The management of gaseous inclusions or particles in glass is a challenge existing throughout the glass industry. Depending on the processes considered, their gravity-driven migration can be altered by natural convection movements. In this context, understanding the dynamics involved within multiphysics convective systems is essential. This subject proposes to model/simulate the transient evolution of a multiphase system representative of a fluid mixture with complex rheology (thermo-dependence and particle effects) containing particles and bubbles subjected to a thermal constraint in a cylindrical geometry. Doing so will allow to estimate the different evolution regimes, their transitions, and the characteristic quantities involved, such as characteristic times of stabilization of particles and bubbles, or their characteristic sizes and distribution zones. These results will be validated by experimental data from the corpus available within the DPME, which also has extensive experience in the development of vitrification processes and their modeling. The knowledge produced will be directly useful to the sector for the production of vitrified nuclear waste, as the processes must be optimized to take into account the presence of solid, liquid, and gaseous phases, which need to be characterized.