Cylindrical Liquid Bridge Research Articles

Inertial capture of aerosol particles using liquid bridges in cross flow

We investigate the inertial capture of aerosol particles, suspended in a laminar cross flow between parallel plates, by stationary cylindrical liquid bridges. We track the motion and fate of individual aerosol droplets (3–9 μm diameter) suspended in a flow moving across a single liquid bridge or two-bridge systems, using high-speed imaging under a microscope. First, we measure the capture efficiency by inertial impaction of an isolated liquid bridge collector as a function of the aerosol particles Stokes number and show that it agrees with the efficiency calculated for rigid particles moving in a potential flow past a solid cylinder. We then consider two-bridge systems, that is two parallel cylindrical collectors oriented perpendicular to the flow, but offset from each other. We investigate the effect that the upstream collector has on the effective capture efficiency of the downstream test collector, depending on the transverse offset and Stokes number. The offset is the relative displacement of the downstream collector normal to its axis and in the direction perpendicular to the freestream flow. For small values of the transverse offset, smaller than the radius of the collector, small (zero) effective efficiencies are measured for the test collector, due to a partial (complete) shielding effect by the upstream collector. The effective efficiency thus increases from zero with increasing transverse offset, above a threshold value for complete shielding. As the transverse offset increases further, effective efficiencies larger than the efficiency of an isolated bridge are measured for the test collector, showing enhanced capture due to the redistribution of particles moving past the upstream collector. The relative increase in the inertial capture efficiency is significant for Stokes numbers around (but below) unity, 0.45≲St≲0.9, with relative gains as large as approximately 50%. There is an optimum offset displaying maximum effective efficiency and larger offset values lead to a slight decrease in measured efficiencies. These results demonstrate the importance of the relative arrangement of individual collectors on the overall inertial capture efficiency of aerosol particles and suggest that an array of bridges in a slanted configuration (with optimal transverse offsets) could be used as a high-performance filtration system for aerosol particles.

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.

Dynamics of elastoviscoplastic filament stretching.

We study theoretically the stretching dynamics of a yield stress material that exhibits both elastic and viscoplastic behavior. The material is confined between two coaxial disks, forming initially a cylindrical liquid bridge and then a neck when the disks are pulled apart. The material follows the Saramito-Herschel-Bulkley constitutive model and yields according to the von Mises criterion. We find that an elongated thin neck is formed when elasticity prevails, connecting the upper and lower parts of the filamentous bridge. This neck has been observed in breakup experiments of yield stress bridges, but this is the first theoretical study that predicts it. Earlier numerical and theoretical studies of filament stretching of yield stress materials failed to do so, because they excluded elasticity from the constitutive model they used in the simulations. Our results indicate that increasing elasticity leads to shorter pinching times and filament length than the viscoplastic case. This is caused by the fact that larger areas of the filament remain unyielded, while they undergo small deformation even before yielding, and only the remaining smaller yielded areas carry the burden of visible deformation. Our findings suggest that the value of the yield strain, defined as the ratio of the yield stress to the elastic modulus, should be used with caution to determine whether elastic effects will affect the filament stretching procedure or not.

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.

Oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids

A series of three-dimensional (3D) transient simulations are carried out to study the onset and features of oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids (Pr = 0.01) with unit aspect ratio (As). The critical conditions for different volume ratios on ground and under microgravity are determined. It turns out that the threshold of the second bifurcation is more sensitive to volume ratio than the first bifurcation. The thresholds in deformed liquid bridges are usually higher than those in cylindrical liquid bridges. The critical azimuthal wavenumber increases from 2 to 3 with volume ratio. Gravity affects the flow through three factors. Compared with the hydrostatic deformation and heating orientation, buoyancy is marginal to the flow transition. It only affects the threshold and the deviation caused by buoyancy is within 2%. The effects of gravity on the critical conditions are dependent on volume ratio. Except for the threshold, gravity may also affect the critical azimuthal wavenumber and the critical oscillation mode. 3D oscillatory flow regimes with two different oscillation modes are present. A new type of “Rocking” mode is discovered in the present study.

Global Linear Stability Analysis of Thermo-solutal Marangoni Convection in a Liquid Bridge Under Zero Gravity

Marangoni convection is induced by the variation of surface tension along a free surface, which depends not only on temperature but also concentration. However, the onset of thermo-solutal Marangoni convection in a liquid bridge system is still unknown. Here, we perform a global linear stability analysis to determine the theoretical onset of the Marangoni convection occurring in the half-zone liquid bridge of the floating zone method of SixGe1−x crystal growth. The cylindrical liquid bridge is heated from the bottom and the highest Silicon concentration is on the top. The thermal and solutal Marangoni forces are in the same direction in this configuration. The stability diagram of the axisymmetric base flow is obtained by solving the large-scale eigenvalue problem using a Jacobian-free Arnoldi method. Oscillatory disturbance patterns appear with different azimuthal wavenumbers for unstable eigenmodes. The present linear stability analysis results explain our previous numerical simulation results.

Characterization of the thermal and solutal Marangoni flows of opposite directions developing in a cylindrical liquid bridge under zero gravity

Numerical simulations of the thermo-solutal Marangoni convection developing in a Si–Ge liquid bridge of a floating-zone system have been performed under zero gravity. Half of the liquid bridge was considered as the three-dimensional (3D) computational domain. In this system, the solutal Marangoni convection develops in the direction opposite to the thermal Marangoni convection along the free surface in the bridge, i.e., the thermal Marangoni number, MaT, is negative and the solutal Marangoni number, MaC, is positive. Since the SiGe melt is a low-Prandtl number (Pr = 6.37 × 10−3) and high-Schmidt number (Sc = 14.0) liquid, the temperature field is almost independent of the convective flow and the concentration field determines the transport structures. When MaC is larger than −MaT, the concentration pattern is steady and two-dimensional (2D) axisymmetric. When MaC is smaller than −MaT, we predict two kinds of flow transitions with the increase in |MaT|. If MaC is sufficiently large (MaC ≳ 530), as |MaT| increases, the flow changes from a 2D-steady pattern to a 3D-chaotic behavior at moderate |MaT| (1050 ≲ |MaT| ≲ 2800). We also predict that a second transition and an oscillatory rotating flow occur as |MaT| increases further. The flow becomes 3D-steady at smaller MaC (MaC ≲ 360) with no transition, and the azimuthal wavenumber (m) decreases with increasing |MaT|. Furthermore, the thermo-solutal Marangoni convection in this system can be suppressed almost completely when MaC is approximately equal to −MaT (MaC ≈ −MaT) and the flow becomes periodically stable with weak fluctuations.

A method for measuring the interfacial tension for density-matched liquids

We propose a method to measure the interfacial tension characterizing the interface between two immiscible liquids of practically the same density. In this method, a cylindrical liquid bridge made of one the liquids is vibrated laterally inside a tank filled with the other. The first resonance frequency is determined and equated to the first eigenfrequency of the m=1 linear mode to infer the interfacial tension value. The method does not involve the density jump across the interface. Therefore, its accuracy is affected neither by the smallness of the Bond number nor by errors of the density difference. The experimental setup is relatively simple, and the procedure does not use image processing techniques. The results satisfactorily agree with those measured by TIFA-AI (Theoretical Fitting Image Analysis-Axisymmetric Interfaces) for the same liquid bridges when the density difference is sufficiently large for TIFA-AI to be valid. We conduct numerical simulations of the Navier-Stokes equations to determine the best parameter conditions for the proposed method. The transfer function characterizing the frequency response of the fluid configuration is measured in some experiments to quantify non-linear effects and to study the role played by the outer bath vibration.

Semiphenomenological model to predict hardening of solid–liquid–liquid systems by liquid bridges

In this study, we developed two semiphenomenological models to quantitatively predict the rheological properties of capillary suspensions. These models can be used to estimate the critical volume fraction of an additional immiscible fluid above which liquid bridges have formed between all neighboring particles, causing gelation. The models can also be used to estimate the numerical value of the yield stress. This model was derived from the mass balance between the net volume of the liquid bridge and the volume fraction of the secondary fluid assuming monodisperse particles and a cylindrical liquid bridge. The yield stress was constructed based on Rumpf’s equation. The calculation results demonstrated good agreement with the experimental data. We also used the developed model to investigate the effect of particle size on the yield stress. This model qualitatively described the experimental data and reference data, and revealed the mechanism of the particle size dependence of the yield stress. The model applies to capillary suspensions in the pendular state.

Effect of Ambient Conditions on Flow and Heat Transfer in a Liquid Bridge

The present study focuses on thermocapillary convection in liquid bridges of high-Prandtl-number fluid (5 cSt silicone oil) under microgravity environment and varying ambient air conditions. The geometry has a cylindrical liquid bridge surrounded by ambient air and is coaxially placed between two differentially heated rods of the same diameter. The commercial software ANSYS Fluent 16.0, based on the finite volume method, is used to solve the governing equations in both the liquid and air domains. An axisymmetric computation is carried out in the liquid bridge by varying control parameters such as the temperature difference between the ends, the ambient air temperature, and the velocity of the ambient air. For passive ambient air around a liquid bridge, the effect of varying the air temperature is considered on the surface tension gradient along the interface, causing thermocapillary flow from the hot to the cold end. To account for the effect of ambient air motion, the velocity values of the axial airstream for codirected and counterdirected flows are chosen to be less than the Marangoni velocity. The study confirms that the percentage change in the maximum local surface velocity near hot corner is found to be around 20% due to the effect of the ambient air velocity as compared to that of stationary air.

Origin of particle accumulation structures in liquid bridges: Particle–boundary-interactions versus inertia

The formation of particle-accumulation structures in the flow in a cylindrical liquid bridge driven by the thermocapillary effect is studied with the aim of determining the physical mechanism which forms the structures. The flow is modeled using the incompressible Navier–Stokes and energy equations with the assumption of constant fluid properties except for surface tension, which is assumed to depend linearly on temperature. Different models for the motion of small non-interacting spherical particles at low concentration are employed, taking into account particle inertia due to density differences between fluid and particles and the restricted particle motion near the boundaries of the flow domain. Attention is focused on differences in formation time between particle-accumulation structures arising as a result of inertial effects only, particle–boundary-interaction effects only, and a combination of the two.

Cylindrical liquid bridges

Cylindrical liquid bridges

Using Parabolic Flights to Examine Quantitatively the Stability of Liquid Bridges under Varying Total Body Force

Liquid bridges were flown aboard a Boeing 727-200 aircraft in a series of parabolic arcs that produced multiple periods of microgravity. During the microgravity portion of each arc, geff, the effective total body acceleration due to external forces became negligibly small so that cylindrical liquid bridges could be suspended across two coaxial support posts. Near the bottom of each arc, geff slowly increased to a maximum of 1.84g, causing the liquid bridges to deform and in some cases collapse. Although the physics of liquid bridges subject to varying total body force is well-established and has been analyzed extensively both theoretically and experimentally, specific hardware was designed to vary geff in a precise way that overcomes the gravity-related limitations and high g-jitter associated with parabolic flights. Bridge-stability was examined for axial and lateral orientations with respect to geff by measuring the slenderness ratio as a function of Bond number at the instant of bridge collapse. Results exhibit remarkable agreement with theory as well as with the experimental results obtained in a magnetic levitation-based experiment. The parabolic flight method offers technical originality and provides experimental insights for researchers in the microgravity field. Here we present hardware development, experimental considerations, and results, and demonstrate that parabolic flight is a viable alternative to extant techniques for quantitative experiments on fluids.

Pseudo-Gaussian cylindrical acoustical beam – Axial scattering and radiation force on an elastic cylinder

Making use of the addition theorem for the cylindrical wave functions and the complex-source-point method in cylindrical coordinates, an exact solution to the Helmholtz equation is derived, which corresponds to a tightly focused (or collimated) cylindrical quasi-Gaussian beam with arbitrary waist. The solution is termed “quasi-Gaussian” to make a distinction from the standard Gaussian beam solution obtained in the paraxial approximation. The advantage of introducing this new solution is the efficient and fast computational modeling of tightly focused or quasi-collimated cylindrical wave-fronts depending on the dimensionless waist parameter kw0, where k is the wavenumber of the acoustical radiation. Moreover, a closed-form partial-wave series expansion is obtained for the incident field, which has the property that the axial scattering (i.e. along the direction of wave propagation) and the axial acoustic radiation force (which is a time-averaged quantity) on a cylinder, can be calculated without any approximations in the limit of linear acoustical waves in a nonviscous fluid. Examples are found where the extinction in the radiation force function plot is found to be correlated with conditions giving reduction of the backscattering from an elastic cylinder. Those results are useful in beam-forming design, particle manipulation in acoustic tweezers operating with focused cylindrical beams, and the prediction of the scattering and radiation forces on a cylindrical particle or liquid bridges.

On the stability of a cylindrical liquid bridge

The problem of the stability of a liquid bridge stretched between parallel plates with a wetting contact angle of 90◦ is revisited. A closed form expression is derived for the height of the bridge, in terms of its volume, within which a cylindrical and various types of unduloidal equilibrium configurations can exist. For a given volume of the liquid and specified height of the bridge, the lateral surface of a uniform cylindrical bridge is smaller than the surface area of any unduloidal equilibrium shape. The lateral surface of the unduloidal shapes increases with the increase in the number of their inflection points. The force required to keep the bridge in equilibrium is evaluated in each case. All unduloidal equilibrium configurations are unstable, the only stable configuration being that of a cylindrical bridge whose height is less than one-half of its circumference. A lower bound estimate is also derived based on a simple energy consideration. The stretching force required for the equilibrium at the onset of instability is compared with its upper bound estimate. For a given height of the bridge, the force required to keep a cylindrical bridge in equilibrium is greater than the force required for equilibrium of any unduloidal configuration of the same height. The opposite is true for the capillary pressure.

Different particle-accumulation structures arising from particle–boundary interactions in a liquid bridge

The formation of particle-accumulation structures in the flow in a cylindrical liquid bridge driven by the thermocapillary effect is studied. The problem is modeled as an incompressible fluid seeded with a low concentration of small spherical particles which are assumed to have a negligible effect on the flow. The particle motion is modeled by pure advection except for small regions at the free surface and walls where the particle interaction with the free surface and walls is modeled by a hard-wall potential which restricts the motion of the particle within some interaction length from the boundaries. The model yields a wide range of particle-accumulation structures when varying the relative interaction length from 5×10-3 to 3×10-2. It is found that in many cases each particle need undergo no more than a handful of returns to the free surface for particle-accumulation structures to arise. It is also found that the shape of the particle-accumulation structures changes qualitatively at certain critical interaction lengths.

Coherent particulate structures by boundary interaction of small particles in confined periodic flows

Rapid demixing of small density-matched particles in the incompressible flow in a cylindrical thermocapillary liquid bridge by the mechanism of particle–boundary interaction is studied. The flow considered is an azimuthally traveling hydrothermal wave which is periodic in time and azimuth. The length scale of the particles relative to that of the liquid bridge ranges from 7×10−4 to 4×10−2. The mechanism of demixing is based on the finite size of a particle, which otherwise perfectly follows the flow, and a specular reflection of the particle from the free-surface bounding the domain. To enable long-time accurate predictions of particle trajectories we consider a model flow which reflects the characteristic features of the hydrothermal wave for which particle accumulation has been detected in experiments. We find that, depending on the size of the particles, the particle–free-surface interaction causes particle attraction to a closed spiral or to the surface of a closed spiral tube. Even in the absence of spiral accumulation patterns very small particles can be completely removed from certain regions of the domain. All structures found are time periodic, exactly as is the underlying flow. The particle-accumulation structures found and the dynamics of the demixing agree qualitatively with experimental data. In the specular reflection model employed, reflected particles are restricted to move on a stream surface which becomes increasingly fragmented as the particles repeatedly interact with the free-surface. Repeated particle–free-surface interactions finally lead to a complex geometry of the stream surface on which particles are restricted to move. The results obtained can explain the line-like and tubular particle accumulation and the characteristic particle-depletion zones observed in experiments.

The dynamics of three-dimensional liquid bridges with pinned and moving contact lines

AbstractLiquid bridges with moving contact lines are relevant in a variety of natural and industrial settings, ranging from printing processes to the feeding of birds. While it is often assumed that the liquid bridge is two-dimensional in nature, there are many applications where either the stretching motion or the presence of a feature on a bounding surface lead to three-dimensional effects. To investigate this we solve Stokes equations using the finite-element method for the stretching of a three-dimensional liquid bridge between two flat surfaces, one stationary and one moving. We first consider an initially cylindrical liquid bridge that is stretched using either a combination of extension and shear or extension and rotation, while keeping the contact lines pinned in place. We find that whereas a shearing motion does not alter the distribution of liquid between the two plates, rotation leads to an increase in the amount of liquid resting on the stationary plate as breakup is approached. This suggests that a relative rotation of one surface can be used to improve liquid transfer to the other surface. We then consider the extension of non-cylindrical bridges with moving contact lines. We find that dynamic wetting, characterized through a contact line friction parameter, plays a key role in preventing the contact line from deviating significantly from its original shape as breakup is approached. By adjusting the friction on both plates it is possible to drastically improve the amount of liquid transferred to one surface while maintaining the fidelity of the liquid pattern.

Heat Transfer Through the Interface and Flow Regimes in Liquid Bridge Subjected to Co-Axial Gas Flow

We present results of an extensive numerical study on the thermocapillary (Marangoni) convection and a heat transfer through the interface in a liquid bridge of Pr = 68. The geometry of the physical problem is a cylindrical and non-deformable liquid bridge concentrically surrounded by an annular gas channel under conditions of zero gravity. The gas flow is co- or counter-directed with respect to the Marangoni flow. The forced gas flow along the interface provides two actions: via shear stresses and heat exchange. Usually the cooling of the interface enhances the flow while the heating slows down. This general trend may not hold when shear and thermocapillary stresses are comparable. The results show that when gas enters from the cold side the heat transfer through the interface is considerably larger than that when gas enters from the hot side.

Preparation of JEREMI Experiment: Development of the Ground Based Prototype

This study has been performed in the frame of preparing the space experiment JEREMI (Japanese and European Research Experiment on Marangoni Instabilities). The use of forced coaxial gas flow is proposed as a way to stabilize the Marangoni convection in liquid bridges, which might have important technological applications in the floating zone technique. A new set-up is under development and all sub-systems have passed severe tests. Here we present the design of this set-up and preliminary results of experiments for shear-driven two-phase flows in a confined volume of liquid under conditions of normal gravity. The geometry corresponds to a cylindrical liquid bridge concentrically surrounded by an annular gas channel with external solid walls. Gas enters into the annular duct, flows between solid walls and upon reaching the liquid zone entrains initially quiescent liquid. The test liquids are ethanol, n-decane and 5 cSt silicone oil, which have different degrees of viscosity and of volatility. The gas flow along the interface strongly enhances the evaporation and, correspondingly, affects the interface shape. Silhouette measurements are used for optical determination of the interface shape. From the digital images the variation of the liquid volume as a function of flow rate is calculated.