Aluminium is a metal sought in the industry because of its various physical properties. It is produced by an electrolysis reduction process in large cells. In these cells, a large electric current goes through the electrolytic bath and the liquid aluminium. This electric current generates electromagnetic forces that set the bath and the aluminium into motion. Moreover, large quantities of carbon dioxide gas are produced through chemical reactions in the electrolytic bath: the presence of these gases alleviates the density of the liquid bath and changes the dynamics of the flow. Accurate knowledge of this fluid flow is essential to improve the efficiency of the whole process. The purpose of this thesis is to study and approximate the interaction of carbon dioxide with the fluid flow in the aluminium electrolysis process. In the first chapter of this work, a mixture-averaged model is developed for mixtures of gas and liquid. The model is based on the conservation of mass and momentum equations of the two phases, liquid and gas. By combining these equations, a system is established that takes into account the velocity of the liquid-gas mixture, the pressure, the gas velocity and the local gas concentration as unknowns. In the second chapter, a simplified problem is studied theoretically. It is shown that under the assumption that the gas concentration is small, the problem is well-posed. Moreover, we prove a priori error estimates of a finite element approximation of this problem. In the third chapter, we compare this liquid-gas model with a water column reactor experiment. Finally, the last chapter shows that the fluid flow is changed in aluminium electrolysis cells when we take into account the density of the bath reduced by carbon dioxide. These changes are quantified as being of the order of 30\% and explain partially the differences between previous models and observations from Rio Tinto Aluminium engineers.