Electrohydrodynamic Phenomena

Abstract

This work is a review article focused on exploring the interactions between external and induced electric fields and fluid motion, in the presence of embedded charges. Such interactions are generally termed electrohydrodynamics (EHD), which encompasses a vast range of flows stemming from multiscale physical effects. In this review article we shall mainly emphasize on two mechanisms of particular interest to fluid dynamists and engineers, namely electrokinetic flows and the leaky dielectric model. We shed light on the underlying physics behind the above mentioned phenomena and subsequently demonstrate the presence of a common underpinning pattern which governs any general electrohydrodynamic motion. Hence we go on to show that the seemingly unrelated fields of electrokinetics and the leaky dielectric models are indeed closely related to each other through the much celebrated Maxwell stresses, which have long been known as stresses caused in fluids in presence of electric and magnetic fields. Interactions between Maxwell Stresses and charges (for instance, in the form of ions) present in the fluid generates a body force on the same and eventually leads to flow actuation. We show that the manifestation of the Maxwell stresses itself depends on the charge densities, which in turn is dictated by the underlying motion of the fluid. We demonstrate how such inter-related dynamics may give rise intricately coupled and non-linear system of equations governing the dynamical state of the system. This article is mainly divided into two parts. First, we explore the realms of electrokinetics, wherein the formation and the structure of the so-called electrical double layer (EDL) is delineated. Subsequently, we review EDL’s relevance to electroosmosis and streaming potential with the key being the presence and absence of an applied pressure gradient. We thereafter focus on the leaky dielectric model, wherein the fundamental governing equations and its main difference with electrokinetics are described. We limit our attentions to the leaky dielectric motion around droplets and flat surfaces and subsequent interface deformation. To this end, through a rigorous review of a number of previous articles, we establish that the interface shapes can be finely tailored to achieve the desired geometrical characteristics by tuning the fluid properties. We further discuss previous studies, which have shown migration of droplets in the presence of strong electric fields. Finally, we describe the effects of external agents such as surface impurities on leaky dielectric motion and attempt to establish a qualitative connection between the leaky dielectric model and EDLs. We finish off with some pointers for further research activities and open questions in this field.

Derivation of Maxwell Stress

The form of the Maxwell stress may be deduced by starting with the force acting on a unit volume due to the simultaneous action of an electric field, ${\mathbf E}$, and a magnetic field, ${\mathbf B}$. The Lorentz force is obtained as49

$$\begin{aligned} {\mathbf F} _e&= q{\mathbf E} + q{\mathbf v} \times {\mathbf B} \end{aligned}$$

(70)

$$\begin{aligned} {\mathbf f} _e&= \rho _e{\mathbf E} + {\mathbf J} \times {\mathbf B} \end{aligned}$$

(71)

where the former is the total force and the latter is the force per unit volume. $\rho _e$ depicts the charge density and ${\mathbf J}$ represents the amperic flux (current). We must now appeal to the Maxwell’s laws to relate $\rho _e$ and ${\mathbf J }$ to the fundamental quantities ${\mathbf E}$ and ${\mathbf B}$. Gauss’s law yields $\rho _e = \epsilon _0\nabla \cdot {\mathbf E}$ while Ampère’s law yields ${\mathbf J} = \frac{1}{\mu _0}\nabla \times {\mathbf B} - \epsilon _0 \frac{\partial E}{\partial t}$. The physical meaning of the above two expressions is clear. Gauss’s law indicates that the flux of the electric field through a surce is due to the charge enclosed by that surface. Ampère’s law indicates that the magnetic field around a contour is proportional to the electric current, ${\mathbf J}$, and the displacement current, $\partial E/\partial t$. Thus, we obtain

$$\begin{aligned} {\mathbf f} _e = \epsilon _0 \nabla \cdot {\mathbf E} {\mathbf E} + \frac{1}{\mu _0}\nabla \times {\mathbf B} \times {\mathbf B} -\epsilon _0\frac{\partial {\mathbf E} }{\partial t}\times {\mathbf B} \end{aligned}$$

(72)

where we can utilize the identity

$$\begin{aligned} \frac{\partial }{\partial t} ({\mathbf E} \times {\mathbf B} ) = \frac{\partial E}{\partial t}\times {\mathbf B} + {\mathbf E} \times \frac{\partial {\mathbf B} }{\partial t} \end{aligned}$$

(73)

along with the Maxwell-Faraday equation $\nabla \times {\mathbf E} = -\frac{\partial B}{\partial t}$ to modify equation (72) to yield

$$\begin{aligned} {\mathbf f} _e = \epsilon _0 \left( \nabla \cdot {\mathbf E} {\mathbf E} - {\mathbf E} \times \nabla \times {\mathbf E} \right) + \frac{1}{\mu _0}\left( -{\mathbf B} \times \nabla \times {\mathbf B} \right) -\epsilon _0 \frac{\partial }{\partial t}\left( {\mathbf E} \times {\mathbf B} \right) . \end{aligned}$$

(74)

In order to symmetrize the above expression, we make use of the Gauss’s law of magnetism which asserts us that there are no magnetic monopoles, i.e. $\nabla \cdot {\mathbf B} = 0$. This can then simply be added to equation (74) to obtain

$$\begin{aligned} {\mathbf f} _e = \epsilon _0 \left( (\nabla \cdot {\mathbf E} ) {\mathbf E} - {\mathbf E} \times \nabla \times {\mathbf E} \right) + \frac{1}{\mu _0}\left( (\nabla \cdot {\mathbf B} ) {\mathbf B} -{\mathbf B} \times \nabla \times {\mathbf B} \right) -\epsilon _0 \frac{\partial }{\partial t}\left( {\mathbf E} \times {\mathbf B} \right) \end{aligned}$$

(75)

upon which further simplification is obtained by noting the vector identity ${\mathbf E} \nabla {\mathbf E} = \frac{1}{2}\nabla (|{\mathbf E} |^2)$ to yield

$$\begin{aligned} {\mathbf f} _e = \epsilon _0\left( (\nabla \cdot {\mathbf E} ){\mathbf E} + ({\mathbf E} \cdot ){\mathbf E} \right) -\frac{1}{2}\epsilon _0\nabla |{\mathbf E} |^2 + \frac{1}{\mu _0}\left( (\nabla \cdot {\mathbf B} ){\mathbf B} + ({\mathbf B} \cdot ){\mathbf B} \right) -\frac{1}{2\mu _0}\nabla |{\mathbf B} |^2 -\epsilon _0\frac{\partial }{\partial t}\left( {\mathbf E} \times {\mathbf B} \right) . \end{aligned}$$

(76)

When the effects of the magnetic field is neglected the above expression may be set in the form of a divergence of a tensor form as

$$\begin{aligned} {\mathbf f} _e = \nabla \cdot \tau ^E;\qquad \tau ^E = \epsilon _0{\mathbf E} \otimes {\mathbf E} -\frac{1}{2}\epsilon _0|{\mathbf E} |^2{\mathbf {I}}\implies \tau ^E_{ij} = \epsilon _0 E_iE_j -\frac{1}{2}\epsilon _0 E_kE_k\delta _{ij} \end{aligned}$$

(77)

where ${\mathbf {I}}$ and $\delta _{ij}$ represent the identity tensor and Kronecker delta respectively.

While the above derivation is done for a volume in vaccum, one can logically extend the same to a medium with a permittivity given by $\epsilon = \epsilon _0\epsilon _r$ where $\epsilon _r$ represents the relative permittivity. In that case the body force may be evaluated by taking the divergence of the Maxwell stress tensor

$$\begin{aligned} \tau ^E = \epsilon {\mathbf E} \otimes {\mathbf E} - \frac{1}{2}\epsilon |{\mathbf E} |^2{\mathbf {I}} \implies \nabla \cdot \tau ^E = (\nabla \cdot \epsilon {\mathbf E} )E + (\epsilon {\mathbf E} \cdot \nabla ){\mathbf E} -\frac{1}{2}\nabla (\epsilon {\mathbf E} \cdot {\mathbf E} ) \end{aligned}$$

(78)

For negligible magnetic effects we may assume that $\nabla \times E = 0$, i.e. the electric field is irrotatonal. We will also make use of the fact that for a medium with permittivity $\epsilon$, the Gauss’s law implies that $\nabla \cdot \epsilon {\mathbf E} = \rho _{e,f}$, where $\rho _{e,f}$ represents the free charge density. Thus we obtain the force as

$$\begin{aligned} {\mathbf f} _e = \rho _{e,f}{\mathbf E} - \frac{1}{2}|{\mathbf E} ^2|\nabla \epsilon \end{aligned}$$

(79)

where we have neglected the additional volumetric body force which is called as the electrostriction force given by

$$\begin{aligned} \nabla \left( \frac{1}{2}|{\mathbf E} |^2 \rho \frac{\partial \epsilon }{\partial \rho }\right) \end{aligned}$$

(80)

which accounts for the force generated by the variation of the permittivity with density. Being a gradient of a quantity, this force may be absorbed in a modified pressure.

The electric body force described in equation (79) is comprised of the force acting due to an electric field on a region of net charge while the second term represents the body force arising due to the spatial inhomogeneity of the permittivity. Typically, the former effect is responsible for electrokinetic phenomena for single phase flows while the latter is important for analyzing the forces at multiphase interfaces.