Numerical investigation of electrohydrodynamic instability and bifurcation in a dielectric liquid subjected to unipolar injection

Abstract

The electroconvection of dielectric liquids subjected to unipolar injection is numerically studied in a two-dimensional cavity. To understand the Coulomb force driven electroconvective instabilities and bifurcations in the closed cavity, a high resolution upwind scheme is applied to carry out linear stability analysis and numerical simulation of nonlinear electrohydrodynamic equations. We focus on the strong injection case, where the non-dimensional injection parameter C is fixed at C=10. The length to height ratio of the cavity is fixed at 0.614. The numerical simulations are performed subject to free and rigid lateral wall boundary conditions. Two dimensionless mobility parameters M=5 and M=10 are considered for each kind of the lateral boundary conditions. Our linear stability analysis result is consistent with previous theoretical predictions. The nonlinear behaviors of the investigated system beyond the convection threshold are carefully examined. Abundant bifurcation phenomena, such as the pitchfork bifurcation, Hopf bifurcation, quasi-periodic Hopf bifurcation and heteroclinic bifurcation, have been observed. Some nonlinear flow features, such as the hysteresis loops, coexistence of multiple solutions and transition to chaos, have also been demonstrated. Our simulation results show that flow structure and bifurcation sequence can be greatly affected by the specified lateral boundary conditions and the chosen mobility parameters. Besides the static state, a series of steady and unsteady solutions has been calculated, like the one-cell steady state, one-cell periodic state, one-cell quasi-periodic state, and the one-cell chaotic state. All these convective states with two-cell structure are also obtained. Moreover, a periodic state oscillating between the one-cell structure and the two-cell structure has been found in flow domain bounded by rigid walls.