## 2 Background
### Exhaled ECA as a Visual Tracer of Respiratory Fluid Flow  
The overwhelmingly submicron droplets that form the particulate phase of ECA have negligible influence on its fluid dynamics, acting essentially as visible tracers or (to a good approximation) as molecular contaminants carried by the fluid. This follows from its basic fluid dynamical characteristic: it is a \"single-phase fluid flow\" (SFF) system [52; 53]. As a consequence, ECA droplets visually mark the actual expiratory flow associated with vaping exhalations (we discuss the optical properties that allow for its visualization in [22]). Exhaled ECA is just one among numerous gas markers and aerosols in a SFF regime that serve (and are widely used) to visualize expired air [54; 55]. This also applies to mainstream exhaled tobacco smoke, whose particulate matter is also made of submicron liquid and solid droplets. In fact, there are studies that have directly used cigarette smoke as a tracer to visualize respiratory airflows [56; 57; 58]. Respiratory droplets potentially carried by exhaled ECA would not change its possible role as a tracer of expiratory flows, since, as we show in Section 3.3, these droplets are also overwhelmingly in the submicron range and their numbers are several orders of magnitude fewer than ECA droplets. Submicron ECA droplets are carried by a fluid made of the gas phase of ECA strongly diluted in exhaled air (in practice, we can think of the carrier fluid as exhaled air at mouth temperature \(\sim\)30-35 \({}^{\circ}\)C and 80-100% relative humidity). Under such conditions, ECA droplets essentially follow the fluid flow because of their little inertia, as they are well within the Stokes regime with Reynolds numbers \(Re_{p}\ll 1\) and negligibly small relaxation times \(t_{\text{rel}}\), the response time of an aerosol particle to adjust to external forces. For \(d_{p}=0.3\,\upmu\)m, we get [59]  
\[t_{\text{rel}}=\frac{\rho_{p}\,d_{p}^{2}\,C_{c}}{18\,\mu}\approx 5.3\times 10^{- 6}\,\text{sec}, \tag{1}\]  
where \(\rho_{p}\sim 1.3\,\text{gm}/\text{cm}^{3}\) (close to VG density), \(\mu=1.895\,\times 10^{-5}\,\text{gm}/(\text{s}\,\text{cm})\) the dynamic viscosity of air at \(35\,^{\circ}\)C and \(C_{c}=1+(\lambda/d_{p})[2.34+1.05\,\text{exp}(-0.39d_{p}/\lambda)]\approx 1.4\) is the Cunningham slip factor with \(\lambda=0.066\,\upmu\)m the mean molecular free path of air. The relaxation time (1) provides the time scale for a particle released into a fluid with velocity \(U\) along a horizontal stream to settle into the fluid velocity (neglecting gravity). In this case (see Chapter 3 of [59]), the velocity of the particle \(v_{p}(t)=U\,(1-e^{-t/t_{\text{rel}}})\) becomes practically identical to \(U\) in about \(10^{-5}\) s (instantaneously in practical terms), thus justifying the notion of particles simply following the fluid flow with (practically) no influence on its dynamics, (effects that characterize the Stokes regime). This behavior also occurs for the larger ECA droplets of \(d_{p}\sim 1\,\upmu\)m whose relaxation times are \(t_{\text{rel}}\sim 10^{-4}\) (since \(t_{\text{rel}}\propto d_{p}^{2}\)). Evidently, these relaxation times are much smaller than macroscopic characteristic times of the carrier fluid (e.g., a 2 s inhalation time or even the tenths of a second the ECA stays in the mouth cavity [33]). The Stokes number is defined as \(St=t_{\text{rel}}/t_{f}\), where \(t_{f}\) is a characteristic fluid time, hence for the exhaled ECA we have \(St\ll 1\), which is another criterion to define the Stokes regime and is characteristic of SFF systems. Evidently, larger droplets (diameters larger than a few \(\upmu\)m) are present in particle diameter distributions (of both ECA and respiratory droplets) and such particles should contain a significant portion of the aerosol mass [35], but they are too few in numbers and deviate from the flow following ballistic trajectories, thus do not affect the dynamics of the carrier fluid to consider ECA as a biphasic fluid flow system (as is the case with violent coughs or sneezes, see [60]).