Abstract

The interface between two immiscible fluids can become unstable under the effect of an imposed tangential electric field along with a stagnation point flow. This canonical situation, which arises in a wide range of electrohydrodynamic systems including at the equator of electrified droplets, can result in unstable interface deflections where the perturbed interface gets drawn along the extensional axis of the flow while experiencing strong charge build-up. Here, we present analytical and numerical analyses of the stability of a planar interface separating two immiscible fluid layers subject to a tangential electric field and a stagnation point flow. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a local linear stability analysis in the vicinity of the stagnation point to study the behaviour of the system in terms of the relevant dimensionless groups of the problem. The local theory is complemented with a numerical normal-mode linear stability analysis based on the full system of equations and boundary conditions using the boundary element method. Our analysis demonstrates the subtle interplay of charge convection and conduction in the dynamics of the system, which oppose one another in the dominant unstable eigenmode. Finally, numerical simulations of the full nonlinear problem demonstrate how the coupling of flow and interfacial charge dynamics can give rise to nonlinear phenomena such as tip formation and the growth of charge density shocks.

Highlights

  • A century ago, Zeleny (1917) photographed instabilities of electrified interfaces, sparking interest into understanding the phenomenon

  • We present results on the stability of the system by comparing predictions from the local linear theory (LT) of § 4 and from the numerical linear stability analysis (Num-LSA) of § 5.2

  • As we show in figure 8 and discuss further below, this approximation results in a decrease in growth rate when compared with (4.9), suggesting that charge convection is destabilizing under the local theory

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Summary

Introduction

A century ago, Zeleny (1917) photographed instabilities of electrified interfaces, sparking interest into understanding the phenomenon. Depending on the electric properties of the fluids, the surface flow is directed either to the poles or to the equator. The latter case is shown in figure 1(a). We develop a two-dimensional model to study the dynamics of a system of two superimposed layers of fluids subject to a tangential electric field and a stagnation point flow. 2. Problem definition and governing equations We study EHD instabilities that arise at the interface S between two immiscible fluids under the combined effects of a tangential electric field E0 and of an imposed stagnation point flow u∞(x), to be specified more precisely later. In all transient nonlinear simulations, we use the periodic array of point forces as background flow

Non-dimensionalization
Local linear stability theory
Boundary element method and numerical stability
Boundary element method
Numerical linear stability analysis
Results and discussion
Effect of tangential electric field
Combined effects of electric field and flow
Mechanisms of charge transport in the dominant mode of instability
Conclusions
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