Electrohydrodynamics deals with fluid motion induced by electric fields. In the mid 1960s GI Taylor introduced the leaky dielectric model to explain the behavior of droplets deformed by a steady field, and JR Melcher used it extensively to develop electrohydrodynamics. This review deals with the foundations of the leaky dielectric model and experimental tests designed to probe its usefulness. Although the early experimental studies supported the qualitative features of the model, quantitative agreement was poor. Recent studies are in better agreement with the theory. Even though the model was originally intended to deal with sharp interfaces, contemporary studies with suspensions also agree with the theory. Clearly the leaky dielectric model is more general than originally envisioned.

ELECTROHYDRODYNAMICS:The Taylor-Melcher Leaky Dielectric Model

Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeter-scale nozzle. The solid fibers are notable for their very small diameters (<1 μm). Recent experiments demonstrate that an essential mechanism of electrospinning is a rapidly whipping fluid jet. This series of papers analyzes the mechanics of this whipping jet by studying the instability of an electrically forced fluid jet with increasing field strength. An asymptotic approximation of the equations of electrohydrodynamics is developed so that quantitative comparisons with experiments can be carried out. The approximation governs both long wavelength axisymmetric distortions of the jet, as well as long wavelength oscillations of the centerline of the jet. Three different instabilities are identified: the classical (axisymmetric) Rayleigh instability, and electric field induced axisymmetric and whipping instabilities. At increasing field strengths, the electrical instabilities are enhanced whereas the Rayleigh instability is suppressed. Which instability dominates depends strongly on the surface charge density and radius of the jet. The physical mechanisms for the instability are discussed in the various possible limits.

https://www.seas.harvard.edu/brenner/I_Stability.pdf

Electrospinning and electrically forced jets. I. Stability theory

Current and droplet size in the electrospraying of liquids. Scaling laws

Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeter-scale nozzle. This article uses the stability theory described in the previous article to develop a quantitative method for predicting when electrospinning occurs. First a method for calculating the shape and charge density of a steady jet as it thins from the nozzle is presented and is shown to capture quantitative features of the experiments. Then, this information is combined with the stability analysis to predict scaling laws for the jet behavior and to produce operating diagrams for when electrospinning occurs, both as a function of experimental parameters. Predictions for how the regime of electrospinning changes as a function of the fluid conductivity and viscosity are presented.

https://www.seas.harvard.edu/brenner/II_Applications.pdf

Electrospinning and electrically forced jets. II. Applications

When a liquid is subject to a sufficiently strong electric field, it can be induced to emit thin fluid jets from conical tip structures that form at its surface. Such behaviour has both fundamental and practical implications, from raindrops in thunderclouds to pendant drops in electrospray mass spectrometry. But the large difference in length scales between these microscopic/nanoscopic jets and the macroscopic drops and films from which they emerge has made it difficult to model the electrohydrodynamic (EHD) processes that govern such phenomena. Here, we report simulations and experiments that enable a comprehensive picture of the mechanisms of cone formation, jet emission and break-up that occur during EHD tip streaming from a liquid film of finite conductivity. Simulations show that EHD tip streaming does not occur if the liquid is perfectly conducting or perfectly insulating, and enable us to develop a scaling law to predict the size of the drops produced from jet break-up.

Electrohydrodynamic tip streaming and emission of charged drops from liquid cones

Many electro-spraying devices raise to a high electric potential a pendant drop of weakly conducting fluid, which may adopt a conical shape from whose apex a thin, charged jet is emitted. Such a jet eventually breaks up into fine droplets, but often displays surprising longevity. This paper examines the stability of an incompressible cylindrical jet carrying surface charge in a tangential electric field, allowing for the finite rate of charge relaxation. The viscosity is assumed to be small so that the shear resulting from the tangential surface stress can be large, even for relatively small fields. This shear can suppress surface tension instabilities, but if too large, it excites electrical ones. For imperfect conductors, surface charge is redistributed by the rapid fluid reaction to variations in tangential stress as well as by conduction. Phase differences between the effects due to the tangential field and the surface charge lead to charge ‘over-relaxation’ instabilities, but the maximum growth rate can still be lower than in the absence of electric effects.

Electrohydrodynamic stability of a slightly viscous jet

The process of levitation melting of metals is examined analytically and numerically for the case of axisymmetric toroidal high-frequency currents. The governing equations for the mean-velocity field and associated free-surface shape are derived under the assumption of low magnetic Reynolds number and the neglect of thermal effects. The form of the solution for high Reynolds number is discussed in general, and particularized to the case of high surface tension, in which limit a perturbation analysis about a spherical shape is presented. Finite-difference techniques are used to solve the Navier–Stokes equations in the sphere, and the surface perturbation is calculated. The asymptotic behaviour of the potential vorticity is illustrated by the numerical experiments.

Magnetic levitation of liquid metals

Fully developed flow in an infinite helically coiled pipe is studied, motivated by physiological applications. Most of the bends in the mammalian arterial system curve in a genuinely three-dimensional way, so that the arterial centreline has not only curvature but torsion and can be modelled by a helix. Flow in a helically symmetric pipe generalizes related problems in axisymmetry (Dean flow) and two-dimensionality, but the geometry ensures that even irrotational flow has a cross-pipe component. Fully developed helical flows driven by a steady pressure gradient are studied analytically and numerically. Varying the radius and pitch of the helical pipe, the effects of curvature and torsion on the flow are investigated.

Steady flow in a helically symmetric pipe

We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

Fully developed flow in a helical pipe is investigated with a view to modelling blood flow around the commonly non-planar bends in the arterial system. Medical research suggests that the formation of atherosclerotic lesions is strongly correlated with regions of low wall shear and it has been suggested that the observed non-planar geometry may result in a more uniform shear distribution. Helical flows driven by an oscillating pressure gradient are studied analytically and numerically. In the high-frequency limit an expression is derived for the second-order steady flow driven by streaming from the Stokes layers. Finite difference methods are used to calculate flows driven by sinusoidal or physiological pressure gradients in various geometries. Possible advantages of the observed helical rather than planar arterial bends are discussed in terms of wall shear distribution and the inhibition of boundary layer separation.

A. J. Mestel

Papers

Electrohydrodynamic stability of a slightly viscous jet

Magnetic levitation of liquid metals

Steady flow in a helically symmetric pipe

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

Unsteady blood flow in a helically symmetric pipe