Numerical Analysis of the Effect of Marangoni Convection in a Shallow Cylinder

A numerical study in a two-dimensional, axisymmetric cylindrical geometry is performed to evaluate the effect of Marangoni convection in shallow liquid layers. The test fluid used is silicone oil. The study aims to understand Marangoni convection for a range of temperature gradients (9.0 K, 11.5 K, and 14.0 K), layer depths (4.0 mm, 4.5 mm, and 5.0 mm) under earth gravity (1.0 g), and microgravity (0.5 ×10 -3 g) conditions. The finite element method is used to solve the present problem and the numerical models were created using COMSOL V5.6 commercial tool. The influence of fluid meniscus on Marangoni convection is analyzed and found out the thermo-capillary force is less significant in driving the fluid. The present investigation in microgravity ensures the Marangoni convection dominates natural convection. For all the test conditions, it is found that Marangoni convection is influenced more by temperature difference than layer depths. It is observed that the velocity magnitude deviates around 45% for the selected temperatures at 4 mm layer depth, whereas, a 3% deviation is only found between the selected ranges of layer depths at a temperature difference of 11.5 K. From the microgravity study, it is clear that for velocity magnitude at z = 2.5 mm the Marangoni convection contributes to almost 90% of the total convection. It was also observed that the presence of Meniscus will reduce the free surface velocity by almost 60 %.

(1) Where Ma and Ra stand for the Marangoni and Rayleigh numbers respectively, σ represents surface tension, g denotes gravitational acceleration, R denotes the characteristic length, and ρ, β represents liquid density, the volumetric expansion coefficient respectively.
The capillary length (lc), the length scaling factor that relates to gravity and surface tension is Substituting the value of in Eq. (2) we get; Eq. (3) shows that Ma is inversely proportional to the characteristic's length (R). Benard-Marangoni convection thus predominates for thin liquid films (R  lc), whilst, Rayleigh-Benard convection dominates for thick liquid films (R  lc). Marangoni convection is not limited to just Benard-Marangoni convection but can be classified based on the change in temperature and concentration that induces gradients of surface tension. Flow-induced by surface tension gradients due to temperature distribution is called thermo-capillary convection and due to concentration gradients is known as solute-capillary convection. In the present study, only thermo-capillary convection is considered to investigate the flow characteristics.
Marangoni convection has a huge scope as it has various engineering applications, such as welding, crystal growth, electron beam melting, etc. Developments in the study of surface tension flows helped to understand the convection process that happens in the meltpool during crystal growth. The earliest studies of Marangoni convection in crystal growth go back to more than 40 years [1]. The studies on the surface tension flows in floating zone melting showed that the radial inhomogeneities were produced due to thermo-capillary forces at smaller driving forces. At large driving forces, the flow is turbulent and oscillatory. The flow driven by surface tension in the floating zone of sodium nitrate (NaNO3) is experimentally observed [2]. Temperature oscillations induced by surface tension have been found inside the zone. Surface tension-driven flow induced by temperature gradient can be of high magnitude at a large free surface and higher values of the temperature gradient. Later, these studies were extended to find the magnitude of thermo-capillary convection in larger melt volumes of 20 mm × 20 mm ×12.5 mm [3]. The numerical analysis of Czochralski (CZ) crystal growth can be validated using the results of [4], who conducted both numerical and experimental investigations with a fluid of Pr = 6.8.
Nakanishi et al. [5] realized that Marangoni flow aids in the growth of crystals with low oxygen concentration. Additionally, they found that a strong surface flow was directed from the crucible wall towards the crystal under the influence of surface tension. The studies on the effect of gradients in surface tension on the flow field and the oxygen content of the melt [6] showed that the Marangoni convection accelerates the flow velocity at the melt surface and, as a result, affects how rapidly oxygen evaporates from the free surface. The time-dependent analysis and calculation of the thermal and flow fields in Czocharalski silicon melt [7] confirmed that the concentration of oxygen in crystals is influenced by Marangoni convection. An increase in Marangoni number will strengthen the radial inward flow which shifts the flow pattern from circumferential to spiral in the Czocharalski silicon melt [8].
Meanwhile, the surface tension strengthens the buoyancy convection and the isotherms were observed to be curved downwards.
Li et al. [9] have numerically studied the surface patterns of silicon melt in CZ furnaces, by using a three-dimensional unsteady simulation of thermo-capillary flow for two Riley et al. [11] have studied the effect of combined buoyancy-Marangoni convection in a thin rectangular geometry. The emphasis of the study was on the formation of hydrothermal wave instabilities. They have studied the transition from steady, unicellular convection to oscillatory convection and confirmed the presence of oscillatory hydrothermal waves. Pure Marangoni convection without the presence of buoyancy force in a cylindrical geometry was studied using Particle Image Velocimetry (PIV) [12]. Studies were conducted at different layer depths of 3.9, 4.0, and 4.3 mm, and temperatures of 9.6, 9.9, and 11.8 K. By heating from above and keeping shallow depths, buoyancy effects were diminished.
Numerical simulations in a differentially heated rectangular cavity [13], accurately predicted the flow in the double-layer system even for large aspect ratios by developing a single-layer problem with a modified tangential-stress condition at the interface. Convective flows in a two-layered system was observed with a horizontal temperature gradient [14]. The nonlinear simulation of the silicone oil-HT70 system showed that the direction to which the wave propagates depends on the ratio of layer thickness and Marangoni number. Hydrothermal wave instabilities in Marangoni flow were studied using the shadowgraph method [15]. Two types of waves that are dependent on fluid depth were observed. At larger fluid depths, the first hydrothermal wave occurs which propagates from the radial direction, while at smaller fluid depths the other hydrothermal wave propagates radially at the onset. Studies to determine the characteristics of thermo-capillary flow in differentially heated shallow and deep annular pools indicated that three-dimensional stationary flow is observed more in deep pools (d = 5 mm) and in shallow pools, (depth = 1 mm) hydrothermal wave with curved spokes predominates [16].
The analysis on the effect of heat dissipation on thermo-capillary convection indicated that the hydrothermal waves are dominant at weak heat dissipation rates, but at higher dissipation rates longitudinal rolls near the outer wall are dominant [17]. The thermocapillary convection in ethanol evaporation at two different layer depths of 55 mm and 60 mm was studied [18]. The experimental study of the temperature distributions along the free surface in both shallow and deep vessels showed that the core holds a large cold region that is home to numerous Marangoni convective cell patterns. The surface velocity is estimated to be in the order of 0.7 cm/s. A shallow annular pool with different aspect ratios containing a binary mixture was chosen to conduct a series of numerical and experimental studies on thermo-capillary convection [19,20]. Results showed the hydrothermal waves and chaos phenomena will orderly appear with an increase in the surface velocity. Also, transition and flow characteristics are dependent on capillary ratio, aspect ratio, and thermal and surface velocity. A shallow annular pool with different evaporation rates was studied experimentally to understand the effect of surface dissipation [21]. In microgravity experiments, no oscillations were observed even at higher Ma numbers.
Experiments were carried out to understand Benard-Marangoni-Instability on surfaces of silicone oil during the microgravity phase of the sounding rocket MAXUS 2 on which 10 centistokes silicone oil was heated from below to observe convection patterns [24]. Benard-Marangoni convection was visualized and the onset of convection as a function of aspect ratio is measured. The importance of the Marangoni effect under microgravity is numerically investigated by Giangi et al. [25]. Results were presented for values of Ma up to 16,120 and Ra of 5. Additionally, they provided typical space gravity levels, which can range from 0.1 to 2,500 microgravity conditions. A two-dimensional, steady thermo-capillary flow in a liquid bridge under microgravity conditions with a wide range of aspect ratios was studied [26]. A liquid bridge is a mass of liquid sustained by the action of the surface tension force between two parallel supporting disks. The stability of a thermo-capillary flow is highly dependent on the normal temperature gradient at the liquid-gas interface. A three-dimensional thermocapillary flow in a liquid bridge under microgravity showed that an increase in Marangoni number will strengthen and accelerate the instability of thermo-capillary convection [27]. A correlation for estimating Reynolds number (Re) in terms of (Ma) and (Pr) is developed [28] and can be used to predict whether Marangoni convection or buoyancy convection is predominant in the flow.
Recently, the effect of Prandtl number and pool curvature on the stability of thermocapillary in annular pools is studied using linear stability and energy analysis [29]. Energy analysis provides information about by which kinetic energy and thermal energy are supplied From the above literature, it is clear that the investigation of Marangoni convection has a huge scope in various engineering applications, such as controlling oxygen concentration in crystal growth [5][6][7], material processing under microgravity [23], [26], studies in shallow annular pools [21], [29], behavior is also examined.

Physical model
A two-dimensional axisymmetric model selected for the present study is shown in (4) to (7).

Marangoni convection boundary condition
The Marangoni stress boundary condition gives the relationship between surface tension gradients and local tangential shear stress and is applied at the top free surface. The kinematic boundary condition [32] is expressed as Where τ, is shear stress at the free surface, μ denotes dynamic viscosity and n is local coordinate normal.
∂σ ∂T , ∇ S T is the surface tension temperature coefficient and tangential surface temperature gradient, respectively.

Methodology
The governing equations are solved by using the finite element method. COMSOL Multiphysics V5.6 is used to solve the two-dimensional steady state, incompressible, laminar flow. In COMSOL "Fluid Heat Transfer" module and "Laminar Flow" module are used to apply the boundary conditions. "Non-isothermal flow" and "Marangoni effect" Multiphysics are used to solve the problem. Non-isothermal flow enables to apply buoyancy effects while "Marangoni effect" apply thermal shear stresses. The single set of algebraic equations for all of the relevant physical models is created using the "Fully coupled technique". These equations are implemented in a single iteration scheme that is repeated until convergence is achieved. For continuity, the residual criterion of 1 x 10 -9 is set in the solution of governing equations. Once the value becomes less than the set tolerance criterion, the solution is assumed to be converged.

Grid Independent test
Grid independent test is carried out from course mesh to fine mesh by varying the total number of elements from 29895, 38688, 44640, 53398, and 96013. Fig. 3 shows the surface velocity of silicon oil in the radial direction for the selected range of mesh sizes. From Fig. 3 it is clear that 53398 elements can be taken for study as the variation from the previous mesh size is minimum.  zoomed view for d = 3.9 mm, ∆T = 9.9 K

Validation
The flow patterns in a cylindrical geometry of layer depth d = 3.9 x 10 -3 m and a temperature difference ∆T = 9.9 K is compared with the results of [12]. The velocity contours of the present study are compared with the literature as shown in Fig. 4. The oil is heated from the top using a copper rod that creates a temperature gradient as the bottom and side walls are kept at a lower temperature. This makes the surface tension of the heated region to be lower than that of the colder regions. Since the flow only moves from a region of lower surface tension to higher surface tension, the oil flows from the hot tip region to colder areas.  In the presence of Marangoni convection, when the surface tension at the center drops, the free surface flow is carried from the hot center towards the colder sidewalls. From Fig. 7 (b) the presence of secondary cells (cell 1 and cell 2) within the main cell is clearly visible.
These cells and the region between are responsible for the sudden deceleration and acceleration of the fluid in the bulk region. Hence the flow is distorted and is not following a uniform path. The sudden drop in velocity afterward can be attributed to the smaller temperature gradients.
From the comparative study, it is evident that the overall velocity for Marangoni flow is much higher than that of non-Marangoni flow and it can be credited to the convection driven by surface tension.  Fig. 7). Further away from the center of cell 1, a marginal increase in the velocity magnitude is observable before dropping again as the flow approaches the sluggish cell 2 regions. Once the flow gets past cell 2 an increase in the velocity is noticeable before decelerating to its minimal value.

The effects of temperature gradient on Marangoni convection
The Streamline contours were plotted for three different temperature values: ∆T = 9.0 K, 11.5 K, and 14.0 K for a depth of 4 x 10 -3 m in Fig. 10. For all temperature differences, along with the main cell, two other cells rotating in the same direction are visible. For 11 K and 14 K, cell 2 is clearly more separated than the temperature difference of 9 K. between the lowest and highest temperature gradients is approximately 45%. Hence it can be concluded that at a higher temperature gradient, the surface velocity will also increase and has a major impact on Marangoni flow. when it touches another material or surface. All fluids possessing surface tension will make a meniscus by adjusting their shape to minimize surface energy. Also with the help of the meniscus, the surface tension of a particular fluid can be calculated. Hence the presence of meniscus can't be ignored. The streamline contour in the presence of a meniscus is presented in Fig. 15. The Meniscus is modeled based on the experimental study of [12], where the oil is making a 90º contact angle with the hot tip. The Presence of the meniscus locally inclines the interface and temperature gradient [22]. Streamline near the tip is more inclined upwards as fluid tends to flow up the meniscus. Thus, it may be inferred that the meniscus plays a significant role in the heat transfer between the hot tip and the surrounding areas.      The influence of temperature gradient and layer depth on Marangoni convection is studied for the selected range of ∆T and d. It is found that the maximum velocity deviation is around 45 % and 3 % respectively.  It can be concluded that for the chosen conditions, the temperature difference is influencing Marangoni convection more than layer depth.  The presence of meniscus causes the surface velocity to be smaller which makes Marangoni convection less effective in driving the flow. The peak velocity magnitude gets reduced by 60 % in the presence of the meniscus.  Microgravity study proved that the layer depths selected are thin enough for Marangoni convection to be dominant.  It was found that pure Marangoni convection dominates in the presence of microgravity that helps in improving the quality of the material produced such as a single crystal, welded joints etc.

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