Direct numerical simulation of the thermo-capillary migration of bubble or drop in a liquid under reduced gravity

Introduction

The Marangoni convection is a well-known phenomenon in which an interfacial flow is generated by surface tension gradients. When such gradients originate from temperature variations, the induced flow is also known as the thermocapillary migration. For most of the liquids, the surface tension decrease with the temperature, inducing therefore a flow from hot zones to cold zones. The Marangoni convection is an active field of research since it may have a significant influence on many applications in materials processing involving bubble or drop in a liquid phase and also in the spatial field because it may drive the behavior of fluids encountered in spacecraft operating in reduced gravity. Indeed, numerous experimental and theoretical works have been realized in the last few years to study some aspects of the behavior of bubbles or drops in liquids (e.g. air bubble in hydrofluoroether or fluorocarbon drop in silicone oil) in micro-gravity conditions.

This work deals with the numerical study of the spontaneous motion of a spherical inclusion (bubble or drop), initially at rest, placed in a liquid undergoing a temperature gradient. The Marangoni shear stress appearing at the inclusion interface by this gradient induces then a flow around the inclusion, propelling it towards the warmer zone.

Modeling

This study is realized thanks to a two-dimensional axisymmetrical model of the momentum and heat transport phenomena in the continuous (liquid) phase and in the inclusion (gas bubble or liquid drop) without gravity. The model equations are written in an inertial reference frame attached to the center of the inclusion. Therefore, the boundary conditions of the computational domain evolve dynamically with the inclusion motion. The equations write in dimensionless form:

∂_tu_c + u_c· ∇u_c = -∇p_c + (Pr/Mg) (∇u_c+∇u_c^T ) + d_tU_s ,

∂_tT_c +u_c· ∇T_c = -(1/Mg)∇²T_c ,

in the continuous phase and

ρ^*(∂_tu_i + u_i· ∇u_i) = -∇p_i + (μ^*Pr/Mg) (∇u_i+∇u_i^T ) + ρ^*d_tU_s ,

∂_tT_i +u_i· ∇T_i = -(α^*/Mg)∇²T_i ,

in the inclusion. Mg is the Marangoni number and Pr is the Prandtl number. ρ^*, μ^* and α^* are the density, viscosity and thermal diffusivity inclusion-to-continuous phase ratio's, respectively. U_s refers to the instantaneous inclusion velocity and d_tU_s corresponds to its instantaneous acceleration.
d_tU_s is deduced from the total force applied on the inclusion interface and U_s is obtained from the time integral of d_tU_s.

At the interface between the inclusion and the continuous phases, it is considered that the temperature and the heat flux are continuous, that the tangent velocities are equals and that the normal velocities are zero. To take into account the Marangoni shear stress, the following conditions on the tangential stress is implemented:

((∇u_c+∇u_c^T )·n - μ^*(∇u_i+∇u_i^T )·n)·τ = ∇_ST_int ,

where τ is the unit tangent vector. The domain boundary towards the inclusion wake is considered as an open boundary regarding to the momentum and heat transports. At the domain boundary towards which the inclusion is moving, the velocity and the temperature are imposed. The velocity corresponds to the velocity inclusion U_s and the temperature is dynamically adapted to the inclusion displacement Z_s, deduced from the time integral of U_s.

Simulation results and discussion

The transient boundary-value problem is solved numerically using the commercial software COMSOL Multiphysics 4.4, considering the case of an air bubble in liquid hydrofluoroether. It is observed that the bubble motion induced by the temperature gradient reaches quickly a steady state, for which the viscous stresses in the gas and liquid phase equilibrate the Marangoni convection created by the actual temperature field realized around the bubble by the so-induced flow. It is especially shown that the gaseous phase plays an active role in dynamic evolution of the temperature field and may affect significantly the terminal velocity of the bubble. Indeed, by comparing with the simulation results obtained when the gas phase is not considered, it is found that the terminal velocity is smaller if the gas phase is taken into account. It is due to the fact that the bubble motion triggers a recirculatory flow inside the bubble. This internal flow tends to homogenize the temperature in the gas phase, decreasing the temperature gradient along the bubble interface and hence hindering the Marangoni convection. A parametric study is therefore realized by varying the values of Mg, Pr, ρ^*, μ^* and α^* to reach a better understanding of the interaction between the transport phenomena.

Perspectives

Another fundamental phenomenon which may affect significantly the Marangoni convection under these conditions is the gas-liquid mass transfer. Indeed, evaporation/condensation of the continuous liquid may take place and be coupled to absorption/desorption of dissolved component. These mass transfer phenomena are associated with their latent heat, increasing or decreasing locally the interfacial temperature and hence influencing the Marangoni dynamics. As a perspective, the present model will be therefore completed by a modeling of the mass transport in both phases and the interfacial boundary conditions will be implemented to take into account the heat source/sink generated by the mass transfers.