Stable and decoupled schemes for an electrohydrodynamics model

Abstract

In this paper, we study numerical solutions of an electrohydrodynamics model. The considered model appears in the description of electric convection dynamics arising from unipolar charge injection on the boundary of insulating liquid, which is a coupling of the Navier–Stokes equations, charge transfer equation, and potential energy equation. A class of stable numerical schemes is proposed and analysed for this coupled equation system. The advantage of the proposed schemes is twofold: (1) they are unconditionally stable, consequently the choice of time step size only concerns the accuracy requirement; (2) they decouple the charge density and potential energy from the Navier–Stokes equations, and therefore can be implemented efficiently. The numerical examples provided in the paper show that the proposed schemes achieve the expected convergence rate, and can be used to accurately simulate the changes of flow field and electric field induced by the electrical convection. We first consider the case of constant density, then extend the construction, analysis, and validation of the schemes to the case of variable density.