**Electrohydrodynamics (EHD)** is a physical discipline that arose at the intersection of hydrodynamics and electrostatics . The subject of its study is the motion processes of weakly conducting liquids (liquid dielectrics, hydrocarbon oils and fuels, etc.) placed in an electric field .

Many EHD effects are unexpected, unpredictable, and remain unexplained to date. This is due to the strongly nonlinear nature of electrohydrodynamic phenomena, which causes difficulties in their study ^{[1]} .

## Content

## History

The foundations of the theory of EHD flows were laid back by M. Faraday , but the intensive development of this area of research began only in the 1960s. In the USA, it was developed by a group led by J. Melcher. In Europe - a number of scientific groups in France, Spain and other countries.

In the USSR, EHD theory was worked at the Institute of Mechanics of Moscow State University and Kharkov State University ; more applied research in this area was carried out at the Institute of Applied Physics of the Moldavian Academy of Sciences and at the Leningrad State University under the direction of G. A. Ostroumov . Currently, these works are continuing at the Scientific and Educational Center at St. Petersburg State University under the leadership of Yu. K. Stishkov. A number of studies were also conducted at Perm State University ^{[1]} .

## The system of EHD equations

### Zoom

The system of equations of electrohydrodynamics can be obtained from the system of Maxwell equations and equations of hydrodynamics , taking into account a number of approximations. First, when considering electrohydrodynamic phenomena, the radiation of a moving charged liquid is neglected and the energy of the magnetic field is neglected in comparison with the energy of the electrostatic field . These approximations can be written using the following inequalities:

- $\frac{\epsilon \omega L}{c}\ll \mathrm{one}\phantom{\rule{2em}{0ex}}\frac{\sigma L}{\epsilon c}\ll \mathrm{one}$

where *ε* , *σ* are the dielectric constant and conductivity of the medium, *ω* is the characteristic frequency of the external field, *L* is the characteristic external size of the medium, and *c* is the speed of light . In addition, the motion of the medium must be nonrelativistic (the speed of its motion$v\ll c$ ), and its density should be sufficiently large (so that the mean free path$\lambda \ll L$ )

### General system

In the case of weakly conducting media, the system of EHD equations is usually written in the SI system in the following form:

- $\rho \left(\frac{\mathrm{\partial}{v}_{i}}{\mathrm{\partial}t}+{v}_{k}\frac{\mathrm{\partial}{v}_{i}}{\mathrm{\partial}{x}_{k}}\right)=\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{k}}\left({p}_{ik}+{T}_{ik}\right)+\rho {f}_{i}$ - the equation of motion that determines the balance of pulses at an arbitrary point in the medium
- $\frac{\mathrm{\partial}\rho}{\mathrm{\partial}t}+\frac{\mathrm{\partial}\rho {v}_{i}}{\mathrm{\partial}{x}_{i}}=0$ - continuity equation
- $-\mathrm{\nabla}\cdot (\epsilon {\epsilon}_{0}\mathrm{\nabla}\varphi )=q$ - Poisson equation
- $\frac{\mathrm{\partial}q}{\mathrm{\partial}t}+\frac{\mathrm{\partial}{j}_{i}}{\mathrm{\partial}{x}_{i}}=0$ - continuity equation for electric current

The following notation is introduced here. *ρ* is the mass density of the medium, *v _{i}* are the velocity components,

*f*is the mass density of the forces acting on the medium,

_{i}*p*,

_{ik}*T*are the components of the tensors of mechanical and maxwell stresses ,

_{ik}*φ*is the electrostatic potential ,

*q*is the volume charge density ,

*j*is components of the electric current density ,

_{i}*ε*is the electric constant .

_{0}The system of equations presented above is open. To close it, it is necessary to write the equations of state . The following conditions are commonly used:

- $p}_{ik}=p{\delta}_{ik}+{\tau}_{ik$
- $T}_{ik}=-\left(\frac{\mathrm{one}}{2}\epsilon {\epsilon}_{0}{E}^{2}-{p}_{str}\right)\delta ik+\epsilon {\epsilon}_{0}{E}_{i}{E}_{k$
- $p}_{str}=\frac{{\epsilon}_{0}}{2}\rho \frac{\mathrm{\partial}\epsilon}{\mathrm{\partial}\rho}{E}^{2$
- $j}_{i}={j}_{i}^{\ast}+q{v}_{i$

Here *p* is the mechanical pressure , *τ _{ik}* is the tensor of viscous stresses ,

*p*is the striction pressure associated with the ponderomotive action of the field,

_{str}**j**

^{*}is the migration current, q

**v**is the convective current,

*E*are the components of the electric field .

_{i}### Incompressible fluid equations

- $\rho \frac{\mathrm{\partial}\overrightarrow{v}}{\mathrm{\partial}t}+\rho (\overrightarrow{v}\cdot \mathrm{\nabla})\overrightarrow{v}=-\mathrm{\nabla}p+\eta \mathrm{\Delta}\overrightarrow{v}-\rho \mathrm{\nabla}\varphi$ - Navier-Stokes equation
- $\frac{\mathrm{\partial}\rho}{\mathrm{\partial}t}+\mathrm{div}(-D\mathrm{\nabla}\rho -\rho \mu \mathrm{\nabla}\varphi )=R-\overrightarrow{v}\cdot \mathrm{\nabla}\rho$ - Nernst-Planck equation
- $-\mathrm{\nabla}\cdot (\epsilon \epsilon \mathrm{\nabla}\varphi )=\rho$ - Poisson equation
- $\mathrm{\nabla}\cdot \overrightarrow{v}=0$

## Electrohydrodynamic phenomena

Electrohydrodynamic phenomena have been known for a long time. In the middle of the XVIII century. an opportunity appeared to work with high voltages (see. Leiden Bank , Electrophore Machine ). The first “mystical experience” associated with EHD phenomena was as follows: a crown was placed opposite the burning candle, and as a result the candle was blown out. Another experience is the franklin wheel . If a high voltage is applied to an electrode in the form of a swastika with needles at the end, then this electrode is set in motion. Electrohydrodynamic phenomena were described by Faraday:

If a pint of well-cleaned and filtered oil is poured into a glass vessel and two wires connected to an electrophore machine are lowered into it, then all the liquid will come into an unusually violent motion.

Original text... if a pint of well-rectified and filtered (1571.) oil of turpentine be put into a glass vessel, and two wires be dipped into it in different places, one leading to the electrical machine, and the other to the discharging train, on working the machine the fluid will be thrown into violent motion throughout its whole mass ...- Michael Faraday^{[2]}

## The use of electrohydrodynamic phenomena

Electrohydrodynamic phenomena are used to intensify heat transfer (for example, when natural convection is difficult - in space). EHD phenomena are also used in electrostatic dust collectors ^{[3]} and ionizers, for the manufacture of thin polymer filaments and capillaries ^{[4]} , for dispersed spraying of liquids ( electric paint surfaces), as well as in inkjet printers ^{[5]} .

## See also

- Magnetohydrodynamics

## Notes

- ↑
^{1}^{2}*A.I. Zhakin.*Electrohydrodynamics // UFN . - 2012 .-- T. 182 . - S. 495-520 . - ↑ Experimental Researches in Electricity, Volume 1 / Faraday, Michael, 1791-1867 (Unavailable link) . Date of treatment May 4, 2009. Archived May 16, 2009.
- ↑
*I.P. Vershchagin et al.*Fundamentals of electro-*gasdynamics of*disperse systems. - M .: Energy, 1974. - ↑
*E. A. Druzhinin.*Production and properties of Petryanov’s filter materials from ultra-thin polymer fibers. - M .: Publishing House, 2007. - ↑
*V.I. Bezrukov.*Basics of electro-jet technology. - St. Petersburg: Shipbuilding, 2001.

## Literature

### Books

*I. B. Rubashov, Yu. S. Bortnikov.*Electro-gasdynamics. - M .: Atomizdat , 1971.*Yu.K. Stishkov, A.A. Ostapenko.*Electrohydrodynamic flows in liquid dielectrics. - L .: Publ. Leningrad University, 1989 .-- 174 p.- Electrohydrodynamics / A. Castellanos. - Wien: Springer, 1998. - (CISM Courses and Lectures No. 380).

### Articles

*A.I. Zhakin.*Electrohydrodynamics // UFN . - 2012 .-- T. 182 . - S. 495-520 .*A.I. Zhakin.*Electrohydrodynamics of charged surfaces // UFN . - 2013 .-- T. 183 . - S. 153–177 .