1 Introduction

COVID-19 has been declared as a pandemic, claiming 1 million deaths to date and still counting. On July 9th, 2020, WHO said that “airborne transmission” of COVID-19 in poorly ventilated locations “cannot be ruled out” [1]. Also, Morawska and Cao [2] emphasized the importance of airborne transmission of COVID-19. While sneezing, the infected person’s droplets can travel over a region of “influence” or could be suspended in the atmosphere, which eventually could transmit the virus (by transporting droplets) to the neighboring fellow. Thus, this sneezing phenomenon is a vital aspect to consider while framing the overall policy regarding public transport and public gathering.

While sneezing, broad size spectra of droplets are being ejected; it could be tiny droplets of the range of a few microns or size in hundreds of microns [3,4,5,6,7,8,9]. The flow behavior of the smaller droplets is primarily governed by the microscopic phenomena like Kelvin and Köhler effects. Consequently, the “likelihood” of the COVID-19 virus depends on these phenomena. This aspect of the “likelihood of survival of coronavirus” was extensively studied by Bhardwaj and Agrawal [4, 10]. But, as expected, the larger droplets’ behavior will be influenced by inertia and gravity. The combined effects of these factors were studied in the work of Munir and Xu [11]. In this work, they found that gravity and surface tension plays an important role in propagating the micro-bubble. In addition to the “likelihood” of the coronavirus, the other aspect that has to be taken into account is the distance traveled by these droplets in the ambient surrounding before the droplets get dissipated in the environment [12, 13]. Concerning this point, Cummins et al [14] using the analytical model, deduced that the intermediate size droplets achieve the minimum horizontal range. Also, they argued that the “bi-modal” distribution of droplet size is a function of the Stokes number. In a similar context, Das et al [15] using Monte-Carlo simulations demonstrated that the bigger droplets travel a considerable distance. However, the smaller droplets remain suspended for a longer time and thus concluded that spatial and temporal isolation is vital for preventing the virus’s spread. Meanwhile, Vadivukkarasan et al [16], through rigorous experiments, concluded that the size distribution of droplets coming out of the respiratory fluids depends on the multiple hydrodynamic instabilities. Furthermore, they pointed out that aerosol generation could be attributed to these instabilities. Interestingly, Prasanna Simha and Mohan Rao [17] using experimental analysis found a universal exponential decaying distance-velocity law in the human coughing flow dynamics. They argued that the viscous vortex rings govern this universal flow dynamics pattern. On the preventive aspect from the COVID-19, Dbouk and Drikakis [18] demonstrated the effectiveness of the masks to prevent the spread of the virus through droplets. In addition to this seminal article, in a separate work, they studied the weather conditions on the spread of the droplets [19]. They pointed out that the local weather conditions, primarily the humidity and temperature, will play a decisive role in the second wave of the COVID-19 pandemic. Pendar and Páscoa [20] numerically investigated different scenarios of sneezing and coughing and found that the shorter people are more prone to get affected by the virus because of the droplet’s trajectory coming out from the sneezing/coughing person.

The flow dynamics associated with the sneezing and/or coughing was modeled or visualized as a free shear flow such as turbulent jet/plume [21,22,23,24,25]. While modeling the flow as a jet with the volumetric heating, the flow dynamics is affected by the initial buoyancy; this effect was numerically studied and modeled by Pant and Bhattacharya [26] and Bhattacharya and Pant [27]. Similarly, cumulus clouds were represented as turbulent jets/plumes [28,29,30,31]. Considering the same modeling analogy, we are trying to incorporate the idea of cloud microphysics in the event of human sneezing. Also, recently, Diwan et al [32] in the section “Two closely related fluid flow problems” postulated that “cumulus cloud flows” and “dynamics of small water droplets” are “relevant flow problems” to understand the coughing/sneezing actions. Thus, the present work is motivated to import the present understanding of the mixing in clouds to interpret and expand the understanding of sneezing under the extreme scenario of in-homogeneous mixing. Although we sincerely acknowledge that these two fields (human sneezing and cloud) are entirely different with respect to scale, Reynolds number, and physical/biological perspectives. In clouds, the mixing is primarily divided into two main types: homogeneous and in-homogeneous mixing. Mathematically quantified as Damköhler number(\( \text {Da}\)), given by:

$$ {\text{Da}}=\frac{\tau_F}{\tau_D}$$
(1)

here, \(\tau _F\) is the mixing/fluid time scale and \(\tau _D\) is the time scale associated for droplets [33, 34]. Depending on the problem scenario, \(\tau _D\) is either referred to as phase relaxation time of droplets or the evaporation time scale of droplets [34]. When the time scale of mixing/fluid (\({\tau _F}\)) is much less than the evaporation time scale of droplets (\(\text {Da}<<1\)), the mixing is referred to as homogeneous mixing. It can be physically interpreted as the scenario when the in-homogeneities in the domain are well mixed before the droplet’s response. On the other hand, when the time scale of mixing/fluid (\({\tau _F}\)) is much larger than the evaporation time scale of droplets (\(\text {Da}>>1\)), the mixing is referred to as in-homogeneous mixing. This situation can be physically understood as the immediate response of droplets via evaporation/condensation to the in-homogeneous domain before the inhomogeneities are well mixed by the fluid mixing scales. The droplets tend to evaporate in a few seconds (depending on the size) [24]; thus, the mixing at the sneezing interface can be regarded as a case of extreme in-homogeneous mixing.

Srivastava [35] reported an extensive survey to demonstrate an interplay between the COVID-19 virus transmission and various atmospheric parameters viz. temperature, humidity, wind speed, and the particulate matter. He emphasized that a direct correlation exists between the particulate/gaseous pollutant and the COVID-19 cases. Furthermore, he noted that humidity/temperature has an inverse effect on the COVID-19 transmission. However, in the works of [4, 19, 36] they showed that with the increasing humidity level of the environment, the possibility of the COVID-19 survival increases. Thus, the present work focuses on the effect of humidity contrast between the sneezing zone of the COVID-19 patient and the ambient environment. The initial size distribution of the saliva droplets exhaled during talking/coughing is assumed to follow Weibull distribution [8, 37]. Furthermore, we assumed no distinction between the (COVID-19) virus-laden droplet distribution and the normal coughing/sneezing distribution. Specifically, categorization of droplet distribution according to the higher or lower viral loadings is beyond the scope of the present work.

2 Problem set up, governing equations, methodology and initial condition

2.1 Problem set up

We are considering a 1-D domain (\(0\le x\le 1\)) in which step function of the variables is considered. Figure 1 shows the schematic of the problem set-up. We assume that the COVID-19 droplets’ sneezing has a domain of influence represented by the dotted purple line. This purple line distinguishes the sneezing (domain of influence, \(0\le x < 0.5\)) region from the surrounding atmosphere (\(0.5\le x\le 1\)). This sneezing region (\(0\le x < 0.5\)) contains droplets of the COVID-19 virus and has saturated humidity. In contrast, the surrounding atmosphere (\(0.5\le x\le 1\)) is devoid of these droplets and has humidity based on the city’s location. We took a section of this phenomenon (square box in the figure 1) and modeled it as a 1-D problem (\(0\le x\le 1\)). Thus, overall the region \(0\le x < 0.5\) represents the domain of influence for the sneezing action while region \(0.5\le x\le 1\) signifies an ambient condition driven by the humidity of the environment. The size distribution of the COVID-19 droplets is assumed as the Weibull distribution, which is given by the equation 2:

Figure 1
figure 1

Schematic of the problem set up. The square box is the area of interest in which a step function for the variables is assumed. The interface between the sneezing and non-sneezing zone is depicted by the dash-dotted purple line. The standard triangular file (STL) of the human face is taken from [38].

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Figure 2
figure 2

Methodology: A straight horizontal line represents a one-dimensional domain, red dots showing the grid points, initially at each grid nodes the DSD is defined, and this DSD is divided into bins of length d\(\lambda \).

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$$\begin{aligned} h=\frac{n}{\lambda _p}\left( \frac{\lambda }{\lambda _p}\right) ^{n-1}e^{-(\lambda /\lambda _p)^n} \end{aligned}$$
(2)

here, \(n=8\), \(\lambda _p=0.26\), and \(\lambda \) is normalized radius square described in section 2.2. These values of n and \(\lambda _p\) are considered from the work of Dbouk and Drikakis [37]. Noting that the value of \(\lambda _p\) is scaled in the range of [0, 1].This size distribution of droplet is shown in figures 1 and 2. Dbouk and Drikakis [37] emphasized that, while coughing, the droplets size distribution is best fitted with the Rosin–Rammler or Weibull distribution. While modeling this simplified problem set up, we assumed that the growth of the droplets is governed by one-dimensional diffusion of vapor, thus neglecting the droplet collision. We are neglecting the surface effects, primarily known as Kelvin and Köhler effects. We assume that the Stokes’ number is relatively small, thus neglecting the sedimentation effects. Furthermore, we are also ignoring the fluid/flow properties change due to the presence of mucus during sneezing. In this work, we are making a further assumption of adiabatic mixing, which implies that the effect of temperature is neglected. A similar set-up of the problem was previously used in a two-dimensional framework by Pant and Bhattacharya [39] and in one-dimensional scope by Pinsky et al [40] from the perspective of clouds.

Figure 3
figure 3

Validation against the work of Pinsky et al [40] for (a) normalized standard droplet size distribution (SDSD), corresponds to the figure 6(b) in Pinsky et al [40] and (b) normalized relative humidity (RH1), referred as normalized supersaturation in Pinsky et al [40], referred as figure 3(a) in Pinsky et al [40].

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Figure 4
figure 4

Variation of normalized relative humidity (RH1) with respect to time for initially (a) RH2 \(=-0.1\) (b) RH2 \(=-0.3\) and (c) RH2 \(=-0.4\).

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Figure 5
figure 5

Variation of SDSD for case \(\text {RH2} =-0.1\) against time at (a) t \(=50\), (b) t \(=100\) and (c) t \(=500\).

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2.2 Governing equations and methodology

We solved the following governing equations in a one dimensional frame-work. These set of equations are primarily mimicking the evaporation and diffusion growth of droplets [40]. Pinsky et al [40] have derived and used these equations to understand, explain and formulate the inhomogeneous mixing in clouds.

$$\begin{aligned} q(x,t)= & {} \int _0^\infty \lambda ^{3/2}h(x,t,\lambda )d\lambda \end{aligned}$$
(3)
$$\begin{aligned} \frac{\partial {\Gamma (x,t)}}{\partial {t}}= & {} \frac{1}{\text {Da}}\frac{\partial ^2 {\Gamma (x,t)}}{\partial {x^2}}\end{aligned}$$
(4)
$$\begin{aligned} \text {RH1}(x,t)= & {} \Gamma (x,t)-q(x,t)\end{aligned}$$
(5)
$$\begin{aligned} \frac{\partial {h(x,t, \lambda )}}{\partial {t}}= & {} \frac{1}{\text {Da}}\frac{\partial ^2 {h(x,t, \lambda )}}{\partial {x^2}}-\frac{2}{3}(\text {RH1(x,t)})\frac{\partial h(x,t, \lambda )}{\partial \lambda }\end{aligned}$$
(6)
$$\begin{aligned} \lambda= & {} R^2 \end{aligned}$$
(7)

here, R is radius. Eqn 3 signifies the liquid water ratio (q(x, t)) that is contained inside the droplets. \(\Gamma (x,t)\) is a conservative variable governed by the Eqn 4. From these two variables (q(x, t) and \(\Gamma (x,t)\)), the \(\text {RH1(x,t)}\) (named as normalized supersaturation in Pinsky et al [40]) referred as normalized relative humidity is computed using Eqn 5. Finally, the evolution of droplets size distribution \(h(x,t,\lambda )\) is governed by Eqn 6. The first term on the right-hand side of the Eqn 6 relates to the diffusion, while the second term relates to the evaporation phenomena. Thus, the DSD is governed by the dual effect of inhomogeneity inside the domain and the droplets’ evaporation due to the humidity contrast. To be noted here that the DSD \(h(x,\lambda ,t)\) is a function of spatial coordinates (x) and droplets radius (\(\lambda =R^2\)). The extent of “mixing” or “flow behavior” is governed by the parameter Damköhler number(\( \text {Da}\)). Under the assumption of quiescent environment the time travel of sneezing is approximately 22 s [41], thus for the droplets of diameter (\(2-3\) \(\mu \)m) the typical value of \(\text {Da}\sim 22/1\sim 22\). In the present work, we consider the COVID-19 droplets distribution in a scenario of extreme in-homogeneous mixing (\(\text {Da}=1000\)), with different humidity levels. This value of Damköhler number (\(\text {Da}=1000\)) is chosen to emphasize that the response of COVID-19 droplets in the ambient surrounding will be relatively much faster as compared to the environmental mixing time scales. However, this specific value of 1000 is considered heuristically to signify that the Damköhler number is comparatively very high than 1 to be safely regarded as inhomogeneous mixing. While considering the \(\text {Da}=1000\) we assume that from small droplets (less than 1 \(\mu \)m) the evaporation time scale if 0.02 s and thus the value of Damköhler number (\(\text {Da}\)) is given by:

$$\begin{aligned} \text {Da}=\frac{\tau _F}{\tau _D}=\frac{22}{0.022}=1000 \end{aligned}$$
(8)

While the influence of ambient conditions is tuned by the term RH1. All these equations are written in the normalized form; for further details, refer to Pinsky et al [40]. Figure 2 shows the methodology of the finite difference method used to solve these set of coupled equations. The length of the domain is 1, and it is discretized into grid points (total points\(=nx\)); at each grid point, the DSD is defined. The DSD is considered according to the Eqn 2 and plotted in figure 2. This DSD at each grid nodes is discretized into bins of \(d\lambda \) divisions (total divisions \(=np\)). Standard central differencing scheme (CDS) is used to discretize the second-order terms while first-order Euler time stepping is used for time marching. The time step is less than the diffusion time scale for each case.

2.3 Initial conditions

The initial conditions for the \(\Gamma (x,t)\) and \(h(x,\lambda ,t)\) (DSD) are defined as step function and mathematically given by:

$$\begin{aligned} \Gamma (x,0)= & {} {\left\{ \begin{array}{ll} 1 &{} \text {if }0\le x < 0.5 \\ RH2 &{} \text {if }0.5\le x \le 1 \end{array}\right. }\end{aligned}$$
(9)
$$\begin{aligned} h(x,\lambda ,0)= & {} {\left\{ \begin{array}{ll} \frac{n}{\lambda _p}(\frac{\lambda }{\lambda _p})^{n-1} e^{-(\lambda /\lambda _p)^n} &{} \text {if }0\le x < 0.5 \\ 0 &{} \text {if }0.5\le x \le 1 \end{array}\right. } \end{aligned}$$
(10)

Noting that the 1D domain is spanning as \(0\le x\le 1\). In the present work we are defining the \(\text {RH2}=\text {RH}-1\) and varying the value of \(\text {RH2}\) as \(-0.1\), \(-0.3\) and \(-0.4\).These \(\text {RH}\) values (varying from \(60\%\) to \(90\%\)) corresponds to the ambient humidity level taken from the online data for the month of August for 6 major cities (viz. Bejing, Mumbai, New York, Sydney, Singapore and London) [42]. Specifically we considered the relative humidity RH = \(90\%\), \(70\%\) and \(60\%\).

3 Validation and grid independence study

For validation purpose, we considered a similar setup described in Pinsky et al [40], in which the mono-dispersed DSD was considered, \(\text {Da}=1\) and \(\text {RH2}\) was fixed at \(-1.5\). The number of bins for the DSD discretization is fixed at \(np=24\). Figure 3 shows the comparison of the present work against the previous work of [40]. Figure 3(a) compares the normalized droplet size distribution (named as SDSD) at t=0.475. Here we are representing SDSD by f(x, R, t) and this quantity is related to DSD (\(h(x,t,\lambda )\)) by Eqn 11

$$\begin{aligned} f(x,R,t)=2R \times h(x,\lambda ,t) \end{aligned}$$
(11)
Figure 6
figure 6

Variation of SDSD for case RH2 \(=-0.3\) against time at (a) t \(=50\), (b) t \(=100\) and (c) t \(=500\).

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Figure 7
figure 7

Variation of SDSD for case RH2 \(=-0.4\) against time at (a) t \(=50\), (b) t \(=100\) and (c) t \(=500\).

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Figure 8
figure 8

Variation of SDSD at \(x=0.75\) for (a) RH2 \(=-0.1\), (b) RH2 \(=-0.3\) and (c) RH2 \(=-0.4\).

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The present work matches well with the previous work of Pinsky et al [40]. Also, we compared the SDSD for different resolutions (\(nx=50,81,100\)) to check our results’ sensitivity with the grid size and found that the results are independent of the grid resolutions. In the rest of the manuscript we have simulations corresponding to \(nx=100\) and \(np=50\). Similarly, in figure 3(b), we are comparing the variation of normalized supersaturation (\(\text {RH1}\)) of present work against the previous work of Pinsky et al [40]. Pinsky et al [40] solved the governing equations using “PDEPE” utility of MATLAB. Thus, difference between the present and the prior work of Pinsky et al [40] could be because of the different methodologies for solving the governing equations.

4 Results and discussions

In this section, we first consider the variation of humidity with time in the domain. Figure 4(a) shows the variation of normalized humidity (RH1) with respect to time for the lowest humidity contrast case (\(\text {RH2} = -0.1\)). From hereon, for simplicity, we are denoting the normalized humidity (RH1) as humidity. Please note that in sections 2.2 and 2.3, we are representing RH2 as normalized initial humidity, RH1 as normalized relative humidity, and RH as normalized ambient relative humidity. With the passage of time, the humidity contrast between the two sections of the domain decreases and finally attains the complete saturation state. While comparing the different initial humidity contrast (figures 4(b) and 4(c)), the time to completely homogenize the humidity level increases with the increase in the humidity contrast. This can be understood by the difference in the initial humidity level; because of the high humidity contrast, the mixing took a long time to homogenize the humidity in the domain. The final saturation state (\(\text {RH1} = 0\)) for all the cases signifies that some droplets have completely evaporated and have liberated the water vapor and consequently have saturated the domain in the process of mixing the gradients. This phenomenon will be explained in the next paragraph.

Figure 9
figure 9

Comparison of final (at \(\hbox {t} =500\)) SDSD (in log scale) for all the humidity contrast.

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Figures 5–7 shows the evolution of normalized standard droplet size distribution (SDSD) against time for different humidity contrast case (\(\text {RH2} =-0.1\), \(\text {RH2} = -0.3\) and \(\text {RH2} = -0.4\)). In these figures, two sections (\(x=0\) and \(x=0.25\)) are considered inside the saturation section, while the other two sections are considered in the saturation deficit region (\(x=0.75\) and \(x=1\)), and a section at the interface (\(x=0.5\)) is considered. Initially at t\(=50\), shown in figure 5(a) the droplets in the saturated zone (\(x=0\)) have evolved in the “bi-modal” distribution which is an important aspect of inhomogeneous mixing [43, 44]. Also, the “bi-modal” distribution of droplet size while sneezing was reported in the experiments of Han et al [3]. They attributed this distribution to biological phenomena. But, recently, the analytical work of Cummins et al [14] credited this distribution as the consequence of droplets’ Stokes number. This “bi-modal” distribution have taller peaks for larger radius and smaller peaks related to a smaller radius. This implies that most droplets grew larger (because of condensation), while a smaller number of droplets have undergone evaporation. A similar trend is followed at \(x=0.25\), while important phenomena occur in the humidity deficit region. To emphasize it, we are considering the growth of the droplets at \(x=0.75\) section of the domain since the SDSD looks similar for all the humidity contrast in the regions of \(x=0\) and \(x=0.5\) or in the saturated regions. This is not surprising since, at these locations, the initial conditions were the same.

Figure 8(a) shows the evolution of SDSD with respect to time in the humid deficit region (at \(x=0.75\)) for the RH2 \(=-0.1\) case. Initially (at t \(=10\)), the droplets evaporate, and thus, the SDSD widens corresponding to the smaller droplets. The evaporation of droplets liberate vapor in the domain, hindering the other droplets from undergoing evaporation, and in fact, experienced condensational growth. Thus, there is simultaneous evaporation and condensation of droplets, resulting in the “bi-modal” distribution (at t \(=30\) and t \(=40\)). The similar pattern is also evident for the higher humidity contrast cases (RH2 \(=-0.3\) and RH2 \(=-0.4\)), shown in figures 8(b) and 8(c). But, the number of completely evaporated droplets are higher for the higher humidity contrast case. Certainly, because of these evaporated droplets, the final SDSD for different humidity contrast also varies and will be discussed in the next paragraph.

On comparing the final SDSD for all the humidity ratio (shown in figure 9), we found that the asymmetric bimodal size distribution is a consistent feature in all the humidity contrast. Asymmetry in the size distribution is due to the fact that large portion of droplets keep growing because of condensational growth while a smaller portion of droplets are evaporated to make the 1D parcel/box saturated. A long-tail corresponding to the smaller radius signifies the droplets’ evaporation for all the humidity cases. Simultaneously, droplets are also prone for condensational growth; this tendency will be highest when the humidity contrast is minimal (RH2 \(=-0.1\)). Now, since the droplets have very large spectra of size distribution, i.e., from very small size to very large size droplets, their time scales of evaporation also vary from fractions of seconds to half a second [24]. The smaller droplets tend to remain suspended in the air for a long time, while the larger droplets will travel a larger horizontal distance [15]. Therefore it can be cautiously concluded that under highly in-homogeneous conditions and with the high humidity contrast between the sneezing zone and environment, the rate of infection would be higher.

5 Conclusion

Using the simplified 1D-diffusion equation, we evolved the droplet size distribution (DSD) of COVID-19 droplets ejecting from the human sneezing under extreme in-homogeneous mixing condition with varying humidity contrast between sneezing influenced zone and the ambient environment. We found that under these conditions, a “bi-modal” distribution is evident, having a long tail of droplets with a lower radius, which signifies the evaporation of droplets. Simultaneously, droplets are prone to condensation, which increases the diameter of the droplets. This wide range of droplets diameter, in turn, will have different evaporation time scales. Consequently the smaller droplets remain suspended in the ambient atmosphere for a longer time while larger droplets travel larger horizontal distance [15]. This effectively implies that the transmissibility of COVID-19 increases for high humidity contrast region.