## 4 Hydrodynamical Modeling of Direct Exposure
The stream wise center-line penetration distance and velocity for the jet and puff stages can be given by [109]:  
\[\text{Starting Jet} z_{j}(t)-z_{j0}= C_{jz}\,\big{(}Q_{0}\,U_{0}\big{)}^{1/4}\,(t-t_{j0})^{1/2}, r_{j}(t)=C_{jr}\,z_{j}(t), \tag{7}\] \[U_{jc}=\frac{dz_{j}}{dt}= \frac{C_{jz}^{2}\,(Q_{0}U_{0})^{1/2}}{2\,(z-z_{j0})},\] (8) \[\text{Puff Stage} z_{p}(t)-z_{pd}= C_{pz}\,(Q_{0}\,U_{0})^{1/4}(t-t_{p0})^{1/4}, r_{p}(t)=C_{pr}\,z_{p}(t),\] (9) \[U_{pc}=\frac{dz_{p}}{dt}= \frac{C_{pz}^{4}\,Q_{0}U_{0}}{4\,(z-z_{pd})^{3}}, \tag{10}\]  
where the constants \(C_{jz},C_{jr},C_{pz},C_{pr}\) are empirically determined, \(z_{j0}\) is the \(z\) coordinate value of the ejection orifice and \(z_{pd}\) is the virtual origins of the puff (see Figure 2), which is an appropriate parameter to separate the starting jet and puff stages even though it lies within the starting jet region (see the detailed explanation in [109]). In addition, notice that the direct proportionality of the radial and axial displacements, \(r(t)\) and \(z(t)\), for the jet and puff regimes (through \(C_{jr}\), \(C_{pr}\)) implies the conical shape depicted in Figure 1, with a distinct, but constant, radial dispersion angle \(\alpha\) for each stage. For the axial geometry of the jet/puff system under consideration, we have \(Q_{0}U_{0}=\pi d_{0}^{2}U_{0}/(8C_{pr}^{2})\) and \(\dot{Q}_{0}U_{0}=(3\pi/4)d_{0}^{2}\,U_{0}^{2}\). Following the distinction in Tables 1 and 2 of [109] (see also [109]) between starting and interrupted jets that are, respectively, applica  
Figure 2: Jet/Puff horizontal displacement and centerline velocity. **(a)** The displacement \(z_{jc}\) of the staring jet (blue) and \(z_{pc}\) of the puff (red) as functions of time (Equations (7)–(10)), for the three vaping topographies described in Section 2.1: DTL (Direct to Lung), MTL (Mouth to Lung) and MP (Mouth Puffing). We assumed as injection (exhalation times) \(3,4\) and \(5\) s. The initial velocities from bottom to top are \(U_{0}=50,100,200,300,400,500\) cm/s. **(b)** The centerline velocities \(U_{jc}\) for the starting jet (Equation (8)), as functions of the horizontal displacement \(z_{jc}\) during the injection times and initial velocities of **(a)** (green for MP, blue for MTL and red for DTL). Notice that once injection stops the jet has reached velocities \(\sim\)15–20 cm/s comparable to those of indoor air currents (bottom rectangle in **(b)**) and evolves into a puff. Computations and graphs were made with the software package Maple [112]. Worksheets are available through email request to the corresponding author. ble to our jet and puff stages, we choose the following numerical values for the constants in (7)-(10):  
\[C_{jz}=2.8,\quad C_{jr}=0.10-0.15,\quad z_{j0}=d_{0}/(2C_{jr}), \tag{11}\] \[U_{z}=U_{c}(t)\,f(\eta),\quad U_{r}=U_{c}(t)\,g(\eta),\quad U_{\phi}=0, \tag{5}\]0  
where the time \(t_{pd}\) and position of the puff virtual origin \(z_{pd}\) are determined numerically from (7) by the condition \(z_{j}(t_{pd})=z_{pd}\) (see [109]). The values of \(C_{jr}\), \(C_{pr}\) above indicate for the jet stage a highly collimated directional evolution with a narrow angle of radial dispersion \(\alpha=\arctan\,C_{jr}\sim 6-9\,^{\circ}\), while for the puff stage we have a wider radial dispersion \(\alpha=\arctan\,C_{pr}\sim 12-16\,^{\circ}\) (these angles roughly fit observational data on radial dispersion of mouth breathing, see [57; 80; 81; 82]). Therefore, since the jet stage is better approximated by an axially symmetric statistically steady jet, this collimated shape implies that the axial displacement is far more relevant than the narrow radial displacement to assess the distance and direction of direct exposure to possibly transported pathogens through vaping exhalations. From the dynamical Equations (7)-(10) with the parameter values in (11)-(12), we display in Section 5.2 the horizontal distance and jet/puff velocities characterizing direct exposure.