## 4 Hydrodynamical Modeling of Direct Exposure
1-5 \(\upmu\)m and over) that initially follow the fluid stream but (depending on their size) exit the main flow to follow ballistic trajectories until they either deposit on surfaces, settle on the ground or evaporate (see [60,86]). Given the distance and time dispersion scales (<3 m and <2-3 min), we can approximate the ECA as an airflow at constant atmospheric pressure, air density and dynamical viscosity \(\rho_{a}\) and \(\mu\). For a jet source (vaper's mouth) approximated as an orifice of 1.5-3 cm\({}^{2}\) area [57] (diameter \(d_{0}\) = 1.25-1.75 cm) and initial velocities \(U_{0}\) given by (4), exhalation fluid Reynolds numbers \(Re_{f}=(\rho_{a}/\mu)U_{0}d_{0}\) = 600-4400 are in the transition between laminar and turbulent, values well below the high Reynolds numbers expected near a jet source [103,104], but we are mostly concerned with the jet evolution and displacement (penetration) along horizontal distances \(z\gg d_{0}\). Other parameters to consider are the injection time \(t_{\text{\tiny{adh}}}\) = 2-5 s and a temperature gradient from exhalation (initial) \(T\) = 30\({}^{\circ}\)-35\({}^{\circ}\) C (mouth temperature) into an assumed \(T\) = 20\({}^{\circ}\) C for the surrounding air. For such values and scales, the starting jet can be regarded as isothermal with thermal buoyancy becoming relevant only in the puff stage (see [107,108]). It is well known (see [103,104,110,111]) that statistically steady and unsteady jet/puff systems can be well approximated by analytic solutions of the Navier-Stokes and conservation equations (assuming incompressible fluids) with axial symmetry and a self similar profile for the average centerline and radial components of the velocity field in cylindrical coordinates \(\vec{U}=[U_{z},U_{r},U_{\phi}]\) (see Figure 1)  
\[U_{z}=U_{c}(t)\,f(\eta),\quad U_{r}=U_{c}(t)\,g(\eta),\quad U_{\phi}=0, \tag{5}\]  
where \(f(\eta)\), \(g(\eta)\) are empiric Gaussian or polynomial functions of the self similar variable \(\eta=r/z\), while the centerline velocity is \(U_{c}=U_{z}\) for \(r=0\) along the \(z\) axis, hence \(f(\eta)\), \(g(\eta)\) must satisfy \(U_{z}=U_{c}\) and \(U_{r}=0\) at \(r=0\). In fact, for statistically steady jets (as we are assuming) the axial and radial similarity functions \(f(\eta)\) and \(g(\eta)\) satisfy [111]  
\[g(\eta)=\eta\,f(\eta)-\frac{1}{\eta}\int_{0}^{\eta}\bar{\eta}\,f(\bar{\eta})\, d\eta, \tag{6}\]  
so that \(g(\eta)\) can be obtained from an empiric choice for \(f(\eta)\) (see examples in [103,104,105,106,107,108,109]). The assumptions we make (statistically steady and axially symmetric self similar pro  
Figure 1: Puff and initial Jet with axial symmetry. The starting jet is propelled by linear momentum parallel to the centerline velocity \(U_{c}\); the arrows above and below represent the entrainment velocity mixing surrounding air with the carrier fluid. As the fluid injection terminates (end of exhalation), the entrained air makes about 40% of the fluid mass transition into an ellipsoidal puff through highly turbulent vortex structures. At this point, the puff is likely to disperse rapidly as horizontal displacement velocities are comparable to velocity fluctuations characterizing high turbulence and thermal buoyancy. file) imply that the jet/puff system fulfills the conservation of linear specific momentum \(Q=V_{p=}\,U_{c}\) (puff) and force \(\dot{Q}=(d/dt)(V_{p=}\,U_{c})\) (jet) where \(V_{p=}\) is the penetration volume [103; 109], hence \(Q=Q_{0}\), \(\dot{Q}=\dot{Q}_{0}\) for an initial time \(t=t_{0}\).