Linear, decoupled, positivity preserving, positive-definiteness preserving and energy stable schemes for the diffusive Oldroyd-B coupled with PNP model

In this paper, we present a first-order finite element scheme for the viscoelastic electrohydrodynamic model. The model incorporates the Poisson-Nernst-Planck equations to describe the transport of ions and the Oldroyd-B constitutive model to capture the behavior of viscoelastic fluids. To preserve the positive-definiteness of the conformation tensor and the positivity of ion concentrations, we employ both logarithmic transformations. The decoupled scheme is achieved by introducing a nonlocal auxiliary variable and using the splitting technique. The proposed schemes are rigorously proven to be mass conservative and energy stable at the fully discrete level. To validate the theoretical analysis, we present numerical examples that demonstrate the convergence rates and the robust performance of the schemes. The results confirm that the proposed methods accurately handle the high Weissenberg number problem (HWNP) at moderately high Weissenberg numbers. Finally, the flow structure influenced by the elastic effect within the electro-convection phenomena has been studied.