A mixture model to take into account the flow of small carbon dioxide bubbles dissolved in a liquid is presented. The model describes the evolution of the velocity fields (mixture and gas), the pressure and the volume fraction of gas. The system of equations is derived from mass and momentum conservation of the mixture and gas. Well-posedness is proved for a simplified problem when the volume fraction of gas is known and small. A priori error estimates are proved for a stabilized finite element approximation. An industrial application pertaining to aluminium electrolysis is presented. Numerical results indicated that the effect of gas bubbles on the flow cannot be neglected.