Instability of electroconvection in viscoelastic fluids subjected to unipolar injection

In this paper, a two-dimensional numerical study on the nonlinear behaviors of electrohydrodynamic flows of Oldroyd-B viscoelastic dielectric liquid subjected to unipolar injection is performed via the finite volume method. The entire set of coupled equations, which includes the Navier-Stokes equations, simplified Maxwell's equations, and conformation transport equations, is solved for the first time. The effects of elasticity on the nonlinear evolution of electroconvection and instability patterns are mainly investigated. Various physical models including free and rigid boundary cases are simulated entirely, and detailed analyses of stability parameters are performed. Convection and fluid motion instability are investigated and explained in detail, with a focus on the onset of motion transitions from a purely conducted state to losing its stability. It is found that the coupling of the electric field with the elasticity field gives rise to new instability and completely new mechanisms. In addition to instabilities such as subcritical bifurcation in electroconvection of Newtonian fluids, supercritical bifurcation and Hopf bifurcation are also possible as the first instability in electroconvection of viscoelastic fluids under free boundary conditions. Under rigid boundary conditions, the system with a large Weissenberg number can also lose its stability via earlier Hopf bifurcation. The stability threshold is not affected by the elastic effect if the Weissenberg number is small enough but decreases when the first instability of the system becomes Hopf bifurcation. Moreover, elasticity promotes the transition from a steady state flow to unsteady convection after the onset of convection. These phenomena are closely related to the elastic parameters.