Decoupled second-order energy stable scheme for an electrohydrodynamic model with variable electrical conductivity

Abstract

The motion of ions in the complex fluids widely appears in hydrodynamics, geodynamics and geophysics. Setting up suitable mathematical model, performing accurate and efficient numerical simulations are essential to understand the underlying physical principle in these phenomena. In this paper, we present a linear, second-order accurate in time, decoupled and unconditionally energy stable numerical scheme for solving an electrohydrodynamic power-law model with variable electrical conductivity. Based on a logarithm transformation for conductivity and the zero energy contribution property of the nonlinear coupling terms in the model, we derive an equivalent system by introducing a nonlocal auxiliary variable. The two-step backward differentiation scheme and the finite element method are used for the temporal and spatial discretizations, respectively. To decouple the system, the nonlocal splitting technique is employed to yield several linear subsystems, which can be solved very efficiently. Numerical simulations are carried out to demonstrate the second-order convergence in time and the unconditional energy stability. Moreover, the effects of Coulombic force and power-law fluid exponents are numerically investigated.