Weakly nonlinear stability analysis of subcritical electrohydrodynamic flow subject to strong unipolar injection

Abstract

We analyse in detail the weakly nonlinear stability of electrohydrodynamic (EHD) flow of insulating fluids subject to strong unipolar injection, with and without cross-flow. We first consider the hydrostatic electroconvetion induced by a Coulomb force confined between two infinite flat electrodes, taking into account the charge diffusion effect. The effects of various non-dimensionalized parameters are examined in order to depict in detail and to understand better the subcritical bifurcation of hydrostatic electroconvetion. In addition, electrohydrodynamics with low-or high-Re cross-flow is also considered for investigating the combined effect of inertia and the electric field. It is found that the base cross-flow is modified by the electric effect and that, even when the inertia is dominating, the electric field can still strengthen effectively the subcritical characteristics of canonical channel flow. In this process, however, the electric field does not contribute directly to the subcriticality of the resultant flow and the intensified subcritical feature of such flow is thus entirely due to the modified hydrodynamic field as a result of the imposed electric field. This finding might be important for flow control strategies involving an electric field. Theoretically, the above results are obtained from a multiple-scale expansion method, which gives rise to the Ginzburg-Landau equation governing the amplitude of the first-order perturbation. The conclusions are deduced by probing the changes of value of the coefficients in this equation. In particular, the sign of the first Landau coefficient indicates the type of bifurcation, being subcritical or supercritical. Moreover, as a quintic-order Ginzburg-Landau equation is derived, the effects of higher-order nonlinear terms in EHD flow are also discussed.

Three-dimensional electroconvective vortices in cross flow

“…In the space-charge-limited (SCL) injection ( C →), the flow stability depends on c T alone [77][78][79]. The effect of charge diffusion was investigated by Zhang et al by employing linear stability analysis on the EC problem with a Poiseuille flow [42], and by non-linear analysis using a multiscale method [47]. The authors found that the charge diffusion has a nonnegligible effect on c T , and the transient behavior depends on the Reynolds number ( Re ) [42,47].…”

“…The effect of charge diffusion was investigated by Zhang et al by employing linear stability analysis on the EC problem with a Poiseuille flow [42], and by non-linear analysis using a multiscale method [47]. The authors found that the charge diffusion has a nonnegligible effect on c T , and the transient behavior depends on the Reynolds number ( Re ) [42,47]. Li et al performed linear analysis to study convective instabilities in EHD-Poiseuille flow and found that the ratio of the Coulomb force to the viscous force has an impact on the transition of transverse rolls from convective to absolute instability [80].…”

Two relaxation time lattice Boltzmann method coupled to fast Fourier transform Poisson solver: Application to electroconvective flow

“…Governing equations yield four non-dimensional parameters that describe the system's state [12,[17][18][19][20][21].…”

“…where H is the distance between the electrodes (distance between the two infinite plates), 0  is the injected charge density at the anode, and 0   is the voltage difference applied to the electrodes. The physical interpretation of these parameters are as follows: M -ratio between hydrodynamic mobility and the ionic mobility; T -a ratio between electric force to the viscous force; C -charge injection level; and Fe -reciprocal of the charge diffusivity coefficient [12,17].…”

“…Electro-hydrodynamics (EHD) studies the interaction of the fluid with electric field [1]. As a subset of EHD, electroconvection (EC) is a phenomenon where convective transport is induced by unipolar discharge into a dielectric fluid [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Felici first performed a stability analysis of EC using a non-linear hydraulic model [22,23].…”