## 2 Electrohydrodynamic Model
### Governing Equations of Fluid Dynamics  
We assume the fluid is incompressible, and the viscous flow is laminar, described by the Navier-Stokes fluid equation. As is well known, this equation is a balance of forces acting on the fluid. Therefore, with an appropriate external force density term, the equation is written in the form  
\[\rho_{f}\frac{D\mathbf{u}}{Dt}=-\
abla p\mathbf{I}+\
abla\cdot\left[\eta\left( \
abla\mathbf{u}+\left(\
abla\mathbf{u}\right)^{T}\right)\right]+\mathbf{f} \tag{11}\]  
where \(p\) is the absolute pressure, and \(\mathbf{I}\) is the identity symmetric rank two tensor; \(\eta\) is the dynamic viscosity, which is a specific property of the fluid, and finally \(\mathbf{f}\) represents the external force density term. The left side of this equation is the total derivative of fluid momentum and represents a sum of inertial forces and convection forces; in addition, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reaction** & **Rate Coefficient** & **References** \\ \hline Argon & & \\ \(e^{-}+Ar\to e^{-}+Ar\) & \(f[\sigma(\varepsilon)]\) & [28] \\ \(e^{-}+Ar\to e^{-}+Ar^{+}\) & \(f[\sigma(\varepsilon)]\) & [28] \\ \(e^{-}+Ar\to 2e^{-}+Ar^{+}\) & \(f[\sigma(\varepsilon)]\) & [28] \\ \(e^{-}+Ar^{+}\to 2e^{-}+Ar^{+}\) & \(f[\sigma(\varepsilon)]\) & [28] \\ \(e^{-}+Ar^{+}\to 2e^{-}+Ar^{+}\) & \(5.06\times 10^{-15}\ T_{e}^{-0.67}\) (\(\blacksquare\)) & [29] \\ \(Ar^{*}+Ar^{*}\to e^{-}+Ar+Ar^{+}\) & \(6.2\times 10^{-16}\) (\(\blacksquare\)) & [30] \\ \(Ar^{*}+Ar^{*}\to e^{-}+2 Ar\) & \(3.0\times 10^{-16}\) (\(\blacksquare\)) & [30] \\ \(Ar^{+}+2 Ar\to Ar^{+}_{2}+Ar\) & \(2.25\times 10^{-43}\) (\(+\)) & [31] \\ Xenon & & \\ \(e^{-}+Xe\to e^{-}+Xe\) & \(f[\sigma(\varepsilon)]\) & [32] \\ \(e^{-}+Xe\to e^{-}+Xe\) & \(f[\sigma(\varepsilon)]\) & [32] \\ \(e^{-}+Xe\to e^{-}+Xe\) & \(f[\sigma(\varepsilon)]\) & [32] \\ \(e^{-}+Xe\to 2e^{-}+Xe^{+}\) & \(f[\sigma(\varepsilon)]\) & [32] \\ \(e^{-}+Xe^{*}\to 2e^{-}+Xe^{+}\) & \(f[\sigma(\varepsilon)]\) & [32] \\ \(Xe^{+}+Xe\to Xe+Xe^{+}\) & \(3.6\times 10^{-10}\) (\(\blacksquare\)) & [33] \\ \(Xe^{*}\to Xe\) & \(3.96\times 10^{6}\) (\(\lx@paragraphsign\)) & [33] \\ \(Xe^{*}+Xe^{*}\to e^{-}+Xe+Xe^{+}\) & \(5.0\times 10^{-16}\) (\(\blacksquare\)) & [33,34] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electron-impact and chemical reactions with rate coefficients for argon and xenon discharges. Reactions with a (\(\blacksquare\)) have units of m\({}^{3}\)/s; (+) in units of m\({}^{6}\)/s, and (*) in units of s\({}^{-1}\). first term on the right-hand side accounts for pressure forces (on the approach of pressure gradient), the second term represents the viscous forces, and the third term is the external electric force per unit volume acting on the charged particles that communicate momentum, per unit volume and per unit time, to the neutrals by the agency of electron-atom or ion-atom collisions. The electric current, represented by a drift-diffusion approximation form of the charged particle fluxes, appears in the Navier-Stokes equation through the force density, written here as in Ref. [22]  
\[\mathbf{f}=e(n_{i}-n_{e})\mathbf{E}-k_{B}T_{i}\mathbf{\
abla}n_{i}-k_{B}T_{e}\mathbf{ \
abla}n_{e} \tag{12}\]  
with \(k_{B}\) denoting the Boltzmann constant, and \(T_{i}\) and \(T_{e}\) the ion and electron temperatures, respectively. Equation (12) is a simplified form of the momentum-transfer equation (see, e.g., Ref. [36]). Eventually, other components of this force appear if a sophisticated form of the momentum transfer equation is used (see, e.g., Refs. [7; 8]). These additional terms may be necessary under certain conditions but are not considered in this work for the sake of simplicity and because Equation (12) contains the more relevant terms (see, e.g., Ref. [37]). This equation for \(\mathbf{f}\) can be significant in some specific regions of the discharge and be practically irrelevant in others.