\mathrm{m} \leq y \leq 0.178\, \mathrm{m}$, $-0.146\, \mathrm{m} \leq z \leq 0.43\, \mathrm{m}$. The 3D-grid consisted of polyhedral cells with prism cells near the wall boundaries, so that there were more than $10$ cells per gap height. The boundary opposite to the mask was the pressure outlet and all other boundaries except the mask surface and the nostrils were impermeable no-slip walls. The computational grid consisted of ca. $0.9$ million cells.

Two further cases were investigated with a slightly modified setup, i.e. under-relaxation factors $0.9$ (velocities) and $0.5$ (pressure), time step $\Delta t = 0.0002\, \mathrm{s}$ and a finer grid with ca. $3\cdot 10^{6}$ cells (cf. Fig. \[FIG3Dgrid2\]).

![Top: Computational grid for 3D-flow simulations with realistic head and mask geometries (top left) and close-up views of mask (top right) and gap (bottom)[]{data-label="FIG3Dgrid2"}](images/3DMnarrowGap/GridSection.png "fig:"){width="0.9\linewidth"} ![Top: Computational grid for 3D-flow simulations with realistic head and mask geometries (top left) and close-up views of mask (top right) and gap (bottom)[]{data-label="FIG3Dgrid2"}](images/3DMnarrowGap/MHeadThinGap0_MeshSect3.png "fig:"){width="0.9\linewidth"}

In one set of simulations, air was allowed to enter the mask through two gaps below the eyes, each with width $B_{\mathrm{g}}=1.08\, \mathrm{cm}$ and average height $H_{\mathrm{g}}=0.66\, \mathrm{mm}$. The total gap length was $L_{\mathrm{g}}=0.71\, \mathrm{cm}$, which corresponds more closely to a typical 