Thermocapillary Liquid Bridges Research Articles

Experimental study on coherent structures by small particles suspended in high aspect-ratio (Γ=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma =$$\\end{document} 2.5) thermocapillary liquid bridges of high Prandtl number

In time-dependent traveling-wave convection, it has been confirmed that small particles of low Stokes number (St≪\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{St}\\ll$$\\end{document} 1) form three-dimensional closed structures known as the particle accumulation structures (PASs). We focus on coherent structures formed in high aspect-ratio thermocapillary liquid bridges (Γ=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma =$$\\end{document} 2.5), where oscillatory convection due to the so-called hydrothermal wave instability of mHTW=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{\ extrm{HTW}}=$$\\end{document} 1 stably emerges, and carry out experimental explorations into the volume ratio dependence of the emergence of coherent structures. Variation in the volume ratio induces the presence of coherent structures with different azimuthal wave numbers, that is, different spatial structures. Among the particles added to the liquid bridge, we perform spatio-temporal tracking of particles to quantify the correlation between the behavior of particles and the spatial structures of coherent structures with different rational wave numbers.

Coherent structures formed by small particles in traveling-wave flows.

We experimentally verify the "phase locking model," which describes the formation of one-dimensional coherent structures by low-Stokes-number particles as proposed by Pushkin etal. [Phys. Rev. Lett. 106, 234501 (2011)0031-900710.1103/PhysRevLett.106.234501] in thermocapillary liquid bridges: When the particles form the coherent structures in time-periodic flows, the synchronous coupling of the toroidal vortex and the azimuthally traveling wave is achieved for a range of azimuthal wave number m. We further propose coherent structures of rational wave numbers in the azimuthal direction and unveil those experimentally.

Scaling and modeling of the heat transfer across the free surface of a thermocapillary liquid bridge

PurposeThis paper aims to derive a reduced-order model for the heat transfer across the interface between a millimetric thermocapillary liquid bridge from silicone oil and the surrounding ambient gas.Design/methodology/approachNumerical solutions for the two-fluid model are computed covering a wide parametric space, making a total of 2,800 numerical flow simulations. Based on the computed data, a reduced single-fluid model for the liquid phase is devised, in which the heat transfer between the liquid and the gas is modeled by Newton’s heat transfer law, albeit with a space-dependent Biot function Bi(z), instead of a constant Biot number Bi.FindingsAn explicit robust fit of Bi(z) is obtained covering the whole range of parameters considered. The single-fluid model together with the Biot function derived yields very accurate results at much lesser computational cost than the corresponding two-phase fully-coupled simulation required for the two-fluid model.Practical implicationsUsing this novel Biot function approach instead of a constant Biot number, the critical Reynolds number can be predicted much more accurately within single-phase linear stability solvers.Originality/valueThe Biot function for thermocapillary liquid bridges is derived from the full multiphase problem by a robust multi-stage fit procedure. The derived Biot function reproduces very well the theoretical boundary layer scalings.

High-Prandtl-number thermocapillary liquid bridges with dynamically deformed interface: effect of an axial gas flow on the linear stability

The linear stability of the axisymmetric steady flow in a thermocapillary liquid bridge made from 2-cSt silicone oil $({Pr}=28)$ is investigated numerically. The liquid bridge is heated either from above or below and exposed to an axial air flow which is confined to a concentric tube surrounding the bridge. At the annular inlet, the air flow is fully developed and has the same temperature as the adjacent support rod. Using an extended Oberbeck–Boussinesq approximation in which the density of both fluids depends linearly on the temperature in all equations, critical thermocapillary Reynolds numbers are obtained depending on the strength of the imposed axial air flow. The critical conditions are sensitive with respect to the direction of a weak air flow, because the air flow changes the plateau value of the interfacial temperature midway between the hot and cold ends. For stronger air flow the critical thermocapillary Reynolds number almost saturates at moderate values. Throughout, the instability arises as a hydrothermal wave with the gas phase being passive. The dynamic interface deformations for axisymmetric flow caused by the thermocapillarity flow in the liquid and by the stresses from the air flow are considered separately. Apart from turning points of the critical curve, the impact of dynamic surface deformations on the critical thermocapillary Reynolds number is moderate.

Flow instability in high-Prandtl-number liquid bridges with fully temperature-dependent thermophysical properties

The axisymmetric steady two-phase flow of a differentially heated thermocapillary liquid bridge in air and its linear stability is investigated numerically, taking into account dynamic interfacial deformations in the basic flow. Since most experiments require a high temperature difference to drive the flow into the three-dimensional regime, the temperature dependence of the material properties must be taken into account. Three different models are investigated for a high-Prandtl-number thermocapillary liquid bridge with nominal Prandtl number ${\textit {Pr}}=28.8$ : the Oberbeck–Boussinesq (OB) approximation, a linear temperature dependence of all material properties and a full nonlinear temperature dependence of all material properties. For all models, critical Reynolds numbers are computed as functions of the volume of the liquid bridge, its aspect ratio, its dimensional size and as a function of the strength of a forced axial flow in the ambient air. Under most circumstances the OB approximation overpredicts and the linear model underpredicts the critical Reynolds number, compared with the model based on the full temperature dependence of the material properties. Among the main influence factors are the proper selection of the reference temperature and, at larger temperature differences, the temperature dependence of the viscosity of the liquid.

Dynamic mode decomposition of thermocapillary convection in GaAs melt liquid bridge between unequal ends

The effects of the radius ratio on the stability of thermocapillary convection in the GaAs melt (Pr=0.068) liquid bridge, situated between unequal ends, are studied by three-dimensional direct numerical simulation. The simulated results reveal that, for the first time, the GaAs melt flow transforms from a stable axisymmetric flow into a three-dimensional steady flow through the first bifurcation at 0.5≤Γr≤0.9. The bifurcation phenomenon was previously common in the range of small Prandtl numbers(0.001≤Pr≤0.057). Additionally, we observe that the stability of the thermocapillary liquid bridge is initially weakened and then slowly improved as the radius ratio Γr decreases. The second bifurcation occurs with the further increase of Ma, and the flow becomes oscillatory. Remarkably, two different oscillation forms are identified, depending on the difference in the degree of symmetry of the flow field structure during its oscillation process. Once the flow reaches a stable periodic oscillation, the Dynamic Mode Decomposition (DMD) technique is used to study the inherent spatio-temporal evolution of the temperature field. By combining the spectrum obtained from the Fast Fourier Transform (FFT), we provide evidence for the existence of mixed forms with different azimuthal wavenumbers from a dynamic perspective, intuitively displaying the different components of the oscillation and examining the contribution of different modes to the oscillation. The dynamic model analysis demonstrates that oscillation form I is a mixed form with m = 1 and m = 2, while oscillation form II belongs to the form where m = 2 is absolutely dominant.

Instability of axisymmetric flow in thermocapillary liquid bridges: Kinetic and thermal energy budgets for two-phase flow with temperature-dependent material properties

AbstractIn numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.

Effect of Prandtl number on the flow instabilities in thermocapillary liquid bridges between two coaxial disks with different radii

The effect of Prandtl number on the instability of thermocapillary liquid bridges has been extensively studied, but the vast majority of efforts were put into typical cylindrical liquid bridge between two equal ends. The crystal growth practices have the propensity to form liquid bridges between unequal disks, thus the dependence of instability patterns and underlying mechanisms on the radius ratio is crucial but remains unclear. By fixing the radius ratio at Γr=0.5, we systematically explore the effects of Prandtl number and heating strategy, namely the upper heating and the bottom heating settings, on flow instabilities in thermocapillary liquid bridges between coaxial disks with different radii under microgravity through linear stability analysis based on spectral element method. In contrast to typical cylindrical liquid bridge (Γr=1), the present analysis indicates that the instability is always an oscillatory bifurcation even for small Prandtl number Pr=0.001 in the bottom heating scenario. What's more, the instability remains a stationary bifurcation and the critical Marangoni number experiences significant increase with the rise of Pr in the region of 0.001≤Pr≤1.2 for the upper heating situation. For all liquid bridges, the energy analysis results show that the instability under different Prandtl numbers can be roughly divided into three types.

Instability mechanisms of thermocapillary liquid bridges between disks of unequal radii

In this paper, we explore thermocapillary liquid bridges between two disks of unequal radii with Prandtl numbers Pr of 0.0258 (mercury) and 0.068 (gallium arsenide) to gain insight into the underlying instability mechanism. In the context of Legendre's spectral element method, we determine critical conditions via linear stability analysis and then identify the instability mechanism through energy analysis. For the mercury bridge (Pr = 0.0258), our analysis suggests that the flow instability undergoes an oscillatory bifurcation for radius ratios in the range of 0.5 ≤ Γr ≤ 0.66. In particular, we found three transitions between two-dimensional steady axisymmetric flow and three-dimensional stationary flow by further increasing the radius ratio to 0.73 ≤ Γr ≤ 0.76. For the gallium arsenide liquid bridge (Pr = 0.068), the instability is always an oscillatory bifurcation in the whole computational interval. Furthermore, our observations identify six instability modes with different mechanisms. All instability modes in the mercury bridge (Pr = 0.0258) are purely hydrodynamic, but the thermocapillary mechanism cannot be ignored in the gallium arsenide liquid bridge (Pr = 0.068) because of the enhanced Pr effect.

Effect of Prandtl Number on the Flow Instabilities in Thermocapillary Liquid Bridges between Two Coaxial Disks with Different Radii

Effect of Prandtl Number on the Flow Instabilities in Thermocapillary Liquid Bridges between Two Coaxial Disks with Different Radii

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.

Finite-size coherent particle structures in high-Prandtl-number liquid bridges

Clustering of small rigid spherical particles in particle accumulation structures (PAS) of various shapes is found with highly resolved numerical simulations in a thermocapillary liquid bridge at a Prandtl number of 68. The intricate Kolmogorov--Arnold--Moser (KAM) structure of the fluid flow is unraveled for several Reynolds numbers for flows periodic in time and space. For these flows, a rich variety of periodic and quasiperiodic attractors for particles nearly density-matched to the fluid is found near closed streamlines and KAM tori approaching the free surface closely. A large parametric study finds the PAS dependence on particle size, particle-to-fluid density ratio, and Reynolds number.

Thermal-flow patterns of [formula omitted] in thermocapillary liquid bridges of high aspect ratio with free-surface heat transfer

We investigate the thermal-flow patterns induced by the thermocapillary effect and their transition conditions in half-zone liquid bridges of high Prandtl number fluid; the effect of heat transfer between the liquid and the ambient gas through the liquid-bridge free surface is considered. We especially focus on the standing-wave-type oscillations after the transition onset, which were observed more frequently in the microgravity experiments (Matsugase et al., Int. J. Heat Mass Trans. (2015)) than the ground experiments, in a tall or high-aspect-ratio liquid bridge. We conduct a series of three-dimensional numerical simulations with the straight liquid bridge against a range of Biot number, and reproduce a pair of thermal waves of azimuthal wave number m=1 propagating in opposite azimuthal directions under large Biot-number conditions by increasing the thermocapillary effect. Through a series of ground experiments with the liquid bridge exposed to the forced flow of the ambient gas, we illustrate the thermal-flow patterns of m=1 and their transition conditions in terms of the Reynolds number for the forced flow of the ambient gas. We succeed in reproducing by both of experimental and numerical approaches the standing-wave-type oscillatory flows with ‘x’-shaped low temperature bands on the free surface, which had been observed only in long-term microgravity experiments. We illustrate that the ‘x’-shaped structure is realized by the higher azimuthal modal structures of the oscillatory convection.

Experimental Study on Coherent Structures by Particles Suspended in Half-Zone Thermocapillary Liquid Bridges: Review

Coherent structures by the particles suspended in the half-zone thermocapillary liquid bridges via experimental approaches are introduced. General knowledge on the particle accumulation structures (PAS) is described, and then the spatial–temporal behaviours of the particles forming the PAS are illustrated with the results of the two- and three-dimensional particle tracking. Variations of the coherent structures as functions of the intensity of the thermocapillary effect and the particle size are introduced by focusing on the PAS of the azimuthal wave number m=3. Correlation between the particle behaviour and the ordered flow structures known as the Kolmogorov–Arnold—Moser tori is discussed. Recent works on the PAS of m=1 are briefly introduced.

Coherent Particle Structures in High-Prandtl-Number Liquid Bridges

Clustering of small rigid spherical particles into particle accumulation structures (PAS) is studied numerically for a high-Prandtl-number (Pr = 68) thermocapillary liquid bridge. The one-way-coupling approach is used for calculation of the particle motion, modeling PAS as an attractor for a single particle. The attractor is created by dissipative forces acting on the particle near the boundary due to the finite size of the particle. These forces can dramatically deflect the particle trajectory from a fluid pathline and transfer it to certain tubular flow structures, called Kolmogorov–Arnold–Moser (KAM) tori, in which the particle is focused and from which it might not escape anymore. The transfer of particles can take place if a KAM torus, which is a property of the flow without particles, enters the narrow boundary layer on the flow boundaries in which the particle experiences extra forces. Since the PAS obtained in this system depends mainly on the finite particle size, it can be classified as a finite-size coherent structure (FSCS).

Flow instability in high‐Prandtl‐number thermocapillary liquid bridges exposed to a coaxial ambient gas stream

AbstractNumerical studies have been conducted in order to investigate the linear stability of thermocapillary flow in a liquid bridge suspended between two coaxial, solid rods of different temperatures. We address the full two‐phase flow problem in which both the rods and the liquid bridge are placed in a concentric cylinder and exposed to a nominally axial flow field. Two features distinguish our linear stability analysis from earlier investigations. Firstly, our computational model allows the liquid–gas interface of the basic flow to be dynamically deformable, being affected by the basic temperature and velocity fields. Beyond that, our model accounts for the full temperature dependence of all thermophysical properties of the fluids. Since the problem involves many parameters, we consider a liquid bridge of 5 cSt silicone oil (Pr = 68) and focus on the effect of the ambient gas (Argon) on the flow instability that leads to the onset of three‐dimensional flow. Critical modes are analyzed for a length‐to‐radius aspect ratio of Γ = 1.

Experimental study of coherent structures of finite-size particles in thermocapillary liquid bridges

We experimentally investigate the formation of coherent structures by finite-size particles in thermocapillary liquid bridges. Special attention is paid to particle accumulation structures in which suspended particles gather to form a solid-like-structure in the rotating frame of reference.

Particle accumulation in high‐Prandtl‐number liquid bridges

AbstractParticle accumulation in high‐Prandtl‐number (Pr = 68) thermocapillary liquid bridges is studied numerically. Randomly distributed small rigid non‐interacting spherical particles are found to cluster in particle accumulation structures. The accumulation is found to be caused by a finite‐particle‐size effect when the particles move close to the impermeable flow boundaries. The extra drag force experienced by a particle near the boundaries creates a dissipation in the dynamical system describing the particle motion. This causes particles to be attracted to regions in or near Kolmogorov‐Arnold‐Moser tori of the unperturbed flow field.

Finite-size Lagrangian coherent structures in thermocapillary liquid bridges

A numerical investigation shows that finite-size Lagrangian coherent structures result from bulk transport near Kolmogorov-Arnold-Moser tori and a dissipative effect that finite-size particles experience near the free surface.

Effect of interfacial heat transfer on basic flow and instability in a high-Prandtl-number thermocapillary liquid bridge

The effects of interfacial heat transfer on the basic flow and instability of thermocapillary convection in high-Prandtl-number (Pr) liquid bridges are investigated experimentally and computationally. Liquid bridges of silicone oil (Pr = 28) are formed in a gap between the lower cooled rod and the upper heated rod, both with a diameter of 5 mm, where the rods are surrounded by a cylindrical enclosure made from an acrylic block. The instability data are collected experimentally for AR = 0.30–0.50 and a wide range of (TC-Ta), where AR is the aspect ratio (= height/diameter) of the liquid bridge, TC is the cooled rod temperature, and Ta is the ambient gas temperature. The data indicate the appreciable effect of interfacial heat transfer on the instability and basic flow pattern of thermocapillary convection in liquid bridges. Each instability curve for AR = 0.35–0.50 shows a local peak of the critical temperature difference and the oscillation frequency at a certain (TC-Ta), where such a peak is associated with the transition of the azimuthal oscillation mode. The heat transfer ratio QLB/QHR is evaluated from the numerical simulation to discuss the effect of interfacial heat transfer, where QLB and QHR are the heat transfer rates at the liquid bridge surface and at the heated rod surface, respectively. It is found that the instability conditions for different AR values are well correlated with QLB/QHR. This correlation is consistent with the effect of QLB/QHR on the basic flow and temperature fields, both in the liquid bridge and in the ambient gas. The resultant change in the basic flow and temperature field inside the liquid bridge leads to the change in the onset conditions of oscillatory thermocapillary convection.