### Electrohydrodynamic Effect  
The creation of electric wind due to electrohydrodynamic force is a known effect due to the charged/neutral particles interaction in electrically charged fluids. Electric wind occurs during the DC corona and dielectric barrier discharge. The electrohydrodynamic (EHD) phenomena, within a pair of asymmetrical electrodes having a potential difference of the order of few of kilovolts, was first studied by [PERSON] in 1928 [26]. However, while the efficiency of such devices for creating air flow is a few unit percents, the EHD devices' interest is mainly in the absence of moving parts and for their potential for miniaturization. Numerical modeling of the electro-induced flow by a filamentary discharge electrode on a plate is a topic that has already been discussed in the literature [26; 27; 28]. The gaseous ions, formed in the corona discharge area, are accelerated by the electric field, resulting in a drag force of the bulk fluid and producing an ionic wind, which also includes neutral molecules (Figure 4). For the sake of simplicity, only three generic charged particles will be considered: electrons, positive ions, and negative ions. The EHD flow generated by the positive corona discharge is described by the [PERSON]'s equation, in which the electric field intensity, \(\overrightarrow{E}\), and the electric potential \(V\) are related according to the Equation (1):  
\[\overrightarrow{\
abla}\cdot\overrightarrow{E}=\
abla^{2}V=-\frac{q}{\epsilon _{0}} \tag{1}\]  
In which \(q\) is the space charge density (C m\({}^{-3}\)) and \(\epsilon_{0}\) is the vacuum permittivity. Figure 3: Schematic of the production of by-products by corona discharge. The electron density is then computed by solving the drift-diffusion equation for the electron density (Equation (2)), using the local field approximation (Equation (3)), in order to relate the mean electron energy to the reduced electric field [29]. \[\frac{\partial\mathrm{n_{e}}}{\partial\mathrm{t}}+\
abla\cdot\tensor{\Gamma}{ \mathrm{e}}=\mathrm{R_{e}}\cdot(\tensor{\mathrm{u}}{\mathrm{u}}\cdot\
abla) \mathrm{n_{e}} \tag{2}\]  
\[\varepsilon=\mathrm{F}\left(\frac{\mathrm{E}}{\mathrm{N}}\right) \tag{3}\]  
The EHD force acting on the neutral gas can be approximated, according to the [PERSON]' approach [30], by the following equation:  
\[\mathbf{F}=\mathrm{e_{0}}\left(\mathrm{N_{p}}-\mathrm{N_{n}}-\mathrm{N_{e}} \right)\mathbf{E} \tag{4}\]  
In which \(\mathrm{N_{i}}\) is the i-charged species density and \(\mathrm{e_{0}}\) is the elementary charge, towards the same direction of the electric field. When charged molecules, generated in the corona plasma region, are accelerated by the electric field and gain kinetic energy, due to their collision with other neutral species, their momentum is transferred to the adjacent molecules, resulting in a bulk gas flow. In particular, according to [PERSON] [30], electric wind velocity increases with the square root of both the electrode gap and the discharge current, according to Equation (5):  
\[v_{G}=D\sqrt{\frac{dI}{\mu}} \tag{5}\]  
where \(d\) is the electrode gap, \(I\) is the discharge current, \(\mu\) is the ion mobility, and \(D\) is a constant. This simplified equation is valid for the hypothesis that the cross-section area of the corona discharge and the gas density are constant. The discharge current is then evaluated as follows [2]:  
\[I=CV(V-V_{0}) \tag{6}\]  
where \(V\) is the applied voltage, \(V_{0}\) is the onset voltage, and C a constant dependent on the electrode geometry and ion mobility. The corona onset voltage is the lowest voltage at which continuous corona occurs, and it is described by the Peek's law. Equation (6) highlights that the discharge current increases in a quadratic manner with the applied voltage. The wire electrode thickness also affects the electric wind velocity. [PERSON] [31] showed that by decreasing the wire radius it is possible to attenuate the presence of micro-discharges on the wire and induce an increase in the maximum electric wind velocity.