*4.1. Thermal Field in Liquid Foam*

For the study case of thermal foam, randomly generated bubbles are initiated at the bottom of a partially filled flask-like domain and left to rise due to the gravitational effect (buoyancy) as shown in Figure 6. No slip boundary condition is set on the four walls. The thermal field model is implemented, where Dirichlet boundary condition is set for the three and bottom (source) walls. All the simulation parameters are shown in Table 3. The D2Q25 interaction model is implemented to obtain stable foam, while the SC potential function for the interaction between the two components are set using the case *B* from Equation (8), following the scope of work by Dollet et al. [45] and Fei et al. [46], as it provided much higher bubble stability than case *A*. For the other two same-component interactions, case *B* is still used. The surface tension is re-estimated for this case following the same procedure as in Section 3.1, to evaluate the Bonds number. Although the interaction parameters presented by Fei et al. [46] to reach the highest positive disjoining pressure were *<sup>G</sup>*<sup>12</sup> <sup>=</sup> *<sup>G</sup>*<sup>21</sup> <sup>=</sup> 3.0, *<sup>G</sup>*<sup>11</sup> <sup>=</sup> *<sup>G</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>8.0 and *<sup>G</sup>*, <sup>11</sup> <sup>=</sup> *<sup>G</sup>*, <sup>22</sup> = 7.0, the required repulsive forces for this bubble rising case between the same components seemed to be higher than the attraction ones in order to maintain stable bubbles.

**Figure 6.** Different time steps starting from the rising of random bubbles to foam generation. Temperature contours are shown, while the bubbles and liquid interfaces are clearly visible.


**Table 3.** Simulation parameters for the thermal foam case study.

\*: Cone angle.

The simulation time evolution shows the generation of foam layers from rising bubbles as shown in the figure. Before imposing the bubbles' flow, a pre-time loop is used to reach the steady state for the thermal field. A one-way coupled temperature field is also shown, where the thermal convection due to the bubbles' motion and rearrangement in the lamella is clearly visible. Thermal diffusion inside bubbles is shown to be much slower than in the liquid due to the different Prandtl number. It is not possible to reach this kind of stable foam using solely the short-range interaction model. The bubbles' deformation without or with much delayed coalescence, either with the upper interface or between bubbles themselves, are obvious through the shown interfaces. Contact angle is not considered in this work, which will affect the upper interface.

This setting of the simulation shows a promising tool to investigate the thermal and flow field in thermal foam, where a wide range of length scales are resolved, starting from the lamellar film and two bubbles' interaction to a full-scale domain with a cluster of foam as presented. If the model is extended to include the phase change in foam, one can expect a breakthrough in the field of simulation and optimization of industrial rectification columns, where foam formation is a critical and an often-encountered issue.
