Finite element approximation of dielectrophoretic force driven flow problems

In this paper, we propose a full discretization scheme for the instationary thermal-electro-hydrodynamic (TEHD) Boussinesq equations. These equations model the dynamics of a non-isothermal, dielectric fluid under the influence of a dielectrophoretic (DEP) force. Our scheme combines an H-1-conformal finite element method for spatial discretization with a backward differentiation formula (BDF) for time stepping. The resulting scheme allows for a decoupled solution of the individual parts of this multi-physics system. Moreover, we derive a priori convergence rates that are of first and second order in time, depending on how the individual ingredients of the BDF scheme are chosen and of optimal order in space. In doing so, special care is taken of modeling the DEP force, since its original form is a cubic term. The obtained error estimates are verified by numerical experiments.