Yousuf Alhendal*
| Mohammad Alhamli
© 2025 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
Thermocapillary migration under gravity and zero-gravity conditions was simulated using the Volume of Fluid (VOF) method in ANSYS Fluent software. The migration of a nitrogen bubble in a container filled with liquid ethanol was investigated numerically. Three distinct conditions were used to study the bubble migration in a compact container: gravity, thermocapillary, or both. According to the findings, buoyancy controls bubble migration in larger containers, while thermocapillary forces govern motion only at small scales (below 10 mm), showing an inverse dependence on container dimension. In thermocapillary flow, the bubble maintained its spherical shape as it traveled. In the buoyancy force, the spherical bubble shape became oblate. The bubble in the combination of thermocapillary and buoyancy forces, in contrast to the other two scenarios, shifted from a spherical to an oblate shape as the container's size expanded. It subsequently ruptures before it reaches the heated surface. The significance of this study lies in its estimation of the Marangoni force's emergence, the gravitational force's effect disappearing, and the neglect of its calculations at extremely small measurements. It also helps build high-quality engineering equipment on a small scale.
bubble migration, gravitational force, convective capillary, Marangoni flow, thermocapillary, two-phase, VOF method
Understanding the dynamics of bubbles and droplets is crucial. It can assist in developing new equipment to either mitigate or control the effects of these bubbles in several industrial processes and even in space devices. Under the influence of gravity, bubbles can change shape and even break apart, complicating their movement and sometimes harming equipment [1, 2]. Therefore, controlling these bubbles can enhance various engineering applications, such as welding on Earth and heat/mass transfer operations in space [3, 4].
At first, bubble migration was investigated by Young et al. [5], who observed that the bubbles were moving to the warmer side when a temperature gradient was applied because of surface tension variation. This occurrence is known as thermocapillary / Marangoni. Hardy [6] also studied bubble motion under Marangoni and buoyancy forces. They presented the required temperature gradients to balance the two forces depending on the bubble diameter. Bratukhin et al. [7] investigated Marangoni migration in normal gravity, with different techniques. Time, heat, and mass diffusion, as well as surfactant absorption characterizing the Marangoni convection, were analyzed. In fact, with the presence of gravity on Earth, natural convection occurs rapidly, which overpowers the Marangoni force [8]. Generally, in order to notice the Marangoni effect clearly, micro or zero gravity environments are used [9-13]. Wozniak et al. [14] described the temperature field surrounding a moving bubble in a reduced gravity environment. Some discrepancies were observed between numerical and experimental results due to some assumptions considered in the numerical study, as well as optical disturbances in the experiment.
Most of the studies were experimental. However, measurements are sometimes difficult to realize and very expensive, which pushed researchers to model and simulate the phenomenon as well as compare both numerical and experimental data to each other. Brackbill et al. [15] suggested a method to model surface tension based on the interface curvature, which simplifies the calculations. Thus, many researchers adopted it to construct their numerical model [16-18]. Moreover, the VOF method was found to be very reliable to describe two fluid flows, as shown in references [19-22]. Many numerical studies investigating one bubble migration as well as several bubbles were produced [23-25]. Mostly, when one bubble is migrating from cold to hot regions, it travels along a vertical path. However, it can have complex behavior, especially when it is subjected to other forces such as rotation [26-28] or vibration [29-32]. Those different trajectories have a direct impact on the arrival time of the bubble to the top; it is either delayed or advanced.
Another interesting parameter that affects bubble migration is the container’s shape. Most of the articles were produced on cylindrical containers [33-35], unlike Balla et al. [36], who investigated a square channel container. Similar to the shape, the dimension of the container plays a crucial role. Both the distance that the bubble will travel and the sides of the container will definitely impact the bubble’s velocity and shape [37]. Therefore, this study examines and contrasts the motion and behavior of bubbles in small containers before progressively expanding their size in three scenarios: with the buoyancy force on its own, with the Marangoni effect, and with the Marangoni effect by itself. We concentrate on the level at which the buoyancy effect becomes apparent, which is noticeable and important, and the Marangoni effect starts to progressively disappear.
Under gravity and in zero-gravity environment, the movement of a nitrogen bubble in an ethanol container is investigated (Figure 1). The container is heated from the top by Thot = 325 K, while the bottom is maintained at Tcold = 300 K. The sidewalls are thermally isolated. The velocity is considered null on the walls. The bubble moves at a slow speed, and the flow remains laminar. Ethanol is an incompressible and Newtonian fluid; its thermophysical properties are considered to be unchanged except surface tension, σ, which depends on temperature. Prior to releasing the bubble, the steady-state temperature distribution is established at first in the container, and then it is considered as the initial condition for studying the unsteady motion of the bubble.
Figure 1. Geometry of the calculated field for thermocapillary migration
The YGB model is based on Young et al. [5] linear model:
$V_{Y G B}=\frac{2\left|\frac{d \sigma}{d T}\right| r_b \lambda \frac{d T}{d x}}{\left(2 \mu+3 \mu^{\prime}\right)\left(2 \lambda+\lambda^{\prime}\right)}$ (1)
$\mu, \mu^{\prime}, \lambda$ and $\lambda^{\prime}$ represent the dynamic viscosity and thermal conductivity, respectively, of the two phases (fluid and bubble).
Thermal Reynolds and Marangoni numbers are expressed as:
$R e_T=\frac{r_b V_T}{v}$ (2)
$\mathrm{Ma}_{\mathrm{T}}=\frac{\mathrm{r}_{\mathrm{b}} \mathrm{V}_{\mathrm{T}}}{\alpha}=\mathrm{Re}_{\mathrm{T}} \cdot \operatorname{Pr}$ (3)
$V_T$ is the determined velocity from balancing tangential stresses at the free surface and is utilized to scale the velocity of migration (m/s) in Eqs. (2) and (3):
$V_T=\frac{\frac{d \sigma}{d T} \cdot \frac{d T}{d x} \cdot r_b}{\mu}$ (4)
with Prandtl number defined as:
$\operatorname{Pr}=\frac{v}{\alpha}$ (5)
$\alpha$: thermal diffusivity and ν: kinematic viscosity:
$v=\frac{\mu}{\rho}$ (6)
with $\rho$, the density of the continuous phase fluid, and $r_b$ is the bubble's radius, which is $\mathrm{d} / 2$, with d the bubble's diameter.
$d \sigma / d T$ or $\sigma_T$ denotes the rate of change of interfacial tension, while $d T / d x$ signifies the temperature gradient applied to the continuous phase fluid.
The governing continuum conservation equations for two-phase flow were solved using Ansys-Fluent software, and the volume of fluid (VOF) method was used to track the liquid/gas interface. The geometric reconstruction scheme, based on the piece-wise linear interface calculation (PLIC) method of Youngs [38] in Ansys-Fluent, was chosen for the current investigation. Geo-reconstruction is utilized to give more accuracy for free surface definition [19]. The movement of bubble-liquid interface is tracked depending on the gas bubble volume fraction distribution, i.e., $\alpha_G$, in a computational cell, where the value of $\alpha_G$ is 0 for the liquid phase and 1 for the bubble phase. Therefore, a gas-liquid interface exists in the cell where $\alpha_G$ lies between 0 and 1.
The momentum equation, expressed below, is solved for all the phases existing in the domain:
$\begin{aligned} \frac{\partial}{\partial t}(\rho \vec{v})+\nabla \cdot(\rho \vec{v} \vec{v}) =-\nabla p+\nabla \cdot\left[\mu\left(\nabla \vec{v}+\nabla \vec{v}^T\right)\right]+\vec{F} +\rho \vec{g}\end{aligned}$ (7)
where, $\vec{v}$ is treated as the mass-averaged variable:
$\vec{v}=\frac{\alpha_G \rho_G \vec{v}_G+\alpha_L \rho_L \vec{v}_L}{\rho}$ (8)
When the zero environment is considered, $\vec{g}$ becomes null.
$\vec{F}$ represents surface tension force per unit volume. It is composed of normal force, $\overrightarrow{F_N}$, and tangential one, $\overrightarrow{F_T}$. Continuum surface force (CSF) model is used to compute it for the cells containing bubble-liquid interface [15]:
$\overrightarrow{F_N}=\sigma \frac{\rho k \vec{n}}{\frac{1}{2}\left(\rho_L+\rho_G\right)}$ (9)
where, $\sigma$ is the coefficient of surface tension.
$\sigma=\sigma_0+\sigma_T\left(T_0-T\right)$ (10)
For the tangential surface tension force, it is defined as:
$\overrightarrow{F_T}=-\sigma_T \nabla_S T$ (11)
$\sigma_0$ represents the surface tension at a reference temperature $T_0, T$ is the temperature of the liquid, $\vec{n}$ is the surface normal estimated from the volume fraction gradient, and $k$ is the local surface curvature, defined as follows:
$k=-(\nabla \hat{n})=\frac{1}{|\vec{n}|}\left[\frac{\vec{n}}{|\vec{n}|} \nabla|\vec{n}|-(\nabla \square \vec{n})\right]$ (12)
The interface tracking between bubble and liquid is achieved by the continuity equation solution for bubble volume fraction:
$\frac{\partial}{\partial t}\left(\alpha_G \rho_G\right)+\nabla \cdot\left(\alpha_G \rho_G \vec{v}_G\right)=0$ (13)
This equation is indirectly determined for the host fluid; instead, the volume fraction of the liquid is calculated as follows:
$\alpha_G+\alpha_L=1$ (14)
with $\alpha_G$ and $\alpha_L$, volume fractions of the bubble and host-fluid, in that order. The density and viscosity of each cell at the interface are determined using the following expressions:
$\rho=\alpha_G \rho_G+\left(1-\alpha_G\right) \rho_L$ (15)
$\mu=\alpha_G \mu_G+\left(1-\alpha_G\right) \mu_G$ (16)
Under gravity force alone, $\vec{F}$ is assumed null. In that case, the velocity will be calculated from (7) in addition to continuity equation, expressed as:
$\frac{\partial \rho}{\partial t}+\nabla(\rho \vec{v})=0$ (17)
For the only gravity case, the Peclet number is defined as:
$\mathrm{Pe}=\frac{r_b V_g}{v \alpha}=R e \cdot P r$ (18)
With Vg, the flow velocity takes into account only gravity.
The equation of energy is also solved for all phases:
$\left.\frac{\partial}{\partial t}(\rho E)+\nabla \cdot[\vec{v}(\rho E)+p)\right]=\nabla \cdot\left(\kappa_{e f f} \nabla T\right)$ (19)
where, specific heat depends on the considered phase. Temperature, density $\rho$, and effective thermal conductivity $k_{e f f}$, are shared by the phases.
VOF model treats energy (E) and temperature (T) as massaveraged variables:
$E=\frac{\sum_{q=1}^n \alpha_q \rho_q E_q}{\sum_{q=1}^n \alpha_q \rho_q}$ (20)
GAMBIT was used to generate the finite-volume mesh before integrating it into Ansys-Fluent. The computations were conducted using a pressure-based, segregated, implicit solver. Pressure–velocity coupling was achieved using the pressure-implicit with splitting of operators (PISO), which applies two corrections for neighboring points and skewness. The pressure-staggering option (PRESTO) scheme was also used for pressure interpolation. Conservation equations were discretized using a second-order upwind differencing scheme. The time step used to obtain convergence is 10−2 s.
A nitrogen bubble was positioned at the center and 2xd from the bottom surface in liquid ethanol medium. In previous works [17, 21], thermocapillary model was already examined and validated properly with Thompson et al. [9]. As shown in Figure 2, the results are in good agreement with only minor discrepancies attributed to measurement and numerical errors. The accuracy of the simulation was attained by testing five grid meshes. Grid independence, mesh sizes, time steps, and extending the geometry to 3D were all checked to assure convergence. In this study, a grid consisting of 324,000 nodes, corresponding to 576 cells per bubble diameter, was considered for all simulations, owing to its accuracy in results while minimizing computing time and memory usage.
Figure 2. Present results validation with study [9]
For a container of 60mm diameter and 120mm height, when a temperature gradient is applied to a bubble N2 (d = 8 mm) inside ethanol in zero gravity, this bubble moves in a vertical translation from the bottom (cold wall) to the top (hot wall). This result was already found by many researchers both experimentally and numerically [5, 9, 17, 29]. To investigate the behavior of thermocapillary bubble flow in zero gravity under a linear temperature distribution between the upper and lower walls, five different temperature differences were examined. Temperature changes of 2.5 K were made to the container’s top, which ranged from 317.5 K to 325 K, while the container’s bottom remained at 300 K. Obviously, for the higher Thot, the bubble has already reached the top, while it is still in almost the middle of the container for the weakest Thot. For all the cases, the isotherms behind the bubble are disturbed. Whenever the bubble is moving, a small recirculation of ethanol is created, which makes the temperature change in the medium. In fact, when Thot increases, the bubble velocity augments. Therefore, for a higher temperature gradient in the container, the bubble reaches the top rapidly, as indicated by the decreasing bubble arrival time (Figure 3). The Marangoni problem or thermocapillary migration problem are terms used to describe this kind of phenomenon.
Figure 3. Bubble arrival time depending on Thot
This section investigates the impact of thermocapillary forces on bubble migration by varying container heights (h) and bubble radius (rb) under three distinct conditions: normal gravity, normal gravity with thermocapillary effects, and zero gravity with thermocapillary effects. These conditions correspond to the following flow patterns: buoyancy-driven flow, combined buoyancy and thermocapillary flow, and thermocapillary-driven flow, as illustrated in Figure 4. Bubbles with a radius smaller than or equal to 0.24 mm exhibit no movement in cylinders less than 10mm in height under buoyancy-driven flow. Conversely, for the same cylinder height, in the presence of thermocapillary forces—whether gravity is applied or not—these bubbles consistently migrate toward the heated surface within the cylinder. The constant MaT, regardless of the presence or absence of gravity, suggests that the flow in these cases is dominated by thermocapillary effects alone. As rb increases to 0.32 mm and h increases to 10mm, the bubble begins to migrate under all three flow conditions. However, under the combined buoyancy and thermocapillary flow, the migration pattern reveals noticeable deformation. As rb and h increase further, this deformation becomes more pronounced in the presence of gravity and even more significant when both forces are acting together. In contrast, in the absence of gravity (thermocapillary flow), aside from slight elongation as the bubble migrates toward the heated surface, no deformation is observed. The interaction between gravity and thermocapillary forces significantly alters the bubble’s shape. However, the effect of gravity alone on bubble deformation becomes more evident in larger bubbles. Due to their weight, these bubbles tend to flatten, as previously demonstrated by Cohen et al. [39] for large bubbles. For the three observed flow patterns, the bubble generally moves in a vertical trajectory. However, when h > 20 mm in buoyancy-driven flow, the bubble's trajectory starts to incline slightly away from the center of the container, which is not observed in the other flow patterns. At h = 45 mm, small bubbles begin to detach from the main bubble in the coupled buoyancy and thermocapillary flow scenario. This detachment becomes more pronounced with increasing h, eventually hindering the bubble's motion. By h = 90 mm, this interference leads to a significant reduction in flow, culminating in a complete cessation of motion at h = 120 mm. This behavior can be attributed to the opposing nature of gravitational and thermocapillary forces in these specific scenarios, where the buoyancy force starts to appear gradually from about h = 45mm. In contrast, at smaller dimensions, these forces tend to work synergistically, enhancing bubble migration. Therefore, the selection of container dimensions plays a critical role in bubble migration dynamics when gravity is a factor, particularly at high MaT values.
For fixed bubble radius (rb = 0.48 mm) and surface temperatures maintained at 300 K (cold) and 325 K (hot), Figure 5 presents a comparison of bubble arrival times at the cylinder top under gravity, with and without thermocapillary effects. At h = 10 mm, the Marangoni effect dominates, enabling efficient bubble migration at such small scales. At h = 15 mm, buoyant forces begin to influence the motion, particularly in the cylinder’s lower section. By h = 20 mm, buoyant forces clearly dominate, facilitating faster bubble migration. The relative contribution of thermocapillary forces diminishes as the container height increases.
In Figure 6, variations in bubble diameter and container dimensions further clarify the interaction between thermocapillary and gravitational forces. In some instances, these forces do not combine effectively, resulting in the absence of their joint influence under specific conditions. For cylinder heights of h = 10 mm and h = 20 mm, the bubble’s diameter has minimal impact on the dominant force driving its motion; instead, the height of the cylinder is the critical factor. The Marangoni effect is particularly pronounced in the presence of gravity, significantly reducing bubble arrival time at the cylinder’s top for h = 10 mm. Beyond h = 20 mm, buoyant forces take precedence, accelerating bubble migration independently of thermocapillary effects.
Across all studied cylinder heights, thermocapillary forces alone consistently result in slower bubble migration compared to cases where buoyant forces are involved. In smaller cylinders, the interplay between buoyant and thermocapillary forces optimizes bubble travel time, emphasizing their synergistic contribution. However, in larger cylinders, buoyant forces alone emerge as the dominant mechanism, ensuring faster bubble motion.
These findings demonstrate the crucial role of container dimensions and force interactions in bubble migration dynamics. For small-scale systems, understanding and leveraging the interplay between thermocapillary and buoyant forces can enhance bubble control and efficiency. In larger systems, where buoyancy dominates, simplified models focusing on gravitational effects may suffice for predicting bubble behavior.
The results of this study have direct implications for several engineering and scientific applications. In microgravity environments, such as spacecraft thermal-control systems and propellant management units, buoyancy is negligible; therefore, thermocapillary forces can serve as an effective driving mechanism for bubble transport and removal. The present simulations show that for containers up to 10 mm in height and $r_b \leq 0.24$ mm, bubbles remain stationary under gravity alone but migrate fully under thermocapillary forces. This finding suggests that even modest temperature gradients (ΔT = 25 K) can ensure continuous bubble removal and stable operation in compact heat-transfer devices in space.
In terrestrial small-scale systems, including microreactors and compact condensers, the dominance of thermocapillary forces below 10 mm and the transition to buoyancy at 20 mm provide a practical design guideline. To promote stable bubble transport, vertical distances should be kept below 10 mm or subdivided into short heated segments where thermocapillary motion assists buoyant rise. Conversely, in taller devices where buoyancy dominates, simpler gravity-based flow management can be applied.
For chemical and biomedical applications, these results highlight how the interplay between thermocapillary and gravitational forces affects both the arrival time and the shape of migrating bubbles. In chemical reactors or separation columns, combined thermocapillary and buoyant effects can enhance phase separation efficiency, while in biomedical microchannels, thermocapillary control could guide microbubbles for targeted drug delivery or localized heating without the excessive deformation observed under gravity-dominated conditions.
The model assumes an incompressible Newtonian liquid with constant properties and a linear dependence of surface tension on temperature. This approximation is appropriate for the small-scale cylinders studied here (height < 120 mm) and the modest temperature difference (ΔT = 25 K), where property variations are minimal. A nonlinear σT relation or temperature-dependent viscosity could slightly alter the predicted bubble velocity but would not change the identified dominance of thermocapillary forces at h ≤ 10 mm or buoyancy at h ≥ 20 mm. Future work may include these refinements and limited experimental verification to confirm the robustness of the present findings.
The present study targets small cylindrical containers (height < 10 mm - 120 mm) with a hot top (325 K), cold bottom (300 K), and adiabatic sidewalls. The liquid was treated as incompressible and Newtonian, with constant thermophysical properties, while surface tension varied linearly with temperature. This simplification is widely used for narrow temperature ranges (ΔT = 25 K) and small domains, and it isolates the thermocapillary mechanism effectively. We note, however, that if σT becomes nonlinear or temperature-dependent viscosity is considered, the quantitative magnitude of bubble velocity could change slightly, though the observed dominance trends (10 mm to 20 mm transition and deformation sequence) are expected to remain valid. Future numerical studies may examine nonlinear σT or temperature-dependent viscosity to confirm sensitivity, and experimental validation under microgravity or confined conditions would help verify these trends.
A 3D model was created in Ansys‑Fluent to investigate the effect of thermocapillary forces on the migration of a nitrogen bubble in small-scale containers under both normal and zero‑gravity environments. The numerical model was validated by demonstrating strong agreement between the results obtained and those found in the literature. The findings demonstrate that:
The scale-dependent thresholds identified here also provide practical guidance for system design. In microgravity or confined geometries (h ≤ 10 mm), thermocapillary forces can be used to accelerate bubble removal and stabilize two-phase heat-transfer processes. For larger configurations (h ≥ 20 mm), buoyancy becomes the controlling mechanism, and device layouts should prioritize vertical alignment and venting paths for buoyant migration. These findings translate directly into improved performance and reliability for microgravity heat-transfer systems, chemical processing units, and biomedical microdevices.
We would like to express our gratitude for the support from the Kuwait Foundation for the Advancement of Sciences (KFAS).
|
CFD |
computational fluid dynamics |
|
VOF |
volume of fluid |
|
PLIC |
piece-wise linear interface calculation |
|
YGB |
young, Goldstein, and block model |
|
CSF |
continuum surface force |
|
div |
divergence ($\nabla \cdot \mathbf{u}=\frac{\partial \mathbf{u}}{\partial \mathbf{x}}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$) |
|
PISO |
Pressure-Implicit with Splitting of Operators |
|
PRESTO |
Pressure-Staggering Option |
|
d |
diameter, m |
|
3D |
three dimensions |
|
r |
radius, m |
|
VT |
thermal velocity, ms-1 |
|
E |
energy, joule |
|
g |
gravitational acceleration, m.s-2 |
|
h |
Heat transfer coefficient, w.m-2. k-1 |
|
k |
thermal conductivity, w.m-1. k-1 |
|
k |
local surface curvature |
|
Pr |
Prandtl number, dimensionless |
|
Pe |
Peclet number, dimensionless |
|
$\vec{n}$ |
surface normal |
|
d |
differential |
|
ReT |
thermal Reynolds number, dimensionless |
|
MaT |
thermal Marangoni number, dimensionless |
|
s |
source term |
|
T |
temperature, K |
|
To |
reference temperature, K |
|
t |
time, s |
|
$\vec{v}$ |
velocity vector, (u1, u2, u3) |
|
x |
x coordinate, m |
|
y |
y coordinate, m |
|
mm |
millimeter, mm |
|
Greek symbols |
|
|
ρ |
density, kg/m3 |
|
a |
thermal diffusivity, m2⸳s-1 |
|
µ |
dynamic viscosity, kg⸳m-1⸳s-1 |
|
λ |
thermal conductivity, W⸳m-1⸳K-1 |
|
∂ |
rate of change |
|
σ |
surface tension, N⸳ m-1 |
|
σo |
surface tension at a reference temperature |
|
Subscripts |
|
|
b |
bubble, m |
|
T |
Temperature, K |
|
x |
local distance, m |
|
G |
gas |
|
o |
reference |
|
eff |
effective thermal conductivity |
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