Instability Of Thermocapillary Flow Research Articles

Instabilities of thermocapillary flows in large Prandtl number liquid bridges between two coaxial disks with different radii

We explore the geometric effects on the thermocapillary flow instabilities in large Prandtl number (Pr = 1.4) liquid bridges between two coaxial disks with different radii under microgravity, focusing on the impacts of radius ratio Γr and aspect ratio Γ. The static deformation of the free surface is concerned by the solution of the Young–Laplace equation, and the linear stability analysis based on spectral element method is conducted for accurate identification of the instability characteristic. We observe that the flow stability is generally improved with the decrease in radius ratio Γr or aspect ratio Γ, especially for the liquid bridge heated from the upper disk. The critical oscillation frequency experiences an abrupt drop around Γr = 0.56 as Γr decreases for the liquid bridge with the bottom disk heated. Moreover, three transitions between two-dimensional axisymmetric steady flow and three-dimensional oscillatory flow are observed within the interval 0.87 < Γ ≤ 0.91 at Γr = 0.50 when the liquid bridge is heated from the upper disk. The energy analysis indicates that the instabilities for all cases are predominantly caused by the hydrothermal wave instability and the phenomenon of three transitions results from the variation of thermal energy transfer efficiency with the growth of the Marangoni number.

Multiple instabilities of thermocapillary flow in a cylindrical pool with a rotating disk on the free surface

The thermocapillary flow instabilities of silicon melt in a cylindrical pool with a rotating disk on the free surface (a simplified model of the Czochralski crystal growth) are numerically investigated by using the linear stability analysis. The complete neutral or critical stability curves are determined. Results show that the neutral stability curves form a closed region in the parameter plane, in which the steady axisymmetric flow is linearly stable. Two types of rotating wave (RW1 and RW2) instabilities and two types of hydrothermal wave (HTW1 and HTW2) instabilities are found. The energy analysis shows that all the instabilities are hydrodynamic (inertial) in nature. Specifically, RW1 and RW2 are caused by the azimuthal shear induced by the crystal rotation, while HTW1 and HTW2 are caused by the radial shear induced by the thermocapillary force.

Flow instabilities in thermocapillary liquid bridges between two coaxial disks with different radii

The primary instability of thermocapillary flow in a liquid bridge between two coaxial disks with different radii is investigated under microgravity for silicon melt with Prandtl number Pr=0.011 via two heating strategies. The static deformation of the free surface is considered by solving the Young-Laplace equation. The physical instability mechanisms are explored by analyzing the energy budgets of the critical modes, which are determined by linear stability analysis based on the Legendre spectral element method. With the decrease of radius ratio Γr, the stability of thermocapillary flow is significantly improved. In contrast to typical cylindrical liquid bridges (Γr=1), the instability is an oscillatory bifurcation for small radius ratios (Γr≤0.672) when the liquid bridge is heated from the bottom disk. Furthermore, the instabilities for all radius ratios and heating strategies are found to be purely hydrodynamic, but the specific instability mechanisms are different.

Volume effect on the instabilities of thermocapillary flow in annular pools filled with silicon melt

The effect of melt volume on the instabilities of thermocapillary flow in annular pools filled with silicon melt (Prandtl number Pr = 0.011) was investigated by the linear instability analysis. With the increase of volume ratio V/V0 from 0.584 to 1.343, the critical Marangoni number decreases steeply when the volume ratios are small, while increases slightly for intermediate volume ratios, and decreases again for large volume ratios. The wavenumber and oscillation frequency keep decreasing and a jump occurs at approximately V/V0 = 1.2 corresponding to the second decrease of the critical Marangoni number. The energy analysis indicates that the flow instability is initiated by the hydrodynamic inertial instability mechanism.

Effect of rotating magnetic field on instabilities of thermocapillary flow in a Czochralski silicon melt pool

The effect of the rotating magnetic field on the thermocapillary flow instabilities in a Czochralski silicon melt pool was numerically investigated by means of linear stability analysis. We considered three cases, namely, the non-rotating crystal, the crystal rotating in the same direction as the magnetic field (co-rotating crystal), and the crystal rotating in the opposite direction of the magnetic field (counter-rotating crystal). The linear stability analysis results show that the basic flow becomes unstable against two oscillatory instability modes. With the increase of the magnetic Taylor number (Ta), the overall trend of the critical Marangoni number decreases in the cases of a non-rotating crystal and counter-rotating crystal. However, in the case of a co-rotating crystal, the rotating magnetic field significantly improves the flow stability, and an inversion of the propagation direction of the oscillatory wave is observed with the increase of Ta. Energy analyses reveal that the instabilities are hydrodynamic in nature, which are caused by the combination of the strain-induced instability mechanism and the centrifugal instability mechanism, and the rotating magnetic field only provides little energy for the flow instabilities even at large Ta.

Effects of Pr and pool curvature on thermocapillary flow instabilities in annular pool

We linearly analyzed the effect of the Prandtl number (Pr) on the stability of thermocapillary flow in shallow annular pools with an aspect ratio Γ = (Ro - Ri)/d = 20 and two radius ratios ΓR = Ri/Ro= 0.50 and 0.98039, where d, Ro, and Ri are the liquid depth and the radii of the heated outer wall and the cooled inner wall, respectively. The results for Pr ∈ [0, 102] show that the steady axisymmetric thermocapillary flows in these annular pools become unstable against oscillatory instability modes, OSC1, OSC2 and hydrothermal wave (HTW). Two co-dimension-two bifurcations occur at Pr*1 and Pr*2. The critical mode for Pr ≤ Pr*1 is OSC2 with almost constant critical Reynolds number Rec, large wave number and high frequency. OSC1 with smaller wave number and lower frequency is the critical mode for Pr*1 ≤ Pr ≤ Pr*2 and Rec slightly decreases with increasing Pr. HTW is the critical mode for Pr ≥ Pr*2 and Rec decreases with increasing Pr. The values of (Pr*1, Pr*2) are (0.00953, 0.03054) for ΓR = 0.50, and (0.01919, 0.01946) for ΓR = 0.98039. Energy-budget analyses reveal that the instabilities in low-Pr range are caused by instabilities in the steady toroidal vortex (or vortices) near the cold wall.

Effect of the crucible/crystal rotation on thermocapillary instability in a shallow Czochralski configuration

To understand the effects of crucible/crystal rotation on the instability of thermocapillary flow in a shallow Czochralski configuration, a series of linear stability analysis were performed by using Legendre spectral element method, and the energy analysis was applied to explore the instability mechanism. The results indicate that there are two types of instabilities with the increase of dimensionless crucible rotation rate ωc. Thermocapillary force and crucible rotation are the main causes of flow instability for low and high crucible rotation rate, respectively. In the interval 1627<ωc < 1822, with the increase of Marangoni number, three turning points were observed, indicating three transitions between two-dimensional axisymmetric steady flow and three-dimensional oscillatory flow due to the interaction between thermocapillary force and crucible rotation. For crystal rotation, two oscillatory modes for flow instability were observed. When the dimensionless crystal rotation rate is small (ωs < 3600), the oscillatory flow rotates in the opposite direction of the crystal rotation, but in the same direction in the case of large crystal rotation rate (ωs > 3600), and the thermocapillary force is the primary inducement of flow instability.

Influence of aspect ratio on the onset of thermocapillary flow instability in annular pool heated from inner wall

Thermocapillary flow instabilities in annular pool with different aspect ratios were investigated by the linear stability analysis based on the Legendre spectral element method. The energy analysis was applied to explore the underlying physical mechanism for the instability of thermocapillary flow. The results reveal two types of instability for different aspect ratios. The onset of instability is an oscillatory bifurcation in the cases of small aspect ratio whereas a stationary bifurcation in the cases of large aspect ratio. Moreover, the energy analysis indicates that the bifurcation of the flow is caused by the hydrodynamic (inertial) instability mechanism for low Prandtl number (Pr = 0.011), while by the coupled hydrodynamic effect and Marangoni effect for moderate Prandtl number (Pr = 1.4).

Convection and instability of thermocapillary flow in a liquid bridge subject to a non-uniform rotating magnetic field

A three-dimensional liquid bridge is considered in this study to numerically investigate the effects of an external non-uniform rotating magnetic field (RMF) on the thermocapillary flow in semiconductor melt under microgravity. Simulations are carried out to examine the convection and instability features of the thermocapillary flow over a range of Marangoni numbers (Ma=15–50) under a non-uniform RMF. The present results show that applying an external non-uniform RMF enhances the maximum tangential velocity and depresses the maximum axial velocity. As a consequence, an approximately axisymmetric flow is maintained in the melt under the effect of the non-uniform RMF, which is beneficial for growing high quality crystal. Further investigation of the thermocapillary flow subject to different non-uniform RMFs (corresponding to Taylor numbers Ta=3.8×102–1.86×104 and Rotating Reynolds number Reω=2.2×104) reveals that the thermocapillary convection may undergo a transition from the approximately axisymmetric steady flow to a periodically oscillatory flow for Ma above a critical value. The critical Ma generally increases with the intensity of the non-uniform RMF.

Thermocapillary flow instabilities of medium Prandtl number liquid in rotating annular pools

The linear stability of thermocapillary flow for a medium Prandtl number liquid (Pr = 6.7) is investigated in the annular pools rotating along counterclockwise direction. The neutral Reynolds numbers for the incipience of instabilities are determined by linear stability analysis and the underlying mechanisms are analyzed by energy budgets. Four types of oscillatory instabilities are predicted for different aspect ratios over a wide range of Taylor number. The energy budgets reveal that all the instabilities are basically driven by the hydrothermal wave instability mechanism of moderately large-Prandtl-number liquid. The pool rotation influences the basic flow and enables the disturbances to receive energy in different ways. When the system rotation is initiated, the axisymmetric steady thermocapillary flow becomes unstable typically by a hydrothermal wave instability (type I) for all aspect ratios. The slow rotation destabilizes the basic flow whereas the fast rotation stabilizes it when the Taylor number exceeds a certain threshold value. For large Taylor numbers, two types of flow instabilities (types II and III) occur in sequence with increasing aspect ratio. While type IV arises in a range of Taylor number between those of the type I and type II for the layer of A = 0.25. The critical Reynolds numbers for the onset of the later three oscillatory bifurcations all increase with increasing Taylor number.

The influence of surfactants on thermocapillary flow instabilities in low Prandtl melting pools

Flows in low Prandtl number liquid pools are relevant for various technical applications and have so far only been investigated for the case of pure fluids, i.e., with a constant, negative surface tension temperature coefficient ∂γ/∂T. Real-world fluids containing surfactants have a temperature dependent ∂γ/∂T &amp;gt; 0, which may change sign to ∂γ/∂T &amp;lt; 0 at a critical temperature Tc. Where thermocapillary forces are the main driving force, this can have a tremendous effect on the resulting flow patterns and the associated heat transfer. Here we investigate the stability of such flows for five Marangoni numbers in the range of 2.1 × 106 ≤ Ma ≤ 3.4 × 107 using dynamic large eddy simulations, which we validate against a high resolution direct numerical simulation. We find that the five cases span all flow regimes, i.e., stable laminar flow at Ma ≤ 2.1 × 106, transitional flow with rotational instabilities at Ma = 2.8 × 106 and Ma = 4.6 × 106, and turbulent flow at Ma = 1.8 × 107 and Ma = 3.4 × 107.

Three‐dimensional unsteady thermocapillary flow under rotating magnetic field

AbstractUnder a rotating magnetic filed (RMF), the instability of thermocapillary flow and its evolution with increasing Marangoni number (Ma) for semiconductor melt (Pr = 0.01) in a floating liquid bridge model (As = 1) are investigated numerically. Under 5 mT RMF, the thermocapillary flow is steady and axisymmetric with Ma &lt; Mac, and the critical Marangoni number Mac for convection instability is 29.5, which is obtained by the direct numerical simulation. When the Ma is a little bit beyond the Mac, the thermocapillary flow loses stability to become a three‐dimensional rotating oscillatory convection, and a periodic oscillation is confirmed by the fast Fourier transform analysis, the oscillatory main frequency decays with increasing Ma. Under 1 mT–6 mT RMF, the Mac increases roughly with the magnetic strength except the Mac at 4 mT, where the corresponding change of flow mode after the instability is observed. The oscillatory convection occurs with a smaller Ma in the RMF than that without magnetic field. In addition, no instability toward a three‐dimensional steady convection, which is the state of thermocapillary flow without magnetic field after the first instability, is observed under the RMF.

Instability of thermocapillary flow in liquid layers under microgravity

The instability of the thermocapillary flow in liquid layers is studied in the present paper using the linear stability analysis. Based on the two-dimensional steady flow state, the three-dimensional disturbance with a wave number in the spanwise direction is considered. The effects of the aspect ratio and free surface shape of the liquid layer on the flow instability are studied, and the results are compared with the case with the two-dimensional disturbance.

Thermocapillary flow instabilities in an annulus under microgravity — results of the experiment magia

We investigated thermocapillary flow in an annular gap with outer heater container of radius R 1 = 40 mm and inner cooled cylinder of R 2 = 20 mm and with an adjustable height h, 2.5 ≤ h ≤ 20 mm. The gap was filled flat up to the rim with the 0.65 cSt silicone oil hexamethyldisiloxane (Prandtl number Pr = 6.7). The temperature differences ΔT, 0 K ≤ ΔT ≤ 40 K between outer and inner wall generated thermocapillary flow in the free upper surface and various flow structures have been observed under microgravity. We identified hydrothermal waves for small h and more complicated oscillations for larger h. For small h and small ΔT the multiroll structure was visible via IR-images of the free surface: concentric steady convection rolls with the same sense of rotation, embedded into the main thermocapillary roll. We measured the critical Marangoni number Ma c for the transition to time-dependent flow in the aspect ratio range A = h/(R 1 − R 2), 0.125 ≤ A ≤ 1, where it was virtually constant Ma c ≈ 5 · 10 4. We report and discuss a steady temperature asymmetry, changing its direction from time to time, as recorded by the IR-camera. This symmetry breaking is most probably due to slowly changing residual acceleration in the satellite.

Diffuse-interface modelling of thermocapillary flow instabilities in a Hele-Shaw cell

In this paper we present the results of a diffuse-interface model for thermocapillary or Marangoni flow in a Hele-Shaw cell. We use a Galerkin-type spectral element discretization, based on Gauss–Lobatto quadrature, for numerical implementation of the governing equations resulting from the diffuse-interface model. The results are compared to classical results for a linear and circular fixed interface. It is found that the diffuse-interface solution converges to the classical solution in the sharp-interface limit. The results are sufficiently accurate if the interfacial thickness is only small compared to the size of the thermocapillary boundary layer, even if the interfacial thickness used is much larger than the real interfacial thickness. We also consider freely movable interfaces with a temperature gradient perpendicular to the interface. It will be shown that this situation can lead to a destabilizing Marangoni convection.

Hydrodynamical instabilities of thermocapillary flow in a half-zone

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.