well below $0.3$ (cf. Ferziger et al., 2020). The pressure within the mask is assumed to be uniform, so $\partial p/ \partial x_{i} \approx  0$ for all directions $x_{i}$; thus the surface area $S_{\mathrm{m}}$ of the mask filter-piece influences the flow through the filter, but the geometrical shape of the mask has a negligible influence. Therefore, the mask geometry can be simplified to a half-sphere e.g. with radius $r\approx 0.0502\, \mathrm{m}$ and surface area $S_{\mathrm{m}}\approx 0.015833\, \mathrm{m^{2}}$. The mask has a uniform gap along a width $B_{\mathrm{g}}$ of its perimeter and otherwise fits tightly to the face. The mask rim has length $L_{\mathrm{g}}$ with constant gap height $H_{\mathrm{g}}$. Face and nose are approximated as a plane pierced by a channel with cross-sectional area $S_{\mathrm{t}}$ (cf. Fig. \[FIG2DtheoryGeom\]). No flow occurs through the face, the nose or the boundaries of the gap.

When inhaling or exhaling, the total (volumetric) flow rate through the nose is $F_{\mathrm{t}}=u_{\mathrm{t}}S_{t}$ with average velocity $u_{\mathrm{t}}$ and cross-sectional area $S_{\mathrm{t}}$. From mass conservation follows $$F_{\mathrm{t}} = F_{\mathrm{g}} + F_{\mathrm{m}}  \quad ,
\label{EQFtFgFm}$$ with flow rate through the gap $F_{\mathrm{g}} = u_{\mathrm{g}}S_{\mathrm{g}}$, average flow velocity $u_{\mathrm{g}}$ within the gap, gap cross-sectional area $S_{\mathrm{g}}$, flow rate through the mask 