Part 4

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Part 4

INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON ONSET OF INSTABILITY IN LIQUID BRIDGE

4.1

INTRODUCTION TO PART 4

To date, the influence of secondary factors, approaching the theoretical model to a more realistic one can be investigated. For example, they may include static [116, 15] or dynamic [114] surface deformations, variable viscosity, etc. The viscosities of liquids, which are often used in liquid bridge experiments, e.g. silicone oils, vary by almost twice within the experimental range of temperatures. Convection in fluids with variable viscosity has received considerable attention because of its importance in geodynamics processes. The thermo-chemically driven convection in the Earth’s mantel reaches a contrast of viscosity of the order of 10⁵. The desire to understand the phenomenon led to numerous studies in fluids layers with a temperature-dependent viscosity [111, 146]. It was obtained in the early works by Liang [69] and Busse and Frick [8], that for a fluid layer with temperature-dependent viscosity in the case of buoyancy induced convection, the critical Rayleigh number is lower than for a constant-viscosity fluid. The similar tendency has been recognized by linear stability analysis by Kozhouharova et al. [52] for Marangoni convection in liquid bridges.

In Part 4, nonlinear evolution of the flow is studied by direct 3D calculations as the Reynolds number is increased. The dependence of the liquid viscosity upon the temperature is taken into account and the related problem of the reference temperature is discussed in detail. Part 4 presents new results for the onset of instability with viscosity contrast up to 10. The simulation of oscillatory flow above the stability limit is performed to investigate the wave properties of the flow and the influence of the initial disturbances on flow organization. Non-linear interactions of the hydrothermal waves are compared for constant and variable viscosity. One of the first studies of the spatial-temporal behavior of oscillatory convection for the high Prandtl number, P r= 35, is presented, so it can be compared with the experimental results by Muehlner et al [77].

4.2

MATHEMATICAL DESCRIPTION OF THE PROBLEM

The governing Navier-Stokes, energy and continuity equations are written in non-dimensional primitive-variable formulation in a cylindrical co-ordinate system (Fig. 3.2). The kinematic

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4.2 Inﬂuence of temperature-dependent viscosity

viscosity is taken as linear function of temperature

ν(T) =ν(T₀) +νT(T −T₀), νT = ∂ν

∂T =const.

∂V_r

∂t + Γ r

∂

∂r(rV_r²) +Γ r

∂

∂ϕ(V_rV_ϕ) + ∂

∂z(V_rV_z) =−Γ∂p

∂r + +ν

∆V_r−Γ²V_r r² −2Γ²

r²

∂V_ϕ

∂ϕ

+ ΓV_ϕ²

r +A_r, (4.1)

∂Vϕ

∂t +Γ r

∂

∂r(rV_rV_ϕ) +Γ r

∂

∂ϕ(V_ϕ²) + ∂

∂z(V_ϕV_z) =−Γ1 r

∂p

∂ϕ + +ν

∆Vϕ−Γ²Vϕ

r² + 2Γ² r²

∂Vr

∂ϕ

−ΓVrVϕ

r +Aϕ, (4.2)

∂V_z

∂t +Γ r

∂

∂r(rVrVz) +Γ r

∂

∂ϕ(VϕVz) + ∂

∂z(V_z²) =−∂p

∂z +ν∆Vz+Gr(Θ +z) +Az, (4.3) Γ

r

∂

∂r(rV_r) +Γ r

∂

∂ϕ(V_ϕ) + ∂

∂z(V_z) = 0, (4.4)

∂Θ

∂t +Γ r

∂

∂r(rV_rΘ) + Γ r

∂

∂ϕ(V_ϕΘ) + ∂

∂z(V_zΘ) +V_z= 1

P r∆Θ, (4.5)

∆f = Γ² 1

r

∂

∂r(r∂f

∂r) + 1 r²

∂²f

∂ϕ²

+∂²f

∂z²

As viscosity is not constant the Navier-Stokes equations ( 4.1) - ( 4.5) include some additional terms:

Ar, Aϕ and Az denote:

A_r = 2Γ²∂ν

∂r

∂V_r

∂r +Γ² r

∂ν

∂ϕ 1

r

∂V_r

∂ϕ +∂V_ϕ

∂r −V_ϕ r

+∂ν

∂z ∂V_r

∂z + Γ∂V_z

∂r

, A_ϕ = Γ²∂ν

∂r 1

r

∂V_r

∂ϕ +∂V_ϕ

∂r −V_ϕ r

+2Γ²

r²

∂ν

∂ϕ ∂V_ϕ

∂ϕ +V_r

+ ∂ν

∂z ∂V_ϕ

∂z + Γ1 r

∂V_z

∂ϕ

, A_z = Γ∂ν

∂r ∂V_r

∂z + Γ∂V_z

∂r

+Γ r

∂ν

∂ϕ ∂V_ϕ

∂z + Γ1 r

∂V_z

∂ϕ

+ 2∂ν

∂z

∂V_z

∂z

whereVr, Vϕ andVz are radial, azimuthal and axial velocities. The non-dimensional form results from scaling the cylindrical co-ordinates (r, z) by radius R and height d accordingly, and the velocity by Vch = ν₀/d. The temperature of the cold disk T₀ = Tcold is used as reference, Θ₀= (T−T₀)/∆T is the dimensionless temperature with respect to the cold wall and Θ is the deviation from the linear temperature proﬁle Θ = Θ₀−z. The scales for time and pressure are tch=d²/ν₀ andPch=ρ₀Vch2, respectively (ρ₀ is density, ν₀ is the kinematic viscosity ).

The following dimensionless parameters arise in the equations: the Prandtl number, defined byP r=ν₀/k(k is the thermal diffusivity), the Grashof number, defined byGr=gβ∆T d³/ν₀² (g is the gravity acceleration), the Reynolds number, defined as Re = σT∆T d/ρ₀ν₀² and the aspect ratio, defined by Γ = d/R. Except this typical set of parameters for the problems of convection an additional parameter appears due to variable viscosity. A new parameter R_ν defines the relative variation of viscosity.

R_ν = ν(T_hot)−ν₀

ν₀ = ν_T∆T

ν₀ , ν = 1 +R_ν(Θ +z) (4.6)

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Here,ν is dimensionless viscosity scaled byν(T₀).

The equations eq.( 4.1) - ( 4.5) have to be solved together with the following boundary conditions. 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, (4.7) on the hot disk: V(r, ϕ, z = 1, t) = 0, Θ(r, ϕ, z = 1, t) = 0. (4.8) On the cylindrical free surface (r= 1, 0≤ϕ≤2π, 0≤z≤1), the stress balances are:

∂V_ϕ

∂r − V_ϕ

r =−Re ν

1 r

∂Θ

∂ϕ, (4.9)

Γ∂V_z

∂r =−Re ν

∂Θ

∂z + 1

, (4.10)

and V_r= 0 (4.11)

where the corresponding Reynolds number is deﬁned asRe=σT∆T d/ρ₀ν₀². The free surface is assumed thermally insulated

∂_rΘ(r= 1, ϕ, z, t) = 0. (4.12)

4.3

REMARKS ON THE PROCESSING OF RESULTS

If the temperature diﬀerence between the disks is small, the ﬂow is axisymmetric and steady.

Beginning at some Re < Re_cr, the ﬂow ﬁeld and temperature start to exhibit time dependent oscillatory behavior. For the Reynolds numbers less than Recr these oscillations decay with time. The value of Re, at which oscillations of Θ(t) ( or V(t)) are sustained is referred to as the critical Reynolds number,Re=Re_cr. To record the oscillations, four equidistant numerical thermocouples were placed inside a liquid bridge at a transversal section.

For a better understanding of the 3D time-dependent flow structure, its axisymmetric component is subtracted from the resulting flow field and the remaining disturbance flow is analyzed, i.e. for visualizing a field q(r, ϕ, z) its average in the azimuthal direction ₂¹_π₀²^πq(r, ϕ, z)dϕ is subtracted. By doing this, the axisymmetric terms in the trigonometric series with zero wave-number and non-linear self interactions are excluded from the resulting fields. See further eqs.(( 4.13)-( 4.16)). The resulting velocity and temperature fields can be described as pure standing or traveling waves with a small azimuthal wave number, e.g. m = 1,2,3.., or mixed mode at the threshold of instability. Unlike the standing wave, generated by two counter propagating azimuthal waves with equal amplitudes, the traveling wave is also the result of counter propagating azimuthal waves but with different amplitudes.

The azimuthal wave number corresponds to the temperature field structure: it is organized in such a manner, that after the axisymmetric component is subtracted there are m hot and m cold spots observed in a transversal section and on the free surface. The perturbation flow, being dependent on a temperature field, demonstrates vortex structure, the number of vortex cells in a transversal section is equal to 2m, i.e. the sum of hot and cold temperature spots.

Moving further to the supercritical area Re > Re_cr higher modes may be excited.

For each calculation, the 2D steady state solution obtained for the same set of parameters is an initial guess. To initiate the oscillations in the system a perturbation of the temperature ﬁeld is added to the stationary 2D ﬂow. Experience has shown that to obtain periodic sustained oscillations with a constant (steady) amplitude, one should wait as long as thermal equilibrium

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Table 4.1: Results of the tests on the convergence on grid forP r= 1, Γ = 1.

grid (nr×nϕ×nz) Recr mode ω= 2π/T¯

25 ×16 ×21 2565 2 64.14

25 ×32 ×21 2545 2 64.49

49 ×32 ×41 2540 2 64.51

is established (τ_th = d²/k). Scaled time by viscosity (t_ch = d²/ν₀), one should perform the calculations for at least as long as aτ_th≈P r·t_ch, or in dimensionless units, up tot≈P r. This time is called thermal time anywhere below.

4.4

CODE VALIDATION

The program is validated by comparison of the results of calculations for a ﬂuid with constant viscosity (Rν = 0) with the previously published results. For P r = 1,3 and 4 the results of calculations are compared with the linear stability analysis by Wanschura et al. [152]. Calcu- lations for P r = 30 and diﬀerent aspect ratios are compared with results published by Savino and Monti [103], Castagnolo and Carotenuto [13]. The more detailed description of the code validation is given in [119].

4.4.1 Medium Prandtl number liquids with constant viscosity

The purpose of the calculations carried out for P r = 1,3 and 4 and Γ = 1 was to investigate convergence on a grid and to deﬁne the critical Reynolds numbers, oscillation frequencies and azimuthal wave numbers. The results of the calculations are listed in Tables 4.1 and 4.2.

Convergence on the grid with respect to the frequency and a critical Reynolds number for a liquid with P r = 1 is shown in Table 4.1. The ﬁrst column represents the grid resolution: the numbers of points respectively in radial, azimuthal and axial directions. The critical Reynolds number decreases slightly with increasing grid resolution, exhibiting good convergence on the grid. Indeed, forP r= 1, by increasing the amount of points in the azimuthal direction from 16 to 32, the critical Reynolds number is shifted from 2565 to 2545, i.e. only about 0.8%. Increasing the amount of points by 4 times (49×32×41) shifts the value ofRe_cr from 2545 to 2540, about 0.2%. The frequency near the threshold of instability is practically constant while the amount of computational points in radial and axial direction is increased by 4 times. See, for example, ω = 64.49 for (25×32×21), Re = 2545 versus ω = 64.51 for (49×32×41), Re = 2540. The frequency is slightly varied with increasing the amount of azimuthal points.

The quantitative comparison of Recr and the frequency ωcr at the threshold of instability with results by Wanschuraet al[152] forP r= 1,3 and 4 (see Table 4.2) shows good agreement.

For P r = 1 the maximal diﬀerence for Re_cr achieves only 1.0 % and for frequency 0.6 %. As a general trend, for P r = 1, with increasing the grid resolution, the critical Reynolds number approaches closer to the value obtained by linear stability analysis.

For higher Prandtl number the agreement is even better. For P r = 3 and 4, the present calculations on typical grid (25×16 ×21) give respectively Recr = 1170 and Recr = 1030 versusRecr = 1177 and Recr = 1047 with linear stability analysis [152]. Our results are also in excellent agreement with 3D calculations by Leypoldt et al. [68]. They obtained Re_cr = 1030

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Table 4.2: Comparison with linear stability analysis for P r= 1,3 and 4, Γ = 1.

Present work Wanschura et al. [152]

Pr grid Recr ωcr Recr mode ωcr

1 49×32×41 2540 64.51 2539 2 63.2

3 25×16×21 1170 32.06 1177 2 32.1

4 25×16×21 1030 28.30 1047 2 27.9

and ω_cr = 28.7 on grid (30×30×14) for P r = 4, which deviates only in frequency by 1.41%

from our value.

(a) (b)

Figure 4.1: P r = 4, Re = 1300, Γ = 1, ν = const. Temperature distribution (a) and disturbance ﬂow (b) at the cross section z=0.5.

To shed the light on the spatial organization of the thermocapillary flow, temperature disturbance (Fig. 4.1(a)) and the velocity fields (Fig. 4.1(b)) in midplane z = 0.5 for P r = 4, Re = 1300, Γ = 1 are shown. One can see that the oscillatory flow is a traveling wave with azimuthal wave number m = 2. This traveling wave was established as the result of strong non-symmetrical perturbations of the basic state.

Two hot and two cold spots (Fig. 4.1(a)) are observed, rotating in azimuthal direction.

There are four vortex cells in a ﬂow ﬁeld (Fig. 4.1(b)), which are caused by the pairs of hot and cold spots on the free surface, see Fig. 4.2. The location of the temperature spots on the surface demonstrates, that for this set of parameters it develops pure azimuthally traveling wave.

The angle of the inclination of isotherms inz- direction is negligible small. The mechanism of oscillation is similar to that, given in [152]. Liquid with higher temperature is driven by a surface tension gradient towards the locations of cold spots. At the same time radial return ﬂow carries liquid with lower temperature to the free surface to suppress the surface hot spot, but with some delay. A phase shift between the temperature and velocity disturbances, see Fig. 4.3, is responsible for the rotation of spots in azimuthal direction. In Fig. 4.3 solid line corresponds to

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Figure 4.2: P r = 4, Re= 1300, Γ = 1, ν =const. Temperature distribution on the free surface.

the velocity and the dashed line corresponds to the temperature, taken with some coeﬃcient for the scaling. The phase shift is not very large, about 0.146π.

4.4.2 High Prandtl number liquids with constant viscosity

The value of the critical parametersRe_cr, ω_cr are not reported in [103, 13], therefore comparison for high Prandtl numbers may only be qualitative. It was shown by Castagnolo and Carotenuto in [13] that for P r = 32, Re = 1223 and Γ = 1 the disturbance flow is a standing wave with an azimuthal wave number m = 2. But changing the aspect ratio to Γ = 2 modifies the flow organization, leading to the running wave with azimuthal wave numberm= 1 for P r= 30 (see [103]) and forP r= 32 (see [13]).

Being in good agreement with results of [103], the disturbance flow for Γ = 2 and P r = 30, Re = 1000 is a running wave with azimuthal wave number m = 1. For another aspect ratio, Γ = 1, andP r= 30, Re= 1000 present simulations show that during the transient period (in our dimensionless variables τ_trans ≈ 4) near the threshold of instability the flow pattern consists of two independent modesm= 1 andm= 2 with two close frequenciesω₁,ω₂and close amplitudes, the so-called mixed mode, which is observed rather often in experiments [125]. At some moment of the oscillation period the wave exhibits the properties of modem= 2 (two hot and two cold spots at the free surface), at another moment it switches to m= 1 (one hot and one cold spot). The harmonic with a frequency 2ω₂, which can also display the property of the modem= 2, has a much smaller amplitude, therefore it does not influence the formation of the flow pattern. After a transient period the mixed mode was found to switch to a standing wave m= 2, which was observed in [103].

The data from two thermocouples placed at the azimuthal cross sectionsϕ= 0 andϕ=π/2 are shown in Fig. 4.4a by solid and dotted lines. One can observe two oscillatory regimes: mixed mode during the time (t ≤ 4.0), which is conﬁrmed by Fourier analysis (see power spectrum in Fig. 4.4(b), and the pure mode m = 2 further in time. For this case of pure standing wave m = 2 the shift between temperature maxima is constant in time and is equal π, half of the

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Figure 4.3: P r = 4, Re = 1300, Γ = 1, ν =const. Phase shift between temperature and velocity for the traveling wavem= 2.

period.

The disturbance of the surface temperature corresponding to the standing wave with azimuthal wave number m= 2, see Fig. 4.5, is in agreement with calculations of Castagnolo and Carotenuto ([13]).

By varying the aspect ratio to Γ = 2 for P r= 30 the structure of the ﬂow is changing. In comparison with the case described above for Γ = 1, the disturbance ﬂow for Γ = 2 is a running wave with azimuthal wavenumberm= 1 (Figs. 4.6, 4.7).

Unlike the standing wave, generated by two counter propagating azimuthal waves with equal amplitudes, running wave is the result of counter propagating azimuthal waves also but having diﬀerent amplitudes. Conﬁrming the modem= 1 found in our numerics, there are one hot and one cold rotating temperature spots in Fig. 4.6(a). It is worth to be mentioned that unlike the case ofP r= 4 (Fig. 4.2) we have so-called oblique traveling waves, see Fig. 4.7.

The velocity disturbance ﬁeld in a horizontal cut at z = 1.0 is shown in Fig. 4.6(b). In agreement with the temperature ﬁeld, when P r = 30, Re = 1000 and Γ = 2, two vortex cells are observed, rotating counter-clockwise.

Moving to higher Prandtl numbers, the calculations were performed for P r= 40 and Γ = 1 to follow the experiment by Petrov et al [90]. Near the threshold of instability the disturbance ﬂow is a traveling wave with azimuthal wavenumber m= 1, this is demonstrated in Fig. 4.8 by data of four thermocouples forRe= 850. The phase shift between two neighbor thermocouples is constant in time and is equalπ/2. In the case of a standing wave m= 1, the phase shift would be either 0 orπ. The present results coincide with scientiﬁc assumptions that further increasing of the Prandtl number leads to observing running waves withm= 1 for unit aspect ratio.

The calculations give the critical Reynolds number Recr ≈ 700 versus Recr ≈ 400 in the experiment [90], although the disturbance ﬂow organization is the same. Among the others, one of the reasons for discrepancy could be a temperature dependent viscosity.

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Figure 4.4: Mixed mode,P r= 30, Re= 1000,Γ = 1. (a) The temperature signals from two thermocouples at diﬀerent azimuthal positions (ϕ= 0 andϕ=π) and (b) power spectrum for the beginning of the process (t≤4.0) conﬁrming existence of the mixed mode.

4.5

CONDITIONS OF GENERATING THE STANDING AND TRAVEL- ING WAVES IN THE LIQUID BRIDGE

One should note that, besides a set of characteristic parameters deﬁning the structure of the oscillatory ﬂow, the type of hydrothermal waves also depends upon the initial perturbations introduced into the dynamical system. For example, forP r= 4 in the case of symmetrical and weakly non-symmetrical initial perturbations standing waves are observed, which remain stable for a long time. Nevertheless, for P r = 4 it is possible to obtain a stable traveling wave by introducing strong non-symmetrical disturbances in the system.

To shed the light on this question some results of the study of the amplitude equation near the threshold of instability in the thin layer, heated from above [22], are applied to the phenomena in a liquid bridge. The analysis of the amplitude equation states that for a given set of parameters near the threshold of instability only one solution is stable, either a standing or a traveling wave. This is true under the following hypotheses: it should be a supercritical type of bifurcation (small amplitude at onset of instability) and it should be the Hopf bifurcation of the pure mode (no competition between modes). The oscillatory convection in a liquid bridge always fullﬁls the ﬁrst hypothesis and the second one is valid for some set of parameters, e.g.

P r= 4. It is proved (see Fig. 4.4) that in some cases instability begins as a mixed mode.

To give a qualitative explanation for the numerical results a schematic phase plane in Fig. 4.9 is suggested. In general, there is single solution for the standing wave and two for the traveling wave, the wave can propagate in clockwise or counterclockwise directions.

Let us assume that a traveling wave is a stable solution for the system, e.g. node (knot) point and the standing wave is unstable, e.g. saddle point. ForP r= 4 this has been proved in [68] by solving the amplitude equations. In the simplest case, when due to some initial perturbations the system is located close to the ﬁnal stable state TW (point ”a” in Fig. 4.9), the traveling wave after saturation of amplitude will reach this state and remain stable despite small disturbances.

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Figure 4.5: P r= 30, Re= 1000, Γ = 1, ν =const. Temperature distribution on the free surface.

It can also happen that due to some initial perturbations the system will be located on the asymptote, dashed line, (point ”c” in Fig. 4.9). In this case it will never become a traveling wave although it is a stable solution.

For other initial conditions (point ”b” in Fig. 4.9) the system can be located near SW at a large distance from the ﬁnal stable position. In the case of a highly symmetrical computer code and with small external disturbances the system will move slowly along the isocline and the standing wave will be observed in 3D calculations for a long duration.

In the case of strong external non-symmetrical disturbances the system, being in point ”b”, can change the trajectory and switch from SW to a TW. The waiting time can be diﬀerent and it depends how far the system is located from the ﬁnal stable position and how strong the perturbations are. It also depends on the numerical code, e.g. how well the code keeps symmetry.

A good example of the development of the instability is described below, when perturbations of a given symmetry but small amplitude are applied for initiating the oscillatory ﬂow. For the particular caseP r= 4, Re= 1300,Γ = 1, Gr= 0 the wave number of the resulting ﬂow is known in advance,m= 2. Following [68], the periodic symmetrical temperature perturbations Θ_p are introduced into the system:

Θ_p = ˆΘrΓ cos(πz) sin(mϕ) = ˆΘrΓ cos(πz) sin(2ϕ), Θ = 10ˆ ⁻³

The data from two thermocouples placed at point (r, z) = (0.9,0.5) and shifted on 3π/4 in an azimuthal direction are shown in Fig. 4.10. One can observe three oscillatory regimes in this ﬁgure. At the very beginning the amplitude of oscillations is growing exponentially with time, Re > Recr. The saturation of the amplitude is achieved during one thermal time (t≈4) and then remains constant. A standing wave with wavenumber m = 2 is a result of the initial symmetry of the perturbations introduced into the liquid bridge. The present code keeps well the symmetry for the solution. The oscillations in the two azimuthal positions have a constant phase shift, but a diﬀerent amplitude, this is particular to the standing wave withm= 2 when thermocouples are not shifted onkπ/2, k= 1,2,3... (see left insertion in Fig. 4.10).

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(a) (b)

Figure 4.6: P r = 30, Re = 1000, Γ = 2, ν = const. Temperature distribution (a) and disturbance ﬂow (b) at the cross section z=1.0.

But, for this set of parameters and initial temperature perturbations a standing wave appears to be unstable and transition to a traveling wave takes place. After a transition period a stable traveling wave with a constant amplitude can be seen (t≥22) for inﬁnite duration. In contrast to the standing wave, the data from the same thermocouples demonstrate the oscillations both with constant phase shift and equal amplitudes (see right insertion in Fig. 4.10).

Diﬀerent kinds of perturbations were applied in simulations for the liquids with constant viscosity (R_ν = 0) andP r= 4. Introducing such perturbation into the system so that only the temperature in one vertical cross sectionj = 1 was disturbed, no transition from a standing to a traveling wave was observed up tot≥34 for the same parameters. Possibly, the system in this case is located in the vicinity of point ”b” on the phase plane in Fig. 4.9. On the other hand, introducing strong non-symmetrical temperature perturbations the traveling wave was observed from the very beginning of the simulation.

Taking into account the above, one can suggest an explanation on why in experiments Monti et al. [76] did not observe transition from SW to TW at the ﬁxed time, which was calculated numerically. The initial perturbations in numerics and in the microgravity experiment are different. Moreover, unlike numerics, it is impossible to predict exactly the type and the amplitude of the perturbations in natural experiment, as they have in general statistical origin.

4.6

TEMPERATURE DEPENDENT VISCOSITY 4.6.1 On the inﬂuence of the choice of the reference temperature

In the majority of the scientific studies of the thermo convective flows, the basic dimensionless parameters are defined through the viscosity ν₀, e.g. P r = k/ν₀, Re = σ_T∆T d/ρ₀ν₀², etc.

When the viscosity varies, scaling the governing equations requires an appropriate choice of the characteristic temperature T₀, at which viscosity ν₀ is deﬁned. In this case the choice of ν₀ becomes very important. The present calculations demonstrate that in a parametric analysis of instability, the values of critical parameters, e.g. Reynolds number and critical frequency,

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Figure 4.7: P r= 30, Re= 1000, Γ = 2, ν =const. Temperature distribution on the free surface.

strongly depend on the choice of the characteristic temperature. Moreover, this dependency is strengthened with a growth of|Rν|; the larger Rν, the stronger the dependence observed.

There are several possibilities, such as taking ν₀ equal to the viscosity at the mean value of the boundary temperaturesT₀ = (T_cold+T_hot)/2 or the viscosity at one of the boundaries (cold Tcold or hotThot rod). Rather often in the numerical studies of the convection in liquid bridges with a constant viscosity the mean temperature is used as reference.

One should understand that the choiceT_cold to define the reference viscosity is justified and suitable for parametric study. Only the temperature of the cold diskTcstays constant during the experiments and simulations, when ∆T is changed (heating from above). The temperature of the hot wall as well as the temperature difference ∆T near threshold of instability are not known in advance and their values will only be determined in carrying out experiments or calculations.

It means that each time, one has to change the Prandtl number for the same liquid, e.g. this takes place when the aspect ratio is varied. That leads to changing the values of the parameters of the problem and may generate some diﬃculties in the presentation of the results. Moreover, presentations of the experimental results are usually done for a constant Prandtl number of a liquid, deﬁned at the room temperature or at the temperature of the cold disk. For instance, in the experiments [103] with a silicone oils 2cSt and 5cSt the well known Prandtl numbers equal to P r = 30 and P r = 74 are stated, but the reference temperature, which is the room temperature, is hidden in the given value of P r.

Let us look at the problem from the view of changing the Prandtl and Reynolds numbers as functions of co-ordinates. Different choices of the characteristic temperature will result in different distributions of viscosity inside the liquid bridge. The ”local” Prandtl and Reynolds numbers, being functions of co-ordinates, will not be the same, i.e. problems with different parameters are considered.

In this section, two choices of the reference temperature are investigated: when the mean value of the boundary temperatures in the systemT₀ = (Tcold+Thot)/2 is taken for definingν₀, and the other is when T₀ = T_cold. For both cases the influence of T₀ on the flow pattern and temperature distribution is discussed. The values ofT₀are equal. The viscosity, being a function

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Figure 4.8: P r= 30, Re= 1000, Γ = 2, ν=const. Data of four thermocouples.

of temperature, is approximated by a linear relation (2.6). Here viscosity is assumed to be a decreasing function of the temperature, which is the case for the most often used experimental liquids. R_ν is negative. It is easy to show that when T₀ is the mean temperature, the value of viscosity at each point inside the liquid bridge is larger by ∆ν =−Rν/2, in comparison with the case whenT₀=T_cold. Again, the problem under two diﬀerent conditions is under consideration, although the Prandtl and Reynolds numbers deﬁned atT₀ are equal.

Steady-state 2D simulations were carried out under zero-gravity conditions forP r= 4, Re= 1000,Γ = 1, either with R_ν = 0.0 or R_ν = −0.5. As R_ν is large, a noticeable difference between the final results is expected. Indeed, it is obtained that the absolute value of minimum stream function increases by 52 % for T₀ = T_cold, ψmin = −3.602, and by 1.5 % when T₀ = (T_cold+T_hot)/2, ψ_min = −2.403, compared to the case of constant viscosity ψ_min = −2.367, when R_ν = 0.0. In the 2-D results of Kozhouharova et al [52] the reference temperature is T₀ = (Tcold+Thot)/2 and the absolute value of minimum stream function forRν =−0.5 increases by 4.1 %, ψ_min =−2.28 in comparison with constant viscosity, ψ_min(R_ν = 0.0) =−2.19. The difference with our results is about 5%.

Basic-state temperature ﬁeld and isolines of stream function are shown in Fig. 4.11, where Fig. 4.11a corresponds to the reference temperature T₀ = T_cold, and Fig. 4.11b corresponds to the reference temperature T₀ = (T_cold+T_hot)/2. The isolines in Fig. 4.11a and Fig. 4.11b have the same levels, and full and dashed lines represent constant and variable viscosity cases respectively.

For temperature-dependent viscosity the most significant changes in the temperature field are observed when T₀ = Tc in comparison to the case of constant viscosity. The isotherms move up near the symmetry axis adjoined to the hot wall, and move down in the region of the cold corner, increasing the temperature gradients in these regions in comparison with the constant viscosity case (see Fig. 4.11a). In the case whenν₀ is the viscosity defined at the mean temperature, the increase of the temperature gradients is not so significant. Near the hot wall and in the vicinity of the cold corner, there are almost no changes in the temperature field. At

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Figure 4.9: Schematic phase plane. Stable state is traveling wave (TW).

the upper part on the free surface the temperature gradient is even smaller, compared to the constant viscosity case ν₀. Such behavior of the temperature ﬁeld has to result in decreasing the critical Reynolds number, stronger forT₀ =T_cold than forT₀ = (T_cold+T_hot)/2.

To be able to compare our 3D results with the results of linear stability [52], the only results existing until now, for variable viscosity in liquid bridges, the simulations have to be done for the same set of parameters P r = 4,Γ = 1, R_ν = −0.5. For this value of R_ν the viscosity in the volume decreases by factor two from the cold to hot wall. For this set of parameters in the case of a constant viscosity fluid Rν = 0.0 the critical Reynolds number is Recr = 1030. When T₀ = (T_cold+T_hot)/2 and R_ν =−0.5 the transition to oscillatory flow with wavenumberm = 2 is observed for Re_cr = 1010. In the case where T₀ = T_cot for the same R_ν = −0.5, the onset of instability takes place for a much lower Reynolds number, Recr = 635. One can see that influence of the variable viscosity onRe_cr is very strong when T₀ =T_cold in comparison to the case where T₀ = (T_cold+T_hot)/2, namely, 38.1% versus 3.8%. But, the imposed constrains are different and correspond to two different physical conditions. For the onset of oscillatory regime the linear stability analysis [52] shows a decrease of the critical Reynolds number by 8.9 %, e.g.

Re_cr(R_ν = 0.0) = 1047 versus Re_cr(R_ν = −0.5) = 954. But for the same choice of reference

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Figure 4.10: Transition from standing to traveling wave. Time proﬁles of temperature in two diﬀerent azimuthal positions. P r = 4, Re= 1300, Gr= 0,Γ = 1

temperature our 3D simulations gives Re_cr = 1010 and this value is about 5.5% higher than from the linear stability analysis. Although the critical Reynolds number obtained by full 3D calculations should be lower, than those from linear stability analysis, the present results are worth to be conﬁdent of. Surprisingly, the 2-D value of the minimum stream function [52] for constant viscosityψmin =−2.19 is about 7% smaller than those reported by the same scientiﬁc group in another paper [68], e.g. ψ_min =−2.35.

In real physical experiments the viscosity is always temperature-dependent. By choosing a different reference temperature for modeling the particular experiment, a different values of critical parameters will be obtained. It is evident that going back from dimensionless to physical parameters, the critical temperature difference ∆T_cr will be the same, as the physics of phenomena is not changed. But the form of representation of the results remarkably depends on the choice of the reference temperature. Moreover, for each particular case it is possible to establish the correlation between characteristic parameters for different choices of reference temperature. However to compare two experiments or perform simulations one should define a reference temperature at a fixed point. As a rule, the goal of scientific study is to present parametric investigations and to work out the general features of phenomena.

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Figure 4.11: Steady-state distribution of temperature and isolines of stream functions when the reference temperature is (a) the temperature of the cold diskT₀ =T_cold, (b) the mean temperature in the system T₀ = (T_hot +T_cold)/2. Full and dashed lines correspond to constant and variable viscosity respectively. P r = 4, Re= 1000,Γ = 1.

4.6.2 Parametric study of the onset of instability for liquids with temperature- dependent viscosity

In this section, results of three-dimensional simulations of Marangoni convection for medium Prandtl number liquids, P r = 3,4 and 5 and temperature-dependent viscosity are reported.

The critical Reynolds number and frequency of oscillations near the threshold of instability are obtained as a function of viscosity variation R_ν for unit aspect ratio Γ = 1. The frequency of oscillations is determined from the time dependent temperature signals on four numerical thermocouples in a transversal section.

The temperature of the cold wall is taken as reference, T₀ =T_cold. With this choice of ν₀= ν(T₀) the largest value of viscosity in the system is ﬁxed on the cold bottom plate. Everywhere in the liquid bridge the value of viscosity is smaller, therefore higher velocities arise for a given stress. In this case, one may expect a strong decrease of the critical Reynolds number for the onset of instability for variable viscosity ﬂuid.

For each Prandtl number the parameterRν is changed from 0.0 to−0.9 and the value of the Reynolds number, at which convection in a liquid bridge becomes oscillatory, is searched. The values of critical Reynolds numbers are determined with accuracy ∆Re_cr = 5, i.e. the Reynolds number is varied with this step until the damping of oscillations. The present calculations, summarized in Table 4.3, demonstrate signiﬁcant inﬂuence of a viscosity variation on the critical Reynolds numberRe_cr and frequency of oscillations in the system. As a general tendency, the critical Reynolds number diminishes with an increase of the viscosity variation |Rν|. For Pr=3 the value of Re_cr decreases by 2.5 times and forP r = 5 more then three times, when viscosity in the liquid bridge between cold and hot walls is decreasing by 10 times (R_ν = 0.0÷ −0.9).

For a ﬁxed Prandtl number the dependence of Recr upon viscosity variation Rν, shown in Fig. 4.12a, with good accuracy can be treated as a linear function in the range ofRν from−0.1 to−0.9. The crosses in Fig. 4.12a correspond to numerical points, the dashed lines connect these

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Table 4.3: Inﬂuence of Rν on Recr and on the critical frequency for medium Prandtl number liquids.

critical Reynolds numbers critical frequencies R_ν P r= 3 P r= 4 P r= 5 P r= 3 P r = 4 P r= 5

0.0 1170 1030 950 32.06 28.72 25.91

-0.1 1070 945 870 31.43 27.52 25.13

-0.2 990 865 800 30.65 26.74 24.40

-0.3 910 785 725 29.92 26.18 23.49

-0.4 820 710 650 29.17 25.12 22.85

-0.5 740 635 575 28.24 24.64 22.24

-0.6 660 560 505 27.32 23.71 21.30

-0.7 585 490 435 26.10 22.34 20.27

-0.8 520 420 365 25.39 21.30 19.19

-0.9 450 350 295 24.40 20.27 17.80

Table 4.4: . The rate of decreasing ofRe_cr with Prandtl number due to variable viscosity.

P r 3 4 5 35

Re_cr(R_ν = 0)/Re_cr(R_ν =−0.5) 1.58 1.62 1.65 1.66

points, and the solid lines represent the linear approximation. The equation for ﬁtted straight lines, can be written as

Re_cr =Re⁰_cr+K_ν·R_ν, R_ν <0.

The coeﬃcient of proportionality K_ν betweenRe_cr and R_ν is decreasing, when the Prandtl number is increasing. For the investigated range of Prandtl numbers 3 ≤ P r ≤ 5 the critical Reynolds number for the small variations of viscosityRν <0.1 is a fraction higher than it follows from linear dependency. And for a very large contrast of viscosity,|R_ν|>0.9, when the viscosity in the volume is decreasing more than by 10 times, the linear dependence does not work.

The decreasing rate of critical Reynolds number depends upon Prandtl number rather weakly.

The decreasing rate is more pronounced for the range of medium Prandtl numbers. Results of the variation ofRe_cr(R_ν = 0)/Re_cr(R_ν =−0.5) for diﬀerent Pr are shown in Table 4.4. Indeed, when Prandtl number is changing fromP r = 5 toP r= 35 the decreasing rate is increasing only from 1.65 to 1.66, but moving fromP r= 3 only toP r = 5 this rate is shifted from 1.58 to 1.65.

Although the flow organization is very different forP r= 4 andP r= 35. To demonstrate the influence of the Prandtl number on the flow field, the surface temperature distribution is shown in Fig. 4.13 above the threshold of instability in the case of traveling wave. The azimuthal wave numbers are different: (a) m = 2 and (b) m = 1. Fig. 4.13a corresponds to medium Prandtl number P r= 4,Re= 1030, Γ = 1 and Fig. 4.13b corresponds to high Prandtl number P r= 35, Re = 370, Gr = 491,Γ = 1. It is worth mentioning that unlike the case ofP r= 4 when the wave propagates practically azimuthally, in the case of high Prandtl numbers the so-called oblique traveling waves are observed.

Another characteristic of the ﬂow, inﬂuenced by temperature-dependent viscosity, is the frequency of oscillations in the system. Like the critical Reynolds number, the critical frequency

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Figure 4.12: Dependence of (a) Recr and (b) ωcr upon viscosity variation Rν for diﬀerent Prandtl numbers, Pr=3, 4 and 5.

of oscillations near the onset of instability decreases with an increasing absolute value ofRν, see Fig. 4.12b. But the amplitude of the variations of the frequency is much smaller, the maximal deviation is about 30% from the initial value at constant viscosity. It decreases fromω = 32.06 to ω = 24.40 for P r= 3, and fromω = 25.91 toω = 17.80 for P r= 5 when the parameter Rν

changes from 0.0 to −0.9 (see Table 4.3). This dependency also can easily ﬁt using straight lines. Surprisingly, the slopedω_cr/dR_ν practically does not depend upon Prandtl number. For P r = 5 the slope is 8.648 versus 8.63 for P r = 3, i.e. the straight lines are parallel. The empirical formula for frequency can be written as

ω_cr =ω_cr⁰ + 8.64·R_ν.

For all the cases mentioned, P r = 1−5, traveling waves with mcr = 2 wave number are observed. To study the influence of the variable viscosity on the structure of the flow, calculations have been made for P r = 5, when Rν = 0.0, Re = 950 and Rν = −0.9, Re = 300 just at the threshold of instability,→0. The distance from the critical point is defined as

= Re−Re_cr Recr

.

It was discovered that the temperature dependence of viscosity has no significant effect on the structure of the thermocapillary flow for the small(not shown).

At some particular moment of time the positions of the temperature spots are diﬀerent due to their rotation, as a traveling wave is established. But they have the same shape and any changes in their locations with respect to the cylinder axis, as well as the shapes and inclination of the spots on the free surface, are indistinguishable.

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Figure 4.13: Surface temperature distribution for diﬀerent Prandtl numbers near onset of instability (a) P r= 4, m= 2 and (b)P r= 35, m= 1.

In order to understand in which way the temperature-dependence of viscosity inﬂuences the temperature perturbations ﬁeld, simulations with P r = 4 at supercritical Reynolds number Re= 3000 are carried out when R_ν = 0.0, = 1.91 andR_ν =−0.5, = 3.72. The temperature isolines of traveling waves are shown in a transversal section at midplane z = 0.5 for constant (Fig. 4.14a) and variable viscosity (Fig. 4.14b). The axisymmetric part is subtracted from the total temperature distribution. In the vicinity of the axis the shapes of the temperature spots and their positions are almost the same. But what is interesting is that near the free surface a thermal boundary layer is formed in the case of variable viscosity.

4.6.3 The inﬂuence of viscosity on the wave properties of the ﬂow.

Let us look at the solution of eqs.( 4.1) - ( 4.5) near the threshold of instability. For the supercritical Hopf bifurcation, slightly above the onset of instability→0, the solution can be expanded as [57]

F(r, z, ϕ, t) =F₀(r, z) +√

F⁽¹⁾+ F⁽²⁾+O(³^/²) (4.13) whereF =F(V , P, T ) and F₀(r, z) is a steady-state solution.

The O(¹^/²) problem is a linear problem with a normal-mode solution. It consists of two

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Figure 4.14: Temperature disturbance ﬁeld in az= 0.5 horizontal cross section forP r= 4, Re= 3000,Γ = 1. (a)constant viscosity, R_ν = 0 and (b) temperature-dependent viscosity, Rν =−0.5. The axisymmetric part is subtracted from the total temperature distribution.

hydrothermal waves propagating in opposite directions F⁽¹⁾ =F_{lef t}⁽¹⁾ +F_right⁽¹⁾ +c.c where

F_{lef t}⁽¹⁾ = ˆAeⁱ^[^mϕ⁻⁽^iσ⁺^ω⁾^t⁺^g⁽^r,z^)] =A(t)f₁(r, z)eⁱ^[^mϕ⁻^ωt⁺^G⁽^r,z^)]

and the azimuthal wave number has an opposite signm_cr → −m_cr for the right wave. Here the functionG(r, z) is the phase describing the inclination of the wave with respect to the vertical axis, σ → 0 and amplitude depends upon slow time A(t). The more precise expression for amplitude A(t) can be determined from a solution of the Landau-Ginsburg equation. The residual ﬁeldF⁽²⁾ is produced by non-linear interactions of left and right waves. It is a spatially periodic function withm =m_cr and eq.( 4.13) can be expressed as a power-series expansion in A, A^∗, B, B^∗.

F(r, z, ϕ, t) = F₀(r, z) +√

[A(t)f₁(r, z)eⁱ^[^mϕ⁻^ωt⁺^G⁽^r,z^)]+B(t)f₁(r, z)eⁱ^[−^mϕ⁻^ωt⁺^G⁽^r,z^)]+c.c]

+ {A²f₂₁e²ⁱ^[^mϕ⁻^ωt⁺^G¹⁽^r,z^)]+B²f₂₁e²ⁱ^[−^mϕ⁻^ωt⁺^G¹⁽^r,z^)]+ (AA^∗+BB^∗)f₂₂ + AB^∗f₂₃e^2imϕ+AB f₂₄e^2i[−ωt+G²^(r,z)]+c.c.}+O(^3/2) (4.14) In such notationsf₁, f₂₁, f₂₂, f₂₃, f₂₄ are real functions for any physical characteristic except an azimuthal velocity. From continuity equation ( 4.4) follows that azimuthal velocity

Vˆ =−(1/im)[U +r(∂_rUˆ+∂_zWˆ) ] =iV /m˜ (4.15)

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here ˜V is an analog of f₁(r, z) and it is real.

Then, using eq.( 4.15) and the condition of symmetry, eq.( 4.14) for azimuthal velocity can be written as

V(r, z, ϕ, t) = √

[A(t)v₁(r, z)eⁱ^[^mϕ⁻^ωt⁺^g⁽^r,z^)] −B(t)v₁(r, z)eⁱ^[−^mϕ⁻^ωt⁺^g⁽^r,z^)]+c.c]

+ {A²v₂₁e²ⁱ^[^mϕ⁻^ωt⁺^g¹⁽^r,z^)]−B²v₂₁e²ⁱ^[−^mϕ⁻^ωt⁺^g¹⁽^r,z^)]

+ (AA^∗−BB^∗)v₂₂+AB^∗v₂₃e^[2^imϕ⁺^iπ/^2]+c.c.}+O(³^/²) (4.16) where the functionsv₁, v₂₁, v₂₂, v₂₃ are real functions.

Let us now look at the solution of the linear problemF⁽¹⁾ for diﬀerent viscosities. Following eq.( 4.13), and this is well known [58], near the onset of instability forRe > Re_crthe amplitudes ofV and Θ, P are proportional to the square root of the distance from the critical point√

.

Figure 4.15: Dependence of (a) temperature amplitude upon√

and (b) net azimuthal ﬂow upon, Pr=4.

The amplitude of temperature versus√

is shown in Fig. 4.15a using the data from numerical thermocouples for P r = 4. The crosses correspond to constant viscosity, and the triangles correspond to a variable one when Rν = −0.5. One can see that the amplitude near the threshold of instability does not depend upon the variation of viscosity, in fact both cases fit well by using a straight line with a slope 0.82, although the difference in the critical Reynolds number is significant Recr(Rν = 0)/Recr(Rν =−0.5) = 1.62. It is coherent to the structure of flow patterns described above whenP r= 5 and R_ν =−0.9.

To look at the eﬀect of the next order of magnitude the mean azimuthal velocity [68] is calculated by averaging the azimuthal component of velocity prescribed by eq.( 4.16)

V_ϕ,mean(r, z, t) = 1

2π V_ϕ(r, z, ϕ, t)dϕ

The azimuthal velocityV_ϕ at a steady state is equal to zero. All linear harmonics, proportional to √

and containing sin and cos values vanish being integrated in an azimuthal direction.

Hence, income from the term proportional to√

also equals zero. The resulting mean velocity for the traveling wave includes only non-linear self-interactions and is a linear function of ,

V_mean(r, z) =(AA^∗−BB^∗)˜v₂₂+O(³^/²)

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Due to symmetry, the mean azimuthal ﬂow of standing wave is equal to zero. The net azimuthal ﬂow, determined as an integral over volume

Φ = V_ϕ,mean(r, z)rdrdz (4.17)

is shown in Fig. 4.15b for the constant, R_ν = 0, and with variable, R_ν =−0.5, viscosity as a function of forP r = 4. Both dependences can be ﬁtted by using straight lines with diﬀerent slopes

Φ =κ_ν×+O(^3/2), where κ_ν = 0.418 when R_ν = 0.0 (4.18) κ_ν = 0.480 R_ν =−0.5

The net azimuthal flow for the case of constant viscosity is weaker than for temperature- dependent. As Rν is negative it means that decreasing viscosity in the volume results in the increase in net azimuthal flow despite the smaller Reynolds number. The net azimuthal flow is the result of non-linear self-interactions, which become stronger moving into the supercritical region > 0. It means that the role of temperature dependent-viscosity is increased with the growth of and its influence on the flow organization will be remarkable far away from the critical point, = 0. It explains, why the temperature disturbance fields for constant and variable viscosities are the same near the onset of instability and they are different for >1.9 (see Fig. 4.14).

The function Φ has been calculated by Leypoldt et al. [68], but for some reason they omitted the coordinate ”r” under integral in eq.( 4.17) and they have obtained a slope 1.04.

For comparison, the value of the slope has been calculated by their method. It gives a slope 0.97, thus the difference is approximately 7%. Whether the viscosity is variable or constant, the direction of the mean azimuthal flow is not dependent upon this. Following the logic described in [68], our calculations show that the net azimuthal flow for P r = 4 is also opposite to the direction of wave propagation. For P r= 4 the wave travels in a clockwise (negative) direction and the net azimuthal flow is positive, Φ > 0. The direction of propagation of the wave is not a property of the system, it depends upon initial perturbations. For example, forP r = 35 the traveling wave propagating in both clockwise and counter clockwise directions have been generated .

The isolines of mean azimuthal velocity are shown in Fig. 4.16a by using lines for constant viscosity and by using shadows for variable viscosity with a viscosity contrast of 10,R_ν =−0.9.

Both distributions represent the mass flow just above the threshold of instability, ≈0.01. For the variable viscosity the center of the positive mean flow is shifted towards the hot corner and the maximal value becomes≈11% higher than for constant viscosity. The value of V_ϕ,mean(r, z), except in the cold corner, is positive for both cases. With increasing , the cores of the mean azimuthal flow (in Fig. 4.16a) for both cases are rapidly shifted towards the symmetry axis r= 0.

For high Prandtl numbers, P r = 35, such characteristics as ”net azimuthal flow” do not contain comprehensive information. Indeed, the value of Φ itself is close to zero, and the slope κ_ν in eq.( 4.18) is twenty times smaller then in the case whereP r= 4. But, the small value of Φ does not mean a weak mean azimuthal flow. The mean azimuthal velocity, shown by isolines in Fig. 4.16b forP r = 35, is practically equivalent in upper and lower parts of cylinder, but has an opposite direction. Moreover, for the same the maximal value of |V_ϕ,mean| is almost five times greater for P r = 35 than for P r = 4. This is valid both for pure Marangoni convection and for combined Marangoni-Rayleigh convection, whenBo=Gr/Re= 1.227 and Bo= 1.328.

According to previous considerations, the mean ﬂow is still opposite to the traveling wave. But

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Figure 4.16: Isolines of mean azimuthal velocity,Vϕ,mean, for (a) P r= 4, Re= 1030, Gr = 0, R_ν = 0 (lines) and R_ν =−0.9 (shadows), and (b) P r= 35, Re= 370, Gr = 491, R_ν = 0

it is diﬃcult to make such kind of conclusions for high Prandtl numbers ﬂuids as with a further increase of the Pr this value approaches to zero and can be comparable to the accuracy of the solution.

From our point of view, it is the mean azimuthal velocity Vϕ,mean(r, z) which gives more physical information. Analyzing the mean azimuthal velocity for different Prandtl numbers, one may suggest that the profile shape of the traveling wave is related to a distribution of the Vϕ,mean(r, z). Let us compare the surface temperature field in Fig. 4.13a for Pr=4 and distribution of a mean azimuthal velocity in Fig. 4.16a. One may say that the wave is propagating practically azimuthally except for the small bend near the cold corner. The position of this bend coincides with the change of sign in the mean flow in Fig. 4.16a. The same conclusion is reached from a comparison (Fig. 4.13b and Fig. 4.16b) using the high Prandtl number,P r= 35.

Eventually, some assumptions can be made concerning obliquity of the traveling wave by comparing the same ﬁgures. For high Prandtl numbers,P r ≥35, an oblique traveling wave was always observed, while for moderate Pr numbers,P r = 1,3,4, it is almost a pure azimuthal wave.

ForP r= 4 the positive part of velocityV_ϕ,meanon the free surface has an almost constant value, except for the cold corner (see Fig. 4.16a). When increasing the uniformity of the velocity on the free surface becomes even better. Contrary to this, the levels of isolines for P r = 35 are rapidly varied moving from top to bottom. This can result in formation of the slope of the traveling wave. The phase g(z, t) does not enter directly into the eq. ( 4.16), but non-linear multi-parametric task is under consideration. Moreover, the inﬂuence ofg(z, t) can be captured by the terms of the next order of expansion, e.g. O(³^/²).