A nonlinear electrohydrodynamic stability analysis of a thermally stabilized plane layer of dielectric liquid

Abstract

The nonlinear stability of a thermally stabilized horizontal plane layer of dielectric liquid subjected to unipolar charge injection at a voltage near the linear instability threshold is investigated using a normal-mode cascade analysis valid for small perturbation amplitudes. In this first analysis, the primary mode is chosen to be a system of parallel rolls whose amplitude varies aperiodically with time. The branching behaviour at the critical voltage is found to reflect the distinction, apparent in the linear instability problem, between an essentially isothermal space-charge instability and an instability dominated by the effects of an ion mobility varying with temperature. The effect of motion on heat and charge transfer through the system is also considered. Furthermore, in certain cases it appears that overstability is the preferred form of linear instability.

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

“…ETHD studies the flow dynamics of fluids subject to at the same time the potential difference and a heat gradient; this combined effect is interesting because proper manipulation of the electric field can lead to increased efficiency of heat transfer, and thus imply various industrial applications. In fact, the weakly nonlinear analysis of ETHD has been performed by Worraker & Richardson (1981) following the linear stability analyses of Turnbull (1968). By deriving the Ginzburg-Landau equation until the third order, Worraker & Richardson (1981) successfully described the flow transition of ETHD from a supercritical bifurcation to a subcritical one when an electric potential of increasing strength is imposed across two differentially heated plates.…”

“…In fact, the weakly nonlinear analysis of ETHD has been performed by Worraker & Richardson (1981) following the linear stability analyses of Turnbull (1968). By deriving the Ginzburg-Landau equation until the third order, Worraker & Richardson (1981) successfully described the flow transition of ETHD from a supercritical bifurcation to a subcritical one when an electric potential of increasing strength is imposed across two differentially heated plates. This result has also been recently obtained and discussed in Traoré et al (2010).…”

“…No detailed parametric study was presented in their article. In the ETHD work by Worraker & Richardson (1981), the pure EHD flow in a weakly nonlinear phase is marginally discussed as a special case when the thermal difference k 1 across two plates is zero. On the other hand, EHD with cross-flow was rarely studied in the past.…”

“…This would be essential for the design of certain flow control strategies employing an electric field as the manipulating agent; for example, the modification to the wake flow in a developed Poiseuille flow using the electric field (Mccluskey & Atten 1988) or turbulence modification using EHD (Soldati & Banerjee 1998). In the end, compared to Worraker & Richardson (1981), who obtained the cubic GLE, we derive the GLE until the fifth order; therefore, a stable solution to the GLE can be obtained for the subcritical EHD when the fifth-order nonlinear term stabilizes the flow.…”

“…By substitution, we obtain where equation E * t E t = 1 has been used since µ is vanishing at the critical condition. Furthermore, we can define at the critical condition, following Worraker & Richardson (1981) …”