the cylindrical layer of the water is about 3mm. In this way the temperature of the ambient gas around the free surface could be kept at a required value. The experiments under such conditions will be referred to as shielded liquid bridge experiments. The temperature of the water inside pipe and the temperature of the cold rod were kept equal throughout this study, although they were not connected. To protect the liquid bridge from the motion fluctuations in the lab air, the whole system was surrounded by a glass box of large volume for all the experiments. This glass box is a kind of semi-sphere with the diameter 330mm.

7.3 NUMERICAL METHOD

The mathematical model for 3D numerical simulation has been chosen as close as possible to the experimental conditions. The characteristic parameters of the problem correspond to the physical properties of the working liquid which are listed in Table 7.1. Both thermocapillary and buoyancy mechanism of convection are taken into account. Basically the geometry of the theo- retical model follows Fig. 7.1. Unlike to the experiment the free surface is assumed cylindrical and non-deformable.

We repeat here the problem difinition. It is defined that two differentially heated horizontal

flat concentric disks have radius R and they are separated by a distance d. The temperatures

T

and T

are prescribed at the upper and lower disks respectively, yielding a temperature difference of ∆T = T

T

. The temperature of the cold disk T

is used as the reference, T

= T

.

The density, the surface tension and the kinematic viscosity are taken as linear functions of a temperature

ρ = ρ

ρ

β(T

T

), where β =

ρ

∂ρ

∂T σ(T ) = σ(T

)

σ

(T

T

), σ

=

∂σ

∂T = const.

ν

(T ) = ν(T

) + ν

(T

T

), ν

= ∂ν

∂T = const.

All other material properties are regarded as constant. The linear dependence of the kine- matic viscosity upon temperature is widely used for theoretical and experimental studies and it is justified. For example, silicone oil 5cSt has the following temperature-dependence

ν = a10

, where a = 7.091117

10

m

/s b =

5.7259

10

1/

C.

As

is a small parameter this expression can be extended as

ν = a10

= a(1 + b ln10 (T

T

) + . . .) = a

1

1.318

10

(T

T

)

, which gives linear law.

Only the thermal boundary conditions at the free surface are changed for the present study.

Before the free surface of liquid bridge was assumed thermally insulated

∂

Θ(r = 1, ϕ, z, t) = 0, (7.1)

but in Chapter 7 it will not be always the case. The validity of his boundary condition will be discussed in next section.

7.4 MODELLING OF HEAT EXCHANGE ON THE FREE SURFACE Typically a numerical model has some limitations in comparison with experimental conditions.

For the liquid bridges one of the limitation can be recovered by correct modelling of the heat transport on the free surface. The majority of the theoretical studies assume zero heat flux on the free surface, defined by eq.( 7.1).

To fit this boundary condition the temperature distribution in the ambient gas should corre- spond to the surface temperature distribution of the liquid. But the type of surface temperature distribution depends upon the Prandtl number. A linear temperature profile along the free surface is observed for the small Prandtl numbers, e.g. P r

10

. For the medium Prandtl numbers, P r

1

7, the temperature profile is of an

-shape. For the large Prandtl numbers, (P r > 30) this

of the free surface with strong variations near the hot and cold walls. The temperature profile, obtained by 3-D calculations for experimental set of parameters, P r = 105, is shown in Fig. 7.2.

Probably, all of these cases cannot be described correctly in the frame of the same model.

On the boundary between liquid surface and gas Newton’s law determines the heat transfer

λ n

T =

h(T

T

), (7.2)

Figure 7.2: Temperature distribution along the free surface. 3D calculations when Pr=107, Re = 110, R

=

0.34969, Bi = 0.

where h is the heat transfer coefficient, λ

is the thermal diffusivity of the liquid, T

is the temperature of the surrounding gas.

Here the dimensionless temperature is chosen as T = T

+ (Θ + z)∆T . Then the Newton’s law can be written as

n

hL

λ [(Θ + z) + T

T

∆T ]

The heat transfer coefficient is called as Biot number Bi = hL

λ

,

here λ

is the thermal diffusivity of the liquid, L is a length scale.

For comparison with experiments two points should be defined: the value of Biot number and the ambient temperature profile. The ambient temperature is usually assumed either constant or having a linear profile. The temperature of the gas far away from the surface (Bi=1) have been used by Neitzel [79] in the linear stability analysis. The study reveals an increase of the critical temperature difference (e.g. Marangoni number M a

) when a non-zero Biot number is taken into consideration.

If T

is considered as linear temperature profile, e.g. T

= T

+

∆T the eq.( 7.2) will be written as

n

Θ =

Bi Θ. (7.3)

If T

is considered as a constant temperature, the Newton’s law will be written as

n

Θ =

Bi (Θ + z + const), (7.4)

where const = (T

T

)/∆T is equal zero when T

= T

. For the determination of the

Biot number value, the main task is to evaluate the coefficient of heat transfer h with a good

accuracy.

The often used model for the evaluation of the Biot number is following: the heat transfer is calculated from the solid hot plate with a constant temperature to the liquid or to the gas.

The values obtained such a way are used for the evaluation of heat transfer on the interface between liquid and gas, for example, from the surface of the falling liquid film to the gas and at numerous technical applications. With the same admissibility this model can be applicable to the liquid bridge.

Following this model the heat transfer from the gas side is described by the Nusselt number N u = Lh

λ

= f (Re

, P r

).

Considering the heat mass transfer in boundary layer (see Landau [58]) it appears that heat flux, e.g. Nusselt number, is proportional to the square root of the Reynolds number

N u = Re

F (P r

). (7.5) Only in this section the Reynolds number is determined in a classical way Re = LU

/ν

, following notations of Landau [58].

Then the heat transfer coefficient will be determined as h = λ

Re

F (P r)/L and Biot number Bi = λ

λ

Re

F (P r). (7.6) One should notice that the Biot number does not contain the length scale. The dependence upon Prandtl number remains undetermined.

For the high Pr numbers and the laminar boundary layer this function can be fitted as F (P r)

0.33 P r

, then Bi

0.33 λ

λ

Re

P r

. (7.7) This formula is widely used in technical and scientific applications even for moderate Prandtl numbers.

For the high Pr numbers and the turbulent boundary layer the dependence upon the Prandtl number can be written as F (P r) = const P r

For the liquids with Prandtl number close to one, 0.6

P r

15, Levich [66] has suggested the relation

F (P r) = 0.5 α

(P r), (7.8)

where values α(P r) are listed as the Table. For the present case of the air, P r = k

/ν

= 0.733, it gives α(P r) = 0.59. The velocity of the gas U

is considered to be equal to the velocity of the liquid on the free surface. The 3D calculations for Bi = 0 near the threshold of the instability, performed by the code described above, give the order of value U

1.0 cm/s.

For the air at the temperature 20

C the kinematic viscosity [58] is ν

= 0.15cm

/s, then Re

= 3.84. Taking into account that

λ

= 2.6

10

erg s

cm

K

λ

= 13.0

10

erg s

cm

K

Substituting these physical values into eq.( 7.5) and using eq.( 7.8) we will get for the present experimental conditions

Bi = λ

λ

Re

F (P r)

0.48

The value of Biot number could be 3 times smaller, Bi

0.117, if we would use eq.( 7.7).

The choice of the temperature distribution in the ambient gas will be discussed in detail later.

Figure 7.3: Surface temperature distribution for different Biot numbers when T

= T

. 3-D calculations for P r = 108, Γ = 1.2.

As it follows form Fig. 7.3, for the large Prandtl numbers, (P r > 30) the temperature of the liquid on the interface is almost constant in the central part of the free surface, T

T

+ 0.7∆T, with strong variations near the hot and cold walls, see solid line in Fig. 7.3. In this case the heat transfer on the interface is well described by the Newton law (the radiated heat is neglected)

λ

n

T =

h(T

T

).

Below the influence of various temperature profiles in the ambient gas will be analyzed. Let us rewrite the temperature boundary conditions on the free surface. In the case of a uniform temperature of the gas near the free surface (non-shielded LB), T

= T

, the eq.( 7.2) in dimensionless form will be written as

∂

Θ(r = 1, ϕ, z, t) =

Bi(Θ + z + ∆Θ

), (7.9) where ∆Θ

= (T

T

)/∆T. Herein, in accordance with the experiment, the calculations are performed when T

= T

= T

, thus ∆Θ

= 0.

If T

= T

= T

+ (z/d)∆T is a linear temperature profile (shielded LB) then

∂

Θ(r = 1, ϕ, z, t) =

Bi Θ. (7.10)

7.5 RESULTS

7.5.1 The Role of the Biot Number

To reveal the role of heat transfer through the interface, the transition from steady to oscillatory

flow has been investigated for different Biot numbers. In this paragraph, a linear temperature

profile is assumed in the ambient gas. The Prandtl number is P r = 108 at the reference temperature T

= T

= 22

C, and the magnitude of the parameter R

changes in accordance with the chosen temperature difference.

The first numerical results for the thermally insulated system, Bi = 0, reveal a transition from the 2D steady state to an oscillatory one at Re = 107

2. Slightly above the threshold the flow pattern is a traveling wave with azimuthal wave number m = 1. As a next step, the flow organization in the liquid bridge has been calculated using eq.( 7.10) and the realistic value of the Biot number, Bi = 0.5. Being rather small, the Biot number does not influence the critical Reynolds number. With further increase of the Biot number, the critical Reynolds number grows slightly. For example, for Bi = 5 it achieves Re

= 112. As shown in Fig. 7.3 the surface temperature diminishes near the cold side with the increase of Biot number. The variation of the surface temperature due to the heat transfer is weaker near the hot corner.

The numerical model deals with the cylindrical free surface, therefore comparison should be done with those experiments when the shape of the free surface was close to the cylindrical one. For the present experiments the best choice of dimensionless volume is [V ] = V /(πR

d)

0.9. The experimental value of critical temperature difference is about ∆T

(42.2

0.4)

C, while numerical result give ∆T

42.6

C (Re

107). This excellent agreement between the experimental and numerical results is due to the fact that the code is able to take into account the dependence of the viscosity upon temperature in the bulk. Indeed, for the same parameters but for constant viscosity, R

= 0, the critical Reynolds number was obtained equal to Re

145

5 which corresponds to ∆T

58.1

C. This is far above the experimental points.

Figure 7.4: Disturbance surface temperature distributions for different profiles in the ambi- ent gas T

= T

. 3D calculations for P r = 108, Γ = 1.2. Steady solution is subtracted.

7.5.2 The Role of the Temperature Distribution in the Ambient Gas

The described above numerical results for Bi = 0.5 have been obtained, when the temperature

distribution in the ambient gas has a linear profile. The surface temperature is not very sensitive

to the T

. To emphasize the difference, the deviation of the surface temperature from steady

linear profile is shown in Fig. 7.4.

What seems more important is the fact that the form of the periodical time signals Θ(t) depends on T

. In the case when T

is a linear profile, the temperature oscillations Θ(t), shown in Fig. 7.5(a), have almost sinusoidal shape. In the case of uniform T

, the oscillations are also periodic with a sustained amplitude, see Fig. 7.5(b), but their shape is distorted and the extremum (minimum) consists of two peaks. This type of time signal indicates the presence of the strong harmonics in the spectrum.

Figure 7.5: The dependence of temperature upon time. 3-D calculations when Pr=108, Re = 120, R

=

0.38148, Bi = 0.5 (a) T

= T

(b) T

= T

.

With increasing the Biot number the temperature profile in the ambient gas already plays a crucial role in the development of the instability. For example, for Bi = 5 in the case T

= T

the hydrothermal wave with m = 1 appears at Re

112 while for the uniform temperature distribution it appears at much higher Reynolds number, Re

170.

7.5.3 Modeling of the Shielding

One of the primarily goals of the numerical modeling is to understand the experimentally ob- served difference between the results for shielded and non-shielded liquid bridges. Definitely, the shielding changes the heat distribution in the gas around liquid bridge, and correspondingly the heat transport through the free surface. What is the principal difference? It appears that shielded and non-shielded experimental conditions in the liquid bridge correspond to different temperature distribution in the ambient gas.

Contrary to the physical intuition experiments with shielded liquid bridges satisfy the linear

temperature profile in the surrounding gas. It follows from the 3D calculations, that the free

surface has almost constant temperature T

T

+ 0.7

∆T, except some domains near the top

and bottom, see Fig. 7.3 for Bi = 0.5. The temperature of shielding is equal to the cold rod,

T

T

. Then the physical problem in the gas phase can be formulated as the following: a

volume of a gas is confined between two differently heated walls (hot interface and cold wall of the

shielding), see Fig. 7.6. For this configuration a buoyancy force will cause convection in the gas

phase, and the gas should move up along the hot side. But the liquid on the free surface is moving down. Thus one more extra vortex will appear in the gas phase near the liquid free surface. Due to the relatively small Grashof number in the gas phase, Gr = gβ

∆T (R

R

)

/ν

= 150, the motion of the gas will follow to the motion of the liquid near the free surface. The final flow organization is shown schematically in Fig. 7.6. These two vortexes in the gas phase will change the temperature near the free surface. Thus, the linear temperature profile will establish near the liquid surface. These assumptions have been verified by numerical modeling.

Figure 7.6: Sketch of the gas circulation in the case of shielding. Two convective vortexes appear in gas phase.

The experimentally recorded temperature oscillations in a shielded liquid bridge is very similar to those, coming from simulations in Fig. 7.5(a) for T

= T

. This signal is quite smooth, and the power spectrum exhibits only a few weak harmonics. An experiment in a non- shielded liquid bridge corresponds to the conditions, when the temperature of the ambient gas is uniform. Actually for the open system the buoyancy convection in a gas is not strong; the gas motion will be driven by the moving liquid adjacent to the interface and the temperature of the gas near the liquid will remain constant along the interface.

An experimental signal, repeatedly observed in non-shielded bridge, is shown in Fig. 7.7.

Unlike to the previous case the signal has two-peaks contour, indicating the presence of strong

harmonics in the power spectrum just near the threshold of the instability. The shape of this

signal corresponds well to the numerical one in Fig. 7.5(b). Their similarity confirms the correct

choice of the temperature distribution in the gas phase. Why the numerous harmonics appear

in the spectrum for these conditions? The temperature on the liquid free surface is large,

T

T

+ 0.7

∆T, while the temperature in the ambient gas is constant and much lower,

T

= T

+ δ, where δ << 0.7

∆T . Hence there is a kind of jump of the temperature through

the free surface, T

T

, which introduces the thermal perturbations in the system. With

increasing the Biot number these perturbations will cause some important modifications of the

flow organization.

Figure 7.7: Result of the experiment in non-shielded liquid bridge, temperature vs. time;

Γ = 1.2, V = 0.9.

7.6 RESULTS OF NUMERICAL SIMULATIONS

Analyzing the shapes of the interfaces for the different volumes having been investigated ex- perimentally under normal gravity, we have found that in the case of V /V

= 0.9 the shape of the liquid bridge approaches closer to right-circular cylinder than of any other smaller or larger volume. It allows us to make a comparison with 3D numerical simulations for the right–circular cylinder. Therefore the experimental results for the V /V

= 0.9 are shown in Fig. 7.8 as a function Re

versus P r instead of (∆T

vs. T

).

Remind, that for re-calculation of the experimental points via Re and Pr Re

= σ

∆T

d

ρ

ν

, P r = ν

k

the value of the viscosity at the temperature of the cold endwall is used, ν

= ν(T

). Usually the value of the viscosity is given in handbooks at room temperature (T

) or for some short range of temperatures. Therefore to make the plot the values of viscosity at T

have been calculated using the linear dependence

ν(T

) = ν(T

) + ν

(T

T

), ν

= ∂

ν = const. (7.11) The 3D numerical calculations have been done for the same dependence of viscosity upon temperature as in eq.( 7.11), which in present case corresponds to R

=

The calcu- lations have been performed for different Biot numbers using the linear temperature profile in the ambient gas, see eq.( 7.3).

The solid line in Fig. 7.8 corresponds to the linear fitting of the experimental points, which

are shown by small crosses. The dashed line displays the linear fitting of the numerical points

obtained for Bi = 0.5, which are shown by stars at P r = 100 and at P r = 107.83. Numerical

Figure 7.8: The dependence of the critical Reynolds number on the Prandtl number, V = 0.9, Γ = 1.2. The crosses correspond to the experimental points and solid line draws the linear interpolation of them; the stars and the dashed line correspond to the numerical results for Bi = 0.48; the circle indicates the numerical result for Bi = 5.0.

and experimental results are in excellent qualitative and quantitative agreement. The slopes of the linear fittings are equal, and the calculated values of Re

are only 5% higher than the experimental ones. This divergence can be explained by the assumptions about the shape of the interface. Note, that the key point for this agreement is that the numerical code takes into account dependence of the viscosity upon the temperature throughout in the bulk. To emphasize this, the calculations with a constant bulk viscosity, R

= 0, and ν

= ν(T

) have been done for P r = 107.83. The determined value of the critical Reynolds number, Re

= 140, exceeds the experimental results by 40%. The divergency is so large, that it is impossible to put this point in Fig. 7.8.

The numerical values of the Re

nearly coincide for different thermal conditions: in the absence of the heat transfer through the interface Bi = 0 and for the value Bi = 0.48 estimated in section 7.4. Although one may find that the Re

for Bi = 0.48 is a little bit higher than for Bi = 0. But in the scale of Fig. 7.8 they are not distinguishable. To clarify the role of the Biot number the simulations have been done with intentionally magnified value of the Biot number, Bi = 5.0. The effect of Biot number on the Re

is much weaker, then variability of the viscosity.

The numerical value for this case, Bi = 5.0, Re

= 110 is shown by circle in Fig. 7.8. The

presence of the Biot number tends to stabilize the thermocapillary flow, although the difference

is not significant, it is only about 3%, e.g. Re

= 107 for Bi = 0 versus Re

= 110 for Bi = 5.0

Note, that the way of the presentation of the experimental results is also important. For

instance, for the same experimental results V = 0.9 the value of the slope ∂Re

/∂P r is almost

twice larger than ∂∆T

/∂T

, e.g. 0.770 versus 0.363.

7.7 CONCLUSIONS TO PART 7

The development of the oscillatory flows is investigated experimentally and numerically in a liquid zone formed by 10 cSt silicone oil. The experimental Biot number, Bi=0.5, practically does not influence the critical temperature difference. The critical parameters for the onset of the instability, coming from the calculations and the experiments, are in excellent agreement.

Different temperature distributions in the ambient gas correspond to the various experimental

conditions (shielded and non-shielded liquid bridges). Parametric study reveals that for the same

surrounding conditions an increased Biot number stabilizes the flow. Temperature distributions

in the ambient gas strongly influence the critical temperature difference for the larger Biot

numbers, e.g. Bi = 5.