Mixed finite element methods for a heat-assisted model of sedimentation
A model for the consolidation-sedimentation process incorporating thermal effects is analysed. The model is motivated by novel applications to heat-assisted sedimentation of mineral suspensions. A system of convection-diffusion equations for the concentration and temperature are coupled to flow equations for the volume-average velocity and pressure. The model is supplemented with Dirichlet and mixed boundary conditions. A quasi-stationary approach of the system is analysed by means of a mixed variational formulation, where additional variables such as the pseudostress tensor and variables for the gradient of the temperature and concentration are introduced. The existence and uniqueness of the system is shown making use of the Babu{\v s}ka-Brezzi theory and Banach's fixed-point theorem, under hypotheses of smallness of the data and additional regularity on the solutions. A discrete formulation based on finite element spaces is proposed and analysed, and the existence of solutions is achieved thanks to Brouwer's fixed-point theorem. Additionally, Céa and a priori estimates are derived for the mixed formulation. Finally, numerical examples are provided to demonstrate the performance of the model and to confirm the theoretical rates of convergence.