Linear, second-order, unconditionally energy stable scheme for an electrohydrodynamic model with variable density and conductivity

Abstract

The motion of ions in the complex fluids widely appears in biofluids, hydrodynamics, geodynamics and geophysics. Performing efficient and accurate computation for electrohydrodynamic flows is challenging due to the multiphysics and complex nature of the problem. In this paper, we describe a new second-order linear energy stable time-stepping scheme for a variable density electrohydrodynamic power-law model, which satisfies an energy dissipation law. Firstly, a square root transformation for the fluid density and a logarithmic transformation for the electrical conductivity are used to enforce the positivity-preserving property for these two physical parameters. Secondly, by using the zero energy contribution property of the nonlinear coupling terms in the model and introducing a nonlocal auxiliary variable, an equivalent continuous formulation of the model is obtained. Finally, a two-step backward differentiation scheme is proposed for the resulting equivalent model to construct the time-stepping scheme, and the finite element method is used for spatial discretization. The scheme is proved to be unconditionally energy stable and a nonlocal splitting technique can be utilized to yield several linear subsystems, which can be solved more easily. Numerical simulations are carried out to demonstrate the accuracy and stability of the scheme. The dispersion of electrical conductivity and the effect of electrical force on the Rayleigh–Taylor instability are studied for both shear-thinning and shear-thicking fluids.