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Electrokinetic phenomena involving the transport of ions in fluids generally appears in biological system, hydrodynamics and geodynamics. The underlining mathematical models involve the nonlinear coupling between the fluid motion and ion transportation. Performing efficient, accurate and structure preserving computation for these electrohydrodynamic models is challenging due to their multiphysics and complex nature. The Navier–Stokes–Poisson–Nernst–Planck (NSPNP) system is a widely used electrokinetic model in the existing literature. In this paper, a linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme is developed for the NSPNP system. First, a square root transformation is employed to ensure the positivity of ion concentrations. Next, a specific ordinary differential equation is introduced to deal with the nonlinear coupling terms satisfying the "zero-energy-contribution" feature, which is combined with the scalar auxiliary variable method for nonlinear potentials to achieve a decoupled energy stable scheme. Therefore, only several linearly independent equations need to be solved at each time step. The staggered grid finite difference method is used for the spatial discretizations, and the unconditional energy stability of the fully discrete scheme is established. Finally, the unique solvability of the scheme is proved. Numerical experiments are carried out to demonstrate the accuracy and performance of the proposed scheme, and the effect of electric Coulomb force on the flow and motion of ions are also illustrated. • A linear, second-order accurate and unconditionally energy stable scheme is developed for the Navier–Stokes–Poisson–Nernst–Planck system. • A square root transformation is employed to ensure the positivity of ion concentrations. • Only several linearly independent equations need to be solved at each time step. • The unique solvability of the scheme is proved. [ABSTRACT FROM AUTHOR]

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. • The proposed scheme obeys an energy dissipation law at the fully discrete level. • The proposed scheme is a linear, second-order accurate in time and unconditionally energy stable. • The proposed scheme can be split into linear subsystems and solved efficiently at each time step. [ABSTRACT FROM AUTHOR]