# A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges
## 2 Model Description  
As shown in Figure 1, the atmospheric DBD considered in the simulation is generated in a gap filled with pure helium between two parallel-plate electrodes, both covered by a thin dielectric layer with a relative permittivity of 7.5, corresponding to the value of mica glass. The upper electrode is connected to a sinusoidal voltage with an amplitude \(V_{\mathrm{am}}\) of 2 kV, whereas the lower one is grounded. The width of the gas gap \(d_{\mathrm{g}}\) is fixed to 4.4 mm and the dielectric layer thickness \(d_{\mathrm{b}}\) is 1 mm. Note that the gap width is not very large (\(<\)5 mm) and much shorter compared with the radius of electrodes (\(\approx\)56.4 mm) [20]; in this case, the radially homogeneous assumption should be reasonable and accepted. On this basis, a one-dimensional fluid model is appropriate to simulate the discharge process, which has also been successfully applied to investigate the nonlinear phenomena in [6; 21; 22]. In the 1-D fluid model, electron properties, including the electron density and energy density, are governed by the Boltzmann equation under drift-diffusion approximation [23; 24; 25], as given by Equations (1)-(4); Equations (5) and (6) are multi-component transport equations [26; 27] determining the flux of heavy species. Besides, in order to determine the electric field distribution, [PERSON]'s equation is firmly coupled. \[\frac{\partial n_{\mathrm{e}}}{\partial t}+\
abla\cdot\mathbf{\Gamma}_{\mathrm{e }}=S_{\mathrm{e}} \tag{1}\]  
\[\frac{\partial n_{\varepsilon}}{\partial t}+\
abla\cdot\mathbf{\Gamma}_{\mathrm{ c}}+\mathbf{EI}_{\mathrm{e}}=S_{\mathrm{en}} \tag{2}\]\[\mathbf{\Gamma}_{\rm e}=-\mu_{\rm e}\mu_{\rm e}\mathbf{E}-D_{\rm e}\
abla\cdot n_{\rm e} \tag{3}\]  
\[\mathbf{\Gamma}_{\rm e}=-\frac{5}{3}\mu_{\rm e}n_{\rm e}\mathbf{E}-\frac{5}{3}D_{\rm e} \
abla\cdot n_{\rm e} \tag{4}\]  
\[\rho\frac{\partial\omega_{k}}{\partial t}=\
abla\cdot\mathbf{J}_{k}+S_{k} \tag{5}\]  
\[\mathbf{J}_{k}=\rho\omega_{k}(D_{k}\frac{\
abla\omega_{k}}{\omega_{k}}+D_{k}\frac {\
abla M_{\rm n}}{M_{\rm n}}-z_{k}\mu_{k}\mathbf{E}) \tag{6}\]  
Here, \(n_{\rm e}\), \(n_{\rm e}\), \(\mathbf{\Gamma}_{\rm e}\), and \(\mathbf{\Gamma}_{\rm e}\) respectively denote the electron number density, electron energy, total electron flux, and electron energy flux; \(\mathbf{J}_{k}\) is the flux vector for heavy species \(k\), and its mass fraction and charge number are \(\omega_{k}\) and \(z_{k}\), respectively; \(S_{\rm e}\), \(S_{\rm en}\), and \(S_{k}\) respectively represent the source terms describing the net changing rate of electron density, electron energy loss/gain, and the source term for heavy species \(k\); the electron mobility \(\mu_{\rm e}\) is solved by Bolsig\({}^{+}\) with the cross-section data from IST-Lisbon Database [28; 29; 30], then the electron diffusion coefficient \(D_{\rm e}\) can be obtained through [PERSON]'s relation [29]; the mobility and diffusion coefficient for heavy species \(k\), i.e., \(\mu_{k}\) and \(D_{k}\), are referenced from [25]; \(\mathbf{E}\) is the electric field intensity; the mixture properties \(\rho\) and \(M_{\rm n}\) stand for the density of mixture and its mean molar mass. The \"wall\" boundary applied at the surfaces sandwiched between plasma and dielectrics provides two sets of equations, describing the particle flux (Equations (7)-(9)) and the charge accumulation (Equations (10) and (11)), respectively.