Influence of gravitational tilt on the thermocapillary convection in a non-axisymmetric liquid bridge

Fluid slosh caused by residual acceleration in microgravity is a common problem encountered in space engineering. To solve this problem, the ground-based experiment research on the influence of gravity jitter and gravitational tilt on the thermocapillary convection (TCC) transition behaviour of non-axisymmetric liquid bridge has become an important issue in microgravity fluid management. Based on a mesoscale liquid bridge experimental platform which can realize gravitational tilt, the effect of gravitational tilt on TCC by using a high-speed camera equipped with a near-focus lens and a self-developed interface image recognition package. The results show that the spatio-temporal evolution of TCC by the influence of gravitational tilt is still divided into steady and oscillatory flow. In the stable TCC, the vortex core distortion of cellular flow caused by the imbalance left and right interface curvature invites cellular flow close to the free surface, and it shrinks to the intermediate height. As gravitational tilt increases, the transverse/longitudinal velocity peaks are significantly reduced, peak velocity has been reduced by 26%–27%. Meanwhile, the longitudinal velocity gradient at the free interface increases significantly. Therefore, gravitational tilt plays an important role in improving the surface flow velocity. In the oscillatory TCC, the position of vortex core is closer to the free interface at the hot/cold corner as the periodic mutual occupation of the left and right cellular flows. The TCC is obviously inhibited due to the gravitational tilt. The critical temperature difference is increased by 25% and the onset of temperature oscillation at the hot corner is delayed by 20% compared with conventional gravity condition.


Introduction
The thermocapillary convection (TCC) has emerged as a fascinating and complex phenomenon with profound implications in various fields such as microfluidics, heat transfer, and materials science in recent years.The interplay between temperature gradients and surface tension forces gives rise to intriguing fluid flow patterns and transport phenomena, making TCC an area of significant interest in both fundamental and applied research.A large number of theoretical and experimental papers have been published over time to study the behaviour of liquid bridges in microgravity environments, but most have only considered the behaviour of axially symmetric liquid bridges [1][2][3], with only a few studies of the behaviour of non-axially symmetric liquid bridges [4,5] and of special forms of liquid bridges [6][7][8].Lowry and Steen [9] extended the application to capillary surfaces from axisymmetric liquid bridges to droplets and non-axisymmetric liquid bridges.The phenomenon of thermocapillary lubricant migration on textured surfaces illustrates that higher temperature gradients require greater angles of inclination to compensate for lubricant migration [10][11][12].Effect of surface internal energy instability due to gravitational tilt angle (GTA) on solution capillary convection.Among them, the spatio-temporal development of non-axisymmetric liquid bridges can be divided into three stages according to the change of GTA: the 'starting stage near the upper angle' , the 'stage of development towards the medium height' and the 'stage of contraction towards the lower angle' [13].There is a large pattern of instability in thermal capillary convection in the semi-floating region, which may transition from steady axisymmetric convection to steady asymmetric convection [14].Yang et al [15] used a new mass conservation level set method to perform direct numerical simulations of oscillatory TCC within a liquid bridge of a non-axisymmetric high Prandtl number (Pr is a dimensionless parameter that describes the proportionality between the dynamic viscosity and the thermal diffusivity, Pr = C p µ/λ), fluid under normal gravity to capture the microscale migration of the free surface.Zhang et al [16] utilized GTA to devise a non-axisymmetric model for liquid bridges with the aim to scrutinise the role of interfacial energy instability triggered by GTAs in the evolution of thermally buoyant capillary convective flow patterns and velocity distributions.Results reveal that the imbalance transformation in interfacial curvature brought about by GTA results in a non-axisymmetric liquid bridge morphology.In the study of liquid-bridge oscillations [17,18], an analytical expression for the velocity field is obtained using the Helmholtz decomposition.The computational procedure can be used to calculate the linear frequencies of various interface oscillations.
The asymmetry due to changes in the volume of the liquid bridge has been the subject of investigation by some researchers.Non-axisymmetric deformation and non-axisymmetric fracture of the interface can result from changes in the volume of the liquid bridge.The degradation of the maximum pressure singularity into two turning points (folds) when a small inclination angle was introduced to the liquid bridge and the isolation of solution branches were investigated by Chen and Chang [19].The results show a hysteresis jump in surface curvature and excess pressure for changes in diameter/length ratio or liquid volume.The instability leading to asymmetry at the interface when the volume of the liquid bridge is at its maximum was studied by Bezdenejnykh et al [20], but in some cases, this phenomenon was difficult to observe.The asymmetric fragmentation of a liquid column and the calculation of the stability limit of the liquid bridge in the axial gravity field rotating around the eccentric shaft were observed by Rodríguez et al [21].Numerical analysis showed that axial gravity had a greater effect on long bridges and eccentricity had a greater effect on short bridges.Wang et al [22] found that as the separation distance and liquid volume increased, the effect of gravity on the shape of the liquid bridge meniscus became more pronounced, mainly in the form of loss of vertical symmetry and upward shift of the liquid bridge fracture location.
The effects of lateral disturbances on the stability and surface behaviour of liquid bridges has been extensively reported.An analysis on the stability of lengthy liquid bridges when subjected to non-axisymmetric perturbations was conducted by Perales [23] using an asymptotic method based on the bifurcation technique.Their findings indicate that non-axisymmetric disturbances generate more significant impacts than axisymmetric disruptions.Sanz and Diez [24] studied non-axisymmetric oscillations of liquid bridges.Platform techniques have been used to obtain the resonant frequencies of bridges when lateral perturbations are applied.The stability of non-axisymmetric liquid bridges under microgravity conditions was studied by Meseguer et al [25].The effect of non-axisymmetric perturbations with small transverse accelerations and non-coaxial support disks on the stability of a near-cylindrical liquid bridge was analysed using the standard bifurcation technique and it was demonstrated that each non-axisymmetric perturbation (e.g.transverse acceleration and eccentricity) can be used as a critical axisymmetric perturbation.María et al [26] considered the problem of linear vibrations of a non-axisymmetric liquid bridge in the limit of large capillary Reynolds number (Re is a dimensionless parameter used to describe the relative strength of inertial and viscous forces in fluid flow, Re = ρuL/µ.)approximating the viscous dissipation in the Stokes boundary layer near the disk.Acero et al [27] theoretically analysed the influence of the shape of the solid support, the volume of the liquid bridge and the disturbance acting on the liquid bridge, such as axial force, lateral force and centrifugal force, on the equilibrium shape of the liquid bridge.Liang et al [28] investigated the effect of transverse vibration on the thermo-solutocapillary convection and surface behaviour in a toluene/hexane liquid bridge under microgravity conditions.The specific frequency of lateral vibrations can enhance and weaken the velocity of surface flow.The interaction between two elastic semi-infinite bodies and a given volume of intermediate liquid was investigated by Zhang et al [29].The coupling between capillary forces and surface elastic deformation was considered and the mechanical stability of the interface was examined.
Some scholars have investigated the nature of asymmetric liquid bridges by changing the structure of the supporting liquid bridge.For example, the stability of preferentially non-wetting fluids with axisymmetric and non-axisymmetric configurations between spheres was considered [30], the formation and rupture of a liquid bridge between two inhomogeneous particles was discussed as having a significant impact on the local fluid distribution and system performance [31].Consider the liquid bridge force between two spherical particles [32] or unequal sized spheres and a flat plate [33].Migration of non-axisymmetric configurations of capillary bridges with axisymmetric instability between two contact spheres is investigated [34].Ataei et al [35,36] investigated the stability of non-axisymmetric liquid bridges formed between non-parallel surfaces in terms of the balance between surface wettability and capillary forces.Zhao et al [37] found that the displacement moment is only related to the surface wettability and is independent of the asymmetric stretching velocity.Dai et al [38] discussing thermo-capillary migration found that there exists a critical angle of inclination at which thermo-capillary migration of the fluid bridge stops.Pang et al [39] effectively modelled two horizontal elongated plates to study the effects of fluid charge, surface tension and viscosity on the formation and dynamics of the bridge.Aziz and Hooman [40] studied liquid Bridges between different fibres in parallel and orthogonal structures.The results of the paper by Wang and Schiller [41] quantified the shape transformation of droplets on fibre orbitals.Sun et al [42] used a surface evolver (SE)-based numerical method investigated the asymmetric structure of capillary bridges and predicted the axial and circumferential expansion of a liquid on two identical fibres in the presence of gravity effects to determine whether gravity effects are significant or negligible.
In summary, the researchers have combined theoretical simulations and experimental analyses, mainly from non-axisymmetric liquid bridges, using the introduction of inclination, the application of lateral perturbations or changes in the structure of the bridge.For example, stability studies have been carried out on liquid bridges composed in various ways, such as between non-parallel flat plates, between plates with applied inclination, and between spherical surfaces and flat plates.Some scholars have also explored the impact of the volume of the liquid bridge on its stability and interfacial changes.Compared to the study of axisymmetric liquid bridges, the study of non-axisymmetric ones is more challenging and uncertain.In this paper, a model architecture for non-axisymmetric liquid bridges is proposed by imposing a gravitational tilting angle on top of the previous model of TCC of axially symmetric liquid bridges.Experimental studies of steady-state and oscillatory TCC at gravity tilting angles are carried out.

Experiment system and experimental method
The TCC of liquid bridge experimental platform comprises the liquid bridge generator, imaging system, temperature control and free interface recognition post-processing software, as shown in figure 1.The liquid bridge generation system is composed of the slide rail with height adjustment, the bracket, the upper and bottom disks of liquid bridge.The image acquisition system is composed of the moving bracket, high-speed camera (Phantom 410 l), micro-lens, (Macro Probe ® LAOWA), laser transmitter (laser wavelength: 525 ± 10 nm; output power: 0-3 W; the thickness of fan-shaped laser plane: 0.5 mm), and the PIV post-processing software.The liquid bridge and high-speed camera can rotate at an angle synchronously.∆T is the temperature difference between the upper disk (hot disk) and bottom disk (cool disk).The heat of the upper disk comes from the coil spring heater, in addition, the upper disk temperature was monitored by the thermocouple (the heating rate of the upper disk is 0.58 • C s −1 ).In order to maintain liquid bridge between the upper and bottom disks, the 10cSt methyl silicone oil with larger surface tension is selected as the liquid bridge medium.The parameters of 10cSt methyl silicone oil were shown in table 1.
The geometry model of the non-axisymmetric liquid bridge of the semi-suspended zone under the gravity inclination angle is shown in figure 2. The upper and bottom disks of the liquid bridge are coaxial with a 45 • chamfer, so the actual diameter of disk is D = 4.88 mm.The actual volume of the liquid bridge is V = 20 µl, the height is H = 2 mm, the volume ratio is V r = V/V s = 0.535 (V s = 37.38 µl), the throat diameter is D ′ = 2.38 mm, and the aspect ratio is Ar = H/R = 0.83.The static Bond number (Bo = ρgL 2 /σ) is |Bo| = 1.833, thus the effect of buoyancy cannot be ignored in the experiments of this paper [43].The dynamic Bond number measures the relative strength of the buoyancy force compared to the thermocapillary force, Bo d = ρgγL 2 /σ ′ .A linearly decreasing function of temperature is assumed for the surface tension acting on the free surface σ ′ = σ 0 (T c )+ σ T (T − T c ), with σ T = −(dσ/dT)|T c .The Bond number, whether static or dynamic, will align its axial component with the axial component of the acceleration vector, due to the vectorial nature of g,

Results and discussion
To non-affecting the image quality, the shooting frequency should be low as possible to shorten the video file size and increase the shooting time.The particles inside the liquid bridge undergo millimetre/micrometre displacement per unit of time, and the particle image could be blurring in the process of capturing the continuous displacement.The shooting frequency is defined as follows: In equation ( 1), the ω is the optimal shooting frequency, fps.The M is the magnification, M = 1.5.The U l max is the maximum moving speed, mm s −1 .The θ is the angle between the target motion plane and the  normal axis of the lens plane, θ = 0 • .The k is the exposure factor.t s is the inverse of the shooting frequency, t s = 1/100 s.The t p is the best exposure time, t p = 0.0109 s.The ideal and the actual shooting frequency were calculated according to equation ( 2), respectively, R = 102 fps and Ri = 100 fps.
In addition, there is a 'motion blur' between the observed particle diameter d and the particle calibration diameter d s .It is necessary to observe the particle motion for a long time and maintain a long exposure time.Excessive exposure time could distort the observed particle shape, resulting in distortion image.In this paper, the image blur length is calculated by the following formula for evaluating the 'motion blur': where d ′ is the blur percentage, d s is the calibration size, and d Blur is the blurring length, see table 2. The initial experimental conditions (temperature difference between upper and bottom disks, the aspect ratio, and the volume ratio) greatly influence the TCC of the mesoscale liquid bridge, and subtle differences in operation could cause large experimental errors.The five parallel experiments verified the repeatability of the experimental operation within a certain error range.The velocity and critical temperature difference at the monitoring point (x = 3.75 mm, y = 1.60 mm), and the oscillation period of TCC obtained were analysed for error, see table 3.
In liquid bridge experiments under GTA, transverse and longitudinal velocities are mainly analysed under steady state and oscillatory conditions in this paper.Therefore, the transverse and longitudinal velocity errors at the intermediate height were evaluated by means and standard deviations, as shown in equation (3), In which, the σ L is sample standard deviation, u n is fluid velocity, ū is the average velocity, n is repeated times of experiment.
As shown in figure 3, the maximum, minimum and average errors of transverse velocity component are Before the formal experimental study, the imaging quality of polyamide resin particles (PRP1 and PRP2), aluminium powder (P1), silver-coated hollow microbead (P2), and fluorescent particles (PF) were evaluated in the case of liquid bridge with the same aspect ratio (Ar) and volume ratio (Vr).The physical parameters of the tracer particles are shown in table 4. Stokes number (St) is used to evaluate the tacking characteristic of tracer particles.The St parameter describes the ratio of inertia to diffusion.The lower the St value, the more effortless it is for the tracer particles to track the fluid motion.The St is described by the following equation,  where ρ p is the particle density, D P is the particle size, U l max is the maximum velocity.Generally, the surface flow velocity of TCC is greater than the inner bulk return flow velocity in the liquid bridge, and the maximum fluid velocity appears near the interface at the hot corner of liquid bridge [44].Therefore, the equation ( 4) can be calculated by the surface flow velocity (U l max ) at the hot corner.H is the height of the liquid bridge, and µ l is the dynamic viscosity of silicone oil.
The image quality of the above tracer particles was shown under the same laser output power (P = 480 mW) and macro lens magnification (1.5X).Compared with figures 4(a) and (b), the PRP1 and PRP2 are uniformly distributed, but the particle size of PRP1 is too small (d = 5 µm), and the particles cannot be imaged at specific positions.The particle size of PRP2 is too large (d = 50 µm), and the particle images are severely overlapped.According to the o St number in table 2 . The St number of PRP1 and PRP2 is much larger than that of P1 and P2 by one order of magnitude.In figure 4(c), because the P1 is irregular and non-spherical, its refractive index is low.The image clarity of P1 is much lower than that of the other four particles under the same laser irradiation intensity ( ).The distribution of P2 is uniform, as shown in figure 4(d).The St number of P2 is the larger than that of P1, and its refractive index is the largest.The image quality and followability of P2 are both good.
However, because the density of P2 is small, severe particle segregation occurs in the liquid bridge under the GTA, as shown in figure 4(f).Considering that the particle size and refractive index of PF are similar to the P2, the St number of PF is less than 1, and the segregation is difficult to form under the GTA, as shown in figure 4(e).Therefore, the PF is used as the tracer particle in this paper.

Stable TCC under the GTA
The distribution of lateral velocities at five heights for different GTAs (φ = 0 • , φ = 5 • , and φ = 10 • ) is shown in figure 5 (the left side of liquid bridge is far-ground and the right side of liquid bridge is near-ground).The transverse velocity component is positive when the direction of velocity coincides with the positive direction of the x-axis, otherwise it is negative.In figure 5(b), the original centrosymmetric TCC morphology (0 • ) is affected by the gravitational field, and its vortex core is distorted and shrunk in the radial/longitudinal direction, especially the lateral development of cellular flow is significantly inhibited.Compared with the GTA of φ = 0 • , the vortex core of the cellular flow is shifted, and the TCC is closer to the free surface.
In addition, the symmetry centre of the cellular flow is shifted to the near-ground side of the liquid bridge, and the stagnation point (u = 0 m s −1 ) is shifted from  resulting in a noticeable increase of the transverse velocity gradient of the interfacial flow near-ground side.In the steady TCC stage, the vertical development of the vortex core is further suppressed under the condition of φ = 10 • , as shown in figure 5(c).Compared with the GTA of φ = 5 • , the vortex core of cellular flow on the far-ground side is still located near the free interface, while the vortex core of cellular flow on the near-ground side is far away from the free interface, and the cellular flow on the side close to the ground is smaller than that on the side away from the ground.The stagnation point position of the transverse velocity component is shifted from x = 2.38 × 10 −3 m (φ = 5 • ) to x = 2.52 × 10 −3 m (φ = 10 • ), as shown in The longitudinal velocity distributions at the five heights (the left side of liquid bridge is far-ground and the right side of liquid bridge is near-ground) are shown in figure 6.When the velocity direction is consistent with the positive direction of the y-axis, longitudinal velocity component is positive, it is negative.In figure 6(b), at the GTA φ = 5 • , the longitudinal velocity gradient near the free surface increases significantly with the left and right cellular flow moving toward the free interface, however, the longitudinal velocity gradient inside the liquid bridge decreases.Under the effect of GTA, the vortex of the cellular flow shrinks to the intermediate height of the liquid and the peak of longitudinal velocity significantly decreases.The longitudinal velocity difference at the same height under the two types of GTAs is as follows (∆v = v height,0 • − v height,5 • , φ = 0 • and φ = 5 • ), ∆v h=0.6 == 0.66 × 10 −3 m s −1 (height at h = 0.6 mm), ∆v h=0.9 = 1.74 × 10 −3 m s −1 (height at h = 0.9 mm), ∆v h=1.2 = 1.16 × 10 −3 m s −1 (height at h = 1.2 mm), ∆v h=1.5 = 2.32 × 10 −3 m s −1 (height at h = 1.5 mm), ∆v h=1.7 = 1.1 × 10 −3 m s −1 (height at h = 1.7 mm).Due to the distorting vortex approaching the free surface, the bulk flow develops along the longitudinal direction at the centre of the liquid bridge.
In figure 6(c), under the GTA of φ = 10 • , the vertical development of the left and right cellular flows is further suppressed, the vortex core of the cellular flow shrinks to the intermediate height of the liquid bridge, and the longitudinal velocity gradient near the free surface increases significantly.The longitudinal velocity difference at the same height with the different GTAs is as follows (∆v = v height,0 • -v height,10 • , φ = 0 • and φ = 10 • ), ∆v h=0.6 = 0.5 × 10 −3 m s −1 (height at h = 0.6 mm), ∆v h=0.9 = 3.2 × 10 −3 m s −1 (height at h = 0.9 mm), ∆v h = 1.2 = 3.9 × 10 −3 m s −1 (height at h = 1.2 mm), ∆v h = 1.5 = 1.9 × 10 −3 m s −1 (height at h = 1.5 mm), ∆v h = 1.7 = 0.8 × 10 −3 m s −1 (height at h = 1.7 mm), it can be seen that the attenuation of longitudinal velocity is the largest near the intermediary height of the liquid bridge.
Comparing the effect of gravitational tilt on transverse and longitudinal velocity, the peak of transverse velocity decreases by nearly 26% and the peak of longitudinal velocity decreases by nearly 27% under the application of gravitational tilt.The gravitational tilt causes the decrease of the transverse and longitudinal velocity gradient inside the liquid bridge, and the stability of bulk flow in the central region of the liquid bridge is enhanced because of the decrease of the velocity gradient.At the same time, the longitudinal velocity gradient near the free interface is enhanced, while the transverse velocity gradient is weakened (This has a similar effect as applying horizontal vibrations.The longitudinal velocity gradient near the free surface of liquid bridge increases as the increased level of transverse acceleration, see figures 8(a) and (b) in the [28].).The longitudinal velocity enhancement accelerates the surface flow, and then accelerates the fluid replenishment to the hot corner.

Oscillating TCC under the GTA
The temperature variation with time at the hot corner under three types of GTA is shown in figure 7 (φ = 0 • , φ = 5 • , and φ = 10 • ).The main reason for the occurrence of oscillatory TCC is the coupling instability of temperature, velocity, and free surface at the thermal corner, which occurs before the occurrence of oscillatory TCC.It can be seen from figure 7(a) that the critical temperature is T cr = 76.1 • C at the hot corner without the GTA.However, under the GTA of φ = 5 • , the onset of the temperature oscillation at the hot corner is delayed, at a time lag of t = 9.6 s, and the critical temperature rises to T cr = 81.5 • C, as shown in figure 7(b).
In other words, as the critical temperature difference of the oscillating TCC at the hot corner increases, the stability of the TCC is improved to some extent.At the same time, the frequency (f ) and amplitude (A) of the oscillatory temperature increase at the hot corner under the GTA of φ = 5 It shows that the GTA can maintain the stability of TCC to a certain extent and increase the critical temperature in the hot corner.However, once the oscillation of TCC has been excited, the instability of temperature is intensified and appears pulsating oscillation.In figure 7(c), under the GTA of φ = 10 • , the onset of the temperature oscillation at the hot corner is delayed with a time lag of t = 18 s (compared with φ = 0 • ), and the critical temperature increase to T cr = 87.9• C. The stability of the TCC has been further improved.Compared with the GTA of φ = 5 • , there is no variation in the frequency of the oscillatory temperature at the hot corner, but its amplitude continues to increase (f 10 Figure 8 shows the velocity vector of oscillatory TCC in a half period.In figures 8(a)-(c), the spatiotemporal evolution of typical oscillatory TCC was presented.The left and right cellular flows exhibit a characteristic of periodic encroachment on each other, and the oscillation period of cellular flow is T 1/2 = 0.46 s.Under the GTA of φ = 5 • , the oscillation period increases slightly, T 1/2 = 0.49 s.Compared with the φ = 0 • , the flow pattern of TCC is still oscillatory.The vortex core of cellular flow shrank obviously, and its lateral and vertical development is significantly inhibited.The symmetry centre of the cellular flow is shifted to the near-ground side, see figures 8(d)-(f).The position of the vortex core is closed to the free interface of the hot or cold corner during the oscillation stage of TCC.In figure 8(e), the area of the longitudinal return flow in the centre of the liquid bridge is increased and the flow uniformity is obviously improved.The velocity vector of the TCC was shown under a GTA of φ = 10 • , from figures 8(g)-(i).
The oscillation period of the cellular flow is T 1/2 = 0.5 s.Compared with the φ = 0 • , the effect of the vortex core of cellular flow on the flow field at the hot or cold corner is more significant, as shown in figure 8(g) and (i).See [45] for a similar conclusion, the vortex core position of the cellular flow changes when a transverse acceleration is applied, which in turn affects the activity degree of the hot corner.Therefore, the initiation of oscillatory TCC is disturbed by the flow field at the hot corner with the gravitational tilt.
The Marangoni number (Ma) is the ratio of heat convection transport to heat diffusion transport in the TCC, defined by Ma = σ ′ ∆TL/ρυk, σ T denotes the variation coefficient of surface tension with the  temperature (experimental results of surface tension of 10cSt silicone was shown in figure 9), ∆T denotes the temperature difference between the upper and bottom disks, where L denotes the characteristic length (the liquid bridge height, L = 2 mm), ρ denotes the density of silicone oil, υ denotes kinematic viscosity of silicone oil, and k denotes the thermal diffusion coefficient.In addition, the critical Marangoni number (Ma cr ) at the critical temperature difference ∆T cr (see figure 7) is used to determine transition behaviour of TCC.Under the three kinds of gravitational tilt conditions, the critical Marangoni numbers (Ma cr ) are Ma cr,0 • = 7.32 × 10 3 , Ma cr,5 • = 7.32 × 10 3 and Ma cr,10 • = 9.01 × 10 3 , respectively.It is obvious that the critical Marangoni number increases with increasing GTA.Therefore, the stability of TCC is enhanced.
Figure 10 shows the distribution of the transverse velocity component at different heights under the GTA during the oscillation phase of TCC.At the intermediate height of the liquid bridge (y = 1.2 mm), the transverse velocity direction changes periodically as the development of oscillatory TCC, see from figure 10(c) to figure 10(m).The inhibition level of the GTA to the transverse velocity at different heights, and the attenuation of the transverse velocity component at the intermediate height is the most obvious.The maximum transverse velocity component at this height has an absolute value of u 0 • max = 12.8 × 10 −3 m s −1 (x = 2.6 mm, y = 1.2 mm), u 5 • max = 5.9 × 10 −3 m s −1 (x = 2.9 mm, y = 1.2 mm), and u 10 • max = 7.5 × 10 −3 m s −1 (x = 2.0 mm, y = 1.2 mm), respectively.Near the cold or hot disk (y = 1.5-1.7 mm, y = 0.6-0.9mm), the attenuation of the transverse velocity component is weaker.With the increasing GTA, the peaks of transverse velocity are shifted to the near-ground side, resulting in the increase of the transverse velocity gradient near the free surface, as shown in figures 10(a), (k), (b) and (l).The larger 2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (f), (g), (h), (i), (j) At the time t2, the transverse velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (k), (l), (m), (n), (o), At the time t3, the transverse velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles.The t i , t2 and t3 is a time series within half an oscillation period under different gravitational tilt angles in the figure 8.
transverse velocity near the free surface accelerates bulk flow returning to the hot corner.Therefore, the onset of temperature oscillation is delayed at the hot corner, but the disturbance of the flow field is intensified.
As shown in figure 11, the attenuation of the longitudinal velocity component is more significant than that of the transverse velocity in the oscillation phase of TCC.The distribution of the longitudinal velocity at different heights shows an irregular 'M' pattern under the angle of inclination due to gravity.Near the hot corner (y = 1.7 mm), the longitudinal velocity component is basically v = 0 m s −1 .As the GTA increases, the longitudinal velocity distribution shifts towards the near-ground side, causing an increase in the longitudinal velocity gradient near the free interface on that side.At the intermediate height, the maximum value of longitudinal velocity under three types of GTAs (φ = 0 The GTA suppresses the longitudinal velocity components with different heights.
In summary, the velocity gradient near the free interface increases significantly with increasing GTA, and the transverse and longitudinal velocities near the intermediate height are more sensitive to the GTA.Therefore, it is necessary to further discuss the velocity distribution of the surface flow at the intermediate height.Furthermore, in analogy with applied horizontal vibrations, the change of transverse acceleration will cause micro perturbance in surface flow and interface morphology, and the perturbation intensity of the liquid bridge interface is the largest only at the resonance frequency [46,47], which is similar to the fluctuation of surface flow velocity curve with the different GTAs in figure 12.
As shown in figure 12, the GTA has a different influence on the increase of the surface flow velocity on the near-and far-ground sides.The surface flow velocity is obviously increased after the gravity inclination was applied.Because the liquid bridge scale is very small, the dynamic Bond number is far less than 1 (|Bo d | ≪ 1, as shown in table 5), the free surface shape of liquid bridge is determined by the balance of pressure difference and surface tension on the interface (as shown in equation ( 5)).With the application of gravitational tilt, the   Note: fluid side pressure changes at the free surface, the dynamic Bond number decreases significantly (|Bo d |cosφ, see table 5), and the effect of surface tension on the interface shape and surface flow is enhanced.The curvature radius decreases on the near-ground side of liquid bridge, and the free interface shape becomes more curved (right side interface of liquid bridge).The curvature radius increases on the far-ground side of liquid bridge, and the free interface shape becomes more flat (left side interface of liquid bridge), where R 1 and R 2 are the main curvature radius of free surface, P 2 is the pressure on the concave side of free interface (air side), and P 1 is the pressure on the convex side of free interface (fluid side).Therefore, under the GTA (φ = 5 • or φ = 10 • ), the surface flow velocity on the near-ground side is greater than that on the far-ground side.In addition, due to the increased velocity gradient near the free surface, the distribution of the surface flow velocity shows large fluctuations.The velocity distribution on the near-ground side shows an 'M'-shape, and its stability is better than that on the far-ground side.From figures 12(b)-(d), the peak position of surface flow velocity is shifted from the bottom to up with the time, y = 0.75 mm(t 1 ) → y = 1.00 mm(t 2 ) → y = 1.60 mm(t 3 ).From figures 12(e)-(h), the velocity distribution of the surface flow shows a severe fluctuation on the far-ground side, and the peak of the surface flow velocity is located near the upper or bottom disk.With a gravity inclination of φ = 10 • , the flow velocity on the surface of the ground nearby is greater than on the surface of the far-ground side.As the cellular flow moves away from the free surface, the flow velocity on the near-ground side decreases marginally.The cellular flow on the far-ground side is still close to the free surface, and the magnitude of surface flow velocity basically does not change.Compared with φ = 5 • , the distribution of surface flow velocity still presents an 'M'-shape on the near-ground side.

Conclusions
A GTA was introduced to a previous axisymmetric liquid bridge foundation, a non-axisymmetric liquid bridge structure was presented and implemented.The model is used to perform experimental studies of stable and oscillatory TCC at GTAs.During the steady TCC phase, the vertical development of the vortex core is significantly suppressed as the GTA increases.The vortex shrinks to the intermediate height.The transverse velocity and the longitudinal velocity gradient of the interfacial flow increase significantly, the peak transverse/longitudinal velocity decreases significantly near the free surface.During the oscillation TCC, the oscillation period increases slightly with increasing gravitational tilt.The symcenter of cellular flow is shifted to the near-ground side, and the vortex core has a important influence on the flow field at the hot/cold corner.The peak transverse velocity is shifted towards the near-ground side, leading to an increase in the transverse velocity gradient near the free surface.Under the effect of GTA, the longitudinal velocities are attenuated more than the transversal velocities, and the longitudinal velocity distribution has a 'M' pattern.The longitudinal velocity dispersion is shifted towards the near-ground side, increasing the longitudinal velocity gradient near the free interface on the near-ground side.In summary, the effect of gravitational tilt on lateral and longitudinal velocities is more sensitive near intermediate heights.Meanwhile, the velocity gradient increases significantly near the free interface with increasing gravitational tilt.The backflow in the hot corners was accelerated by the distortion of the cellular flow.Therefore, the onset of TCC was delayed.

Figure 2 .
Figure 2. Physical model of a liquid bridge under the gravitational tilt angle.The upper disk is hot (T h ), and the bottom disk is cold (Tc).The direction of the temperature gradient and the direction of the gravitational field form a certain angle φ (φ = 0 • , φ = 5 • , φ = 10 • ).The left side of liquid bridge is far-ground and the right side of liquid bridge is near-ground.
42 × 10 −5 , respectively.The maximum, minimum and average errors of longitudinal velocity component are σ L max-l = 4.03 × 10 −4 , σ L min-l = 4.59 × 10 −5 , σ L av-l = 1.25 × 10 −5 , respectively.The maximum errors of transverse and longitudinal velocity component occur at the free surface, due to the greater interface curvature and velocity gradient near the free surface of liquid bridge.

Figure 3 .
Figure 3. Transverse and longitudinal velocity errors at the middle high position under steady state conditions.(a) Transverse velocity component; (b) longitudinal velocity component.(Vr = 0.669, H = 2 mm).

Figure 7 .
Figure 7. Variation of temperature at the hot corner of TCC with the deferent gravitational tilt angles.(a) φ = 0 • ; (b) φ = 5 • ; (c) φ = 10 • .The critical temperature Tcr refers to the temperature when temperature oscillates in the hot corner.The critical temperature difference ∆Tcr refers to the temperature difference between the critical temperature and the temperature of bottom disk (or ambient temperature).

Figure 8 .
Figure 8. Spatiotemporal evolution of flow patterns in the oscillatory TCC under gravitational tilt angle of φ = 0 • , φ = 5 • , and φ = 10 • .The temperature difference ∆T refers to the temperature difference between the upper and bottom disks (or ambient).

Figure 9 .
Figure 9. Variation of surface tension of 10cSt silicone oil with temperature.

Figure 10 .
Figure 10.Variation of transverse velocity component at different heights with the gravitational tilt angle.(a), (b), (c), (d), (e) At the time t1, the transverse velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (f), (g), (h), (i), (j) At the time t2, the transverse velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (k), (l), (m), (n), (o), At the time t3, the transverse velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles.The t i , t2 and t3 is a time series within half an oscillation period under different gravitational tilt angles in the figure8.

Figure 11 .
Figure 11.Variation of longitudinal velocity component at the different heights with the gravitational tilt angle.(a)-(e) At the time t1, the longitudinal velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (f)-(j) At the time t2, the longitudinal velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles; (k)-(o) At the time t3, the longitudinal velocity component at the height of y = 1.7 mm/y = 1.5 mm/y = 1.2 mm/y = 0.9 mm/y = 0.6 mm under the different gravitational tilt angles.The t i , t2 and t3 is a time series within half an oscillation period under different gravitational tilt angles in the figure 8.

Figure
Figure Distribution of surface flow velocity with the time under the gravitational tilt angle of φ = 0 • , φ = 5 • , and φ = 10 • .

Table 2 .
Display error of different moving particle size.Blur , µm Calibration size ds, µm Blur percentage d ′ , % Absolute error δ, µm Absolute error δ is the difference between the observed diameter and the calibration size. Note:

Table 5 .
The variation of dynamic Bond number.