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Electrohydrodynamics
Electrohydrodynamics
Electrohydrodynamics
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Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids.[1][2] Electrohydrodynamics (EHD) is a joint domain of electrodynamics and fluid dynamics mainly focused on the fluid motion induced by electric fields. EHD, in its simplest form, involves the application of an electric field to a fluid medium, resulting in fluid flow, form, or properties manipulation. These mechanisms arise from the interaction between the electric fields and charged particles or polarization effects within the fluid.[2] The generation and movement of charge carriers (ions) in a fluid subjected to an electric field are the underlying physics of all EHD-based technologies.

Electrohydrodynamics employed for drying applications (EHD Drying)[2].


The electric forces acting on particles consist of electrostatic (Coulomb) and electrophoresis force (first term in the following equation)., dielectrophoretic force (second term in the following equation), and electrostrictive force (third term in the following equation):

[2]

This electrical force is then inserted in Navier-Stokes equation, as a body (volumetric) force.

Electrohydrodynamics employed for Airflow control and Electrospinning applications.

EHD covers the following types of particle and fluid transport mechanisms: electrophoresis, electrokinesis, dielectrophoresis, electro-osmosis, and electrorotation. In general, the phenomena relate to the direct conversion of electrical energy into kinetic energy, and vice versa.

In the first instance, shaped electrostatic fields (ESF's) create hydrostatic pressure (HSP, or motion) in dielectric media. When such media are fluids, a flow is produced. If the dielectric is a vacuum or a solid, no flow is produced. Such flow can be directed against the electrodes, generally to move the electrodes. In such case, the moving structure acts as an electric motor. Practical fields of interest of EHD are the common air ioniser, electrohydrodynamic thrusters and EHD cooling systems.

In the second instance, the converse takes place. A powered flow of medium within a shaped electrostatic field adds energy to the system which is picked up as a potential difference by electrodes. In such case, the structure acts as an electrical generator.

Electrokinesis

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"Electrokinesis" redirects here. For the ability to manipulate electricity, see List of psychic abilities.

Electrokinesis is the particle or fluid transport produced by an electric field acting on a fluid having a net mobile charge. (See -kinesis for explanation and further uses of the -kinesis suffix.) Electrokinesis was first observed by Ferdinand Frederic Reuss during 1808, in the electrophoresis of clay particles [3] The effect was also noticed and publicized in the 1920s by Thomas Townsend Brown which he called the Biefeld–Brown effect, although he seems to have misidentified it as an electric field acting on gravity.[4] The flow rate in such a mechanism is linear in the electric field. Electrokinesis is of considerable practical importance in microfluidics,[5][6][7] because it offers a way to manipulate and convey fluids in microsystems using only electric fields, with no moving parts.

The force acting on the fluid, is given by the equation where, is the resulting force, measured in newtons, is the current, measured in amperes, is the distance between electrodes, measured in metres, and is the ion mobility coefficient of the dielectric fluid, measured in m2/(V·s).

If the electrodes are free to move within the fluid, while keeping their distance fixed from each other, then such a force will actually propel the electrodes with respect to the fluid.

Electrokinesis has also been observed in biology, where it was found to cause physical damage to neurons by inciting movement in their membranes.[8][9] It is discussed in R. J. Elul's "Fixed charge in the cell membrane" (1967).

Water electrokinetics

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In October 2003, Dr. Daniel Kwok, Dr. Larry Kostiuk and two graduate students from the University of Alberta discussed a method to convert hydrodynamic to electrical energy by exploiting the natural electrokinetic properties of a liquid such as ordinary tap water, by pumping fluid through tiny micro-channels with a pressure difference.[10] This technology could lead to a practical and clean energy storage device, replacing batteries for devices such as mobile phones or calculators which would be charged up by simply compressing water to high pressure. Pressure would then be released on demand, for the fluid to flow through micro-channels. When water travels, or streams over a surface, the ions in the water "rub" against the solid, leaving the surface slightly charged. Kinetic energy from the moving ions would thus be converted to electrical energy. Although the power generated from a single channel is extremely small, millions of parallel micro-channels can be used to increase the power output. This streaming potential, water-flow phenomenon was discovered in 1859 by German physicist Georg Hermann Quincke. [citation needed][6][7][11]

Electrokinetic instabilities

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The fluid flows in microfluidic and nanofluidic devices are often stable and strongly damped by viscous forces (with Reynolds numbers of order unity or smaller). However, heterogeneous ionic conductivity fields in the presence of applied electric fields can, under certain conditions, generate an unstable flow field owing to electrokinetic instabilities (EKI). Conductivity gradients are prevalent in on-chip electrokinetic processes such as preconcentration methods (e.g. field amplified sample stacking and isoelectric focusing), multidimensional assays, and systems with poorly specified sample chemistry. The dynamics and periodic morphology of electrokinetic instabilities are similar to other systems with Rayleigh–Taylor instabilities. The particular case of a flat plane geometry with homogeneous ions injection in the bottom side leads to a mathematical frame identical to the Rayleigh–Bénard convection.

EKI's can be leveraged for rapid mixing or can cause undesirable dispersion in sample injection, separation and stacking. These instabilities are caused by a coupling of electric fields and ionic conductivity gradients that results in an electric body force. This coupling results in an electric body force in the bulk liquid, outside the electric double layer, that can generate temporal, convective, and absolute flow instabilities. Electrokinetic flows with conductivity gradients become unstable when the electroviscous stretching and folding of conductivity interfaces grows faster than the dissipative effect of molecular diffusion.

Since these flows are characterized by low velocities and small length scales, the Reynolds number is below 0.01 and the flow is laminar. The onset of instability in these flows is best described as an electric "Rayleigh number".

Misc

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Liquids can be printed at nanoscale by pyro-EHD.[12]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Electrohydrodynamics

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Electrohydrodynamics (EHD), also known as electro-fluid-dynamics, is the study of the interaction between and fluid flows, encompassing the effects of electric forces on the motion, deformation, and stability of electrically conducting or polarizable fluids such as liquids and gases. This interdisciplinary field combines principles from , , and continuum physics to describe phenomena where electric fields induce charge separation, accumulation at interfaces, and resultant stresses that drive bulk flows, interfacial deformations, and instabilities. The foundational theoretical framework for EHD was established in the mid-20th century, building on early observations dating back to the ; William Gilbert documented fluid motion under electric influence in 1600, while coined the term "electrohydrodynamics" in 1964 and developed the leaky dielectric model in 1966 to explain droplet deformation in insulating liquids under uniform . In this model, fluids are treated as imperfect dielectrics with finite conductivity, leading to free charge accumulation at fluid interfaces that generates tangential electro-osmotic stresses and normal Maxwell stresses, causing prolate or oblate deformations depending on ratios of (S) and conductivity (R) between phases. Key governing equations include the Navier-Stokes equations augmented by electric body forces (via and dielectrophoretic terms) and for charge conservation, often simplified under the regime for low Reynolds numbers. Notable EHD phenomena include electrohydrodynamic instabilities such as tip streaming (where cones form at droplet poles leading to jet ejection), Quincke rotation (spontaneous droplet spinning above a critical field strength), and varicose or whipping modes in charged jets that affect breakup into monodisperse droplets. These effects are quantified by dimensionless numbers like the electric Reynolds number (Re_E = \frac{\varepsilon^2 E^2}{\mu \sigma}, measuring charge convection relative to conduction and viscous effects, where \sigma is conductivity) and the electric capillary number (Ca_E = \varepsilon E^2 a / \gamma, comparing electric to surface tension stresses), where \varepsilon is permittivity, E is field strength, \mu is viscosity, \gamma is interfacial tension, and a is characteristic length. EHD has diverse applications across and , including microfluidic pumping without mechanical parts (via ion-drag or conduction mechanisms), high-resolution electrohydrodynamic jet printing for micro/nanofabrication (achieving feature sizes below 100 nm through Taylor cone-jet modes), emulsion stabilization or breakup for and , and systems like ionic thrusters that convert directly to kinetic in liquids or gases. As of 2020, advances leverage EHD for enhanced in cooling, particle manipulation in (e.g., cell sorting), and sustainable energy technologies such as electrostatic precipitators for air purification. As of 2025, further developments include multimodal EHD printing for high-resolution sensor fabrication and resilient flexible EHD pumps for human-machine interfaces.

Introduction

Definition and Scope

Electrohydrodynamics (EHD) is the study of the interactions between and fluid motion in electrically conducting or polarizable fluids, encompassing phenomena such as charge injection, polarization, and the resulting hydrodynamic effects. This interdisciplinary field integrates principles from hydrodynamics, , , and thermophysics, primarily focusing on weakly conducting liquids and gases where electric forces couple effectively with viscous forces. The scope of EHD extends to both and conducting fluids, with particular emphasis on liquid dielectrics like oils exhibiting conductivities in the range of 10^{-12} to 10^{-7} S/m, where induce flows, instabilities, and enhanced without requiring mechanical components. Central to EHD are the electric forces acting on the fluid: the Coulomb force, which exerts on free charges within the fluid (q\mathbf{E}, where q is and \mathbf{E} is the ); the dielectrophoretic force, arising from gradients in the electric field acting on induced dipoles in polarizable media; and , a volumetric force due to electric field-induced density changes in the fluid. These forces drive fluid motion by coupling with the Navier-Stokes equations, often at micro- and nanoscale regimes where surface effects dominate. EHD phenomena are broadly classified into injection-induced flows, such as ion-drag effects where charges are injected from electrodes to propel the fluid, and polarization-induced flows, exemplified by dielectrophoresis where non-uniform fields manipulate neutral or weakly charged particles via . Unlike (MHD), which addresses fluid motion under magnetic fields in highly conducting plasmas or liquids (with magnetic Reynolds numbers of order unity), EHD emphasizes in weakly conducting media where magnetic effects are negligible (σ ε_0 c^2 L^2 ≪ 1, with L as characteristic length). , such as and , constitute specific subsets of EHD involving relative motion between immiscible fluid phases or solids and electrolytes near charged interfaces.

Historical Development

The earliest observations of electrohydrodynamic phenomena date back to the late 16th and early 17th centuries, when scientists began documenting the motion of liquids and particles under . In 1600, William Gilbert described the attraction and movement of liquid droplets toward charged objects, such as rubbed , in his seminal work , marking one of the first recorded instances of electric forces influencing fluid behavior. Similarly, in 1629, Niccolò Cabeo observed the attraction of small particles, like , to electrified bodies, followed by contact and repulsion, providing early evidence of electrodynamic interactions with particulate matter in air. During the 18th and 19th centuries, foundational work on laid the groundwork for understanding electrokinetic effects in fluids. Alessandro Volta's invention of the in 1800 enabled sustained electric currents, facilitating experiments that revealed electrochemical reactions in liquids. advanced this in the 1830s through his studies of in water and other electrolytes, where he quantified the decomposition of fluids under and observed associated motion of charged species, establishing key principles of electrokinetic transport. The saw the formal development of electrokinetics, with quantitative studies on the of colloidal particles providing insights into particle migration in within fluids. The field of electrohydrodynamics was rigorously defined in 1969 through a landmark review by J.R. Melcher and , which integrated electrodynamics and hydrodynamics to explain interfacial shear stresses and fluid motion driven by . had coined the term "electrohydrodynamics" in 1964. Following 1969, research emphasized electrohydrodynamic instabilities using the leaky dielectric model developed by Taylor in 1966 and extended by Melcher, which accounted for finite conductivity in fluids and predicted deformation and circulation in droplets under . In the 1990s and 2000s, electrohydrodynamics integrated with and , enabling precise control of fluid flows at microscales for applications like droplet manipulation and atomization. Recent developments as of 2024 have advanced computational modeling and applications of electrohydrodynamic flows in systems, with key reviews highlighting their role in biological contexts such as vesicle dynamics and cellular processes, as well as enhanced and pumping technologies.

Fundamental Principles

Governing Equations

Electrohydrodynamics couples electromagnetic fields with fluid motion through the interaction of electric charges and fields within conducting or polarizable fluids. The governing equations are derived under the quasi-electrostatic approximation, suitable for low-frequency phenomena where magnetic effects are negligible. The electric field E\mathbf{E} is irrotational, satisfying ∇×E=0\nabla \times \mathbf{E} = 0, which allows representation as E=−∇ϕ\mathbf{E} = -\nabla \phi with ϕ\phi the electric potential. Gauss's law for the electric displacement D=εE\mathbf{D} = \varepsilon \mathbf{E}, where ε\varepsilon is the permittivity, takes the form ∇⋅D=ρf\nabla \cdot \mathbf{D} = \rho_f, with ρf\rho_f denoting the free charge density. Charge conservation is expressed as ∂ρf∂t+∇⋅J=0\frac{\partial \rho_f}{\partial t} + \nabla \cdot \mathbf{J} = 0, where the J\mathbf{J} includes ohmic conduction, convection by fluid velocity v\mathbf{v}, and diffusive contributions: J=σE+ρfv−D∇ρf\mathbf{J} = \sigma \mathbf{E} + \rho_f \mathbf{v} - D \nabla \rho_f, with σ\sigma the electrical conductivity and DD the diffusion coefficient (often negligible in macroscopic EHD flows). The fluid momentum is governed by the Navier-Stokes equations augmented with electric body forces: ρ(∂v∂t+v⋅∇v)=−∇[p](/page/Pressure)+μ∇2v+ρfE−12E2∇ε+∇[12E2(∂ε∂ρ)ρ],\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla [p](/page/Pressure) + \mu \nabla^2 \mathbf{v} + \rho_f \mathbf{E} - \frac{1}{2} E^2 \nabla \varepsilon + \nabla \left[ \frac{1}{2} E^2 \left( \frac{\partial \varepsilon}{\partial \rho} \right) \rho \right], where ρ\rho is mass density, pp , μ\mu dynamic viscosity, the term ρfE\rho_f \mathbf{E} is the force, −12E2∇ε-\frac{1}{2} E^2 \nabla \varepsilon the dielectrophoretic force, and the final term the electrostrictive force (relevant in compressible fluids). The ∇⋅v=0\nabla \cdot \mathbf{v} = 0 holds for incompressible flows. Boundary conditions include no-slip v=0\mathbf{v} = 0 at solid walls and, at fluid interfaces, continuity of tangential E\mathbf{E} and normal D⋅n\mathbf{D} \cdot \mathbf{n} (adjusted for surface charge), alongside kinematic conditions for interface tracking. Non-dimensionalization reveals key regimes via the electric ReE=εE02/(μU)\mathrm{Re}_E = \varepsilon E_0^2 / (\mu U), comparing electric to viscous stresses (with E0E_0 characteristic and UU scale), and the charge relaxation time τε=ε/σ\tau_\varepsilon = \varepsilon / \sigma, contrasting conduction and convection timescales. Low ReE\mathrm{Re}_E and rapid relaxation (τε≪\tau_\varepsilon \ll flow time) yield ohmic conduction dominance, while high ReE\mathrm{Re}_E or slow relaxation favor injection-dominated flows. These parameters delineate behaviors such as leaky dielectric models in non-aqueous systems versus electrokinetic effects in confined aqueous geometries.

Electric Forces in Fluids

In electrohydrodynamics, electric forces acting on fluids originate from the coupling between electromagnetic fields and the material properties of the fluid, such as , , and conductivity, resulting in both volumetric body forces and interfacial surface forces that induce or modify fluid motion. These forces are fundamental to EHD phenomena and can be derived from combined with thermodynamic considerations of the electric energy in the fluid. Body forces include the Coulomb force, which acts on free charges within the fluid volume and is expressed as fC=ρfE\mathbf{f}_C = \rho_f \mathbf{E}, where ρf\rho_f is the volume of free charge and E\mathbf{E} is the vector. In nonuniform , an additional dielectrophoretic force arises from the interaction with , particularly relevant for AC fields, given by fDEP=12ℜ[∇(α∣E∣2)]\mathbf{f}_{DEP} = \frac{1}{2} \Re \left[ \nabla (\alpha |\mathbf{E}|^2) \right], where α\alpha denotes the complex of the fluid, accounting for both real and imaginary components related to and conductivity. Surface forces at fluid interfaces or electrodes are primarily described by the divergence of the Maxwell stress tensor, τM=ε(E⊗E−12∣E∣2I)\boldsymbol{\tau}_M = \varepsilon \left( \mathbf{E} \otimes \mathbf{E} - \frac{1}{2} |\mathbf{E}|^2 \mathbf{I} \right), where ε\varepsilon is the , I\mathbf{I} is the identity tensor, and additional polarization terms may contribute at discontinuities in or conductivity. This tensor yields tangential stresses that drive shear flows along interfaces and normal stresses that promote deformation or rupture, such as in electrospraying. Electrostriction introduces a volumetric body force in fluids where permittivity depends on density, formulated as fES=−12E2∇ε\mathbf{f}_{ES} = -\frac{1}{2} E^2 \nabla \varepsilon, arising from field-induced compression that alters the fluid's dielectric response. In typical liquid EHD systems, this force is often negligible compared to Coulomb or dielectrophoretic effects due to the low compressibility of liquids. EHD processes operate in distinct regimes based on charge generation mechanisms: the injection regime, where free charges are directly emitted from electrodes through processes like , leading to space-charge layers and strong Coulomb-driven flows; and the conduction regime, where charges result from dissociation of neutral molecules under the electric field, with transport dominated by ohmic conduction and heterocharge layers near electrodes. In dielectric fluids with low conductivity, polarization-based forces such as dielectrophoresis dominate the force balance, whereas in electrolytes with significant free charge carriers, Coulomb forces prevail and can lead to intense electroconvection. The total electrohydrodynamic can be derived from the variation of the electric free energy density with respect to fluid displacement, yielding fEHD=ρfE−12E2∇ε+12∇(E2∂ε∂ρρ)\mathbf{f}_{EHD} = \rho_f \mathbf{E} - \frac{1}{2} E^2 \nabla \varepsilon + \frac{1}{2} \nabla \left( E^2 \frac{\partial \varepsilon}{\partial \rho} \rho \right), where the first term captures effects, the second dielectrophoretic contributions from gradients, and the third electrostrictive corrections involving density ρ\rho. These forces enter the Navier-Stokes momentum equation as source terms to govern the overall .

Electrokinetic Phenomena

Electrokinesis

Electrokinesis refers to the bulk motion of a neutral fluid induced by the transfer of from electrically charged ions to neutral molecules, occurring without the presence of phase boundaries or interfaces. This phenomenon arises in liquids where free charges are introduced via unipolar injection from electrodes, leading to forces that drive the overall fluid flow. Unlike interfacial effects, electrokinesis involves volumetric forces distributed throughout the fluid phase. The primary mechanisms of electrokinesis include ion-drag and electroconvection. In the ion-drag process, ions injected at the electrode are accelerated by the applied electric field and collide with surrounding neutral molecules, imparting momentum and thereby entraining the bulk fluid. Electroconvection arises from the Coulomb force on space charge generated by mechanisms such as ion injection or dissociation in applied electric fields, often uniform between electrodes, leading to hydrodynamic instabilities that induce convective patterns, such as plumes or rolls. These mechanisms dominate in poorly conducting liquids, where charge relaxation times are long enough to sustain significant space charge densities. A key dimensionless parameter governing electrokinesis is the injection strength C=ρ0L2εVC = \frac{\rho_0 L^2}{\varepsilon V}, where ρ0\rho_0 is the injected at the , LL is the gap width, ε\varepsilon is the , and VV is the applied voltage. This parameter quantifies the relative importance of injected charge to the field-induced charge; regimes with C>10C > 10 indicate strong injection, where electrokinetic flows become dominant and transition from linear to nonlinear behaviors. Experimental investigations of electrokinesis typically employ insulating oils, such as , confined between parallel plate under high voltages (often exceeding 10 kV). Ion injection occurs at a sharp emitter , creating a that propagates across the gap to a collector. Velocity profiles in these setups exhibit parabolic or plume-like structures, with characteristic speeds scaling as u∼εμ(VL)2u \sim \frac{\varepsilon}{\mu} \left( \frac{V}{L} \right)^2, derived from balancing the electric against viscous dissipation in the Navier-Stokes equations. Electrokinesis fundamentally differs from dielectrophoresis, as it relies on the conduction of free charges rather than the induction of dipoles in neutral molecules, enabling sustained bulk flows without requiring alternating fields. In confined geometries, electrokinesis shares conceptual similarities with , where charge-driven slips occur at walls rather than in open bulk.

Electroosmosis and Electrophoresis

Electroosmosis describes the motion of a liquid relative to a stationary charged surface under the influence of an applied tangential , where the field exerts on the counterions within the diffuse part of the electrical double layer at the solid-liquid interface. This arises in aqueous systems, where the charged surface attracts oppositely charged ions, forming a mobile diffuse layer that shears under the electric field, dragging the bulk fluid. The classical theoretical framework for this flow was established by Helmholtz, who introduced the concept of the double layer, and refined by Smoluchowski, leading to the Helmholtz-Smoluchowski relation for the electroosmotic velocity. Under the thin double-layer approximation, where the Debye length is much smaller than the channel dimension, the electroosmotic velocity is uniform across the channel cross-section and given by ueo=−εζμE,u_{eo} = -\frac{\varepsilon \zeta}{\mu} E, where ε\varepsilon is the electrical of the fluid, ζ\zeta is the at the slipping plane, μ\mu is the dynamic , and EE is the applied strength. This slip velocity represents the electrokinetic coupling coefficient, which quantifies the linear relationship between the flow and the driving field in low-conductivity aqueous electrolytes. Electrophoresis, conversely, involves the motion of charged colloidal particles or macromolecules through a quiescent fluid under an applied , with the velocity formula mirroring that of but with opposite sign due to the relative motion of the particle. In the thin double-layer limit, the electrophoretic velocity is uep=εζμEu_{ep} = \frac{\varepsilon \zeta}{\mu} E, assuming no polarization effects or hydrodynamic interactions dominate. This reciprocity between and electrophoresis stems from the underlying electrokinetic mechanism, where the electric force on the particle's double layer drives its translation relative to the surrounding fluid. In aqueous systems, the λD\lambda_D, which characterizes the thickness of the diffuse double layer, is given by λD=εkT2ne2\lambda_D = \sqrt{\frac{\varepsilon kT}{2 n e^2}}
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