key: cord-0615956-lt3n032f
authors: Wang, Hongping; Li, Zhaobin; Zhang, Xinlei; Zhu, Lixing; Liu, Yi; Wang, Shizhao
title: The motion of respiratory droplets produced by coughing
date: 2020-10-24
journal: nan
DOI: nan
sha: ab61b705a1f06e09c2d68084b972418a1b82d705
doc_id: 615956
cord_uid: lt3n032f

Coronavirus disease 2019 (COVID-19) has become a global pandemic infectious respiratory disease with high mortality and infectiousness. This paper investigates respiratory droplet transmission, which is critical to understanding, modeling and controlling epidemics. In the present work, we implemented flow visualization, particle image velocimetry (PIV) and particle shadow tracking velocimetry (PSTV) to measure the velocity of the airflow and droplets involved in coughing and then constructed a physical model considering the evaporation effect to predict the motion of droplets under different weather conditions. The experimental results indicate that the convection velocity of cough airflow presents the relationship $t^{-0.7}$ with time; hence, the distance from the cougher increases by $t^{0.3}$ in the range of our measurement domain. Substituting these experimental results into the physical model reveals that the small droplets (initial diameter $D leq$ 100 $mu$m) evaporate to droplet nuclei and that the large droplets with $D geq$ 500 $mu$m and initial velocity $u_0 geq$ 5 m/s travel more than 2 m. Winter conditions of low temperature and high relative humidity can cause more droplets to settle to the ground, which may be a possible driver of a second pandemic wave in the autumn and winter seasons.

Coronavirus disease 2019 , caused by the novel coronavirus SARS-CoV-2, has become a global pandemic infectious respiratory disease. According to the coronavirus resource center at Johns Hopkins University, more than 30 million people worldwide have been confirmed to have COVID-19, and there have been more than 1,000,000 deaths globally. Other infectious respiratory diseases such as flu, tuberculosis, severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS) still threaten people's health and life. Understanding the transmission of pathogens is critical to modelling and controlling epidemics (Bourouiba, Dehandschoewercker, and Bush, 2014) . In general, virion-laden respiratory fluid droplets are released into the environment by infected people through expiratory events such as talking, coughing and sneezing. There are three possible transmission pathways between infectious and susceptible individuals (Bourouiba, Dehandschoewercker, and Bush, 2014; Huang et al., 2020; Mittal, Ni, and Seo, 2020) : large droplet transmission, contact transmission and airborne transmission. In the first two pathways, healthy people can be infected by physically inhaling droplets or contacting contaminated surfaces within close proximity. These two pathways are termed direct short-range routes of pathogen transmission, as stated by Bourouiba, Dehandschoewercker, and Bush (2014) .

Airborne transmission has been suggested to be an additional and important route for respiratory diseases (Morawska and Milton, 2020; Liu et al., 2020) . The microdroplet (or droplet nuclei) generated via small droplet evaporation can travel a very long distance within ambient air (Matthews et al., 1989; Dbouk and Drikakis, 2020a) . Liu et al. (2020) investigated the aerodynamic nature of SARS-CoV-2 by measuring the viral ribose nucleic acid (RNA) in aerosols in different areas of two Wuhan hospitals during the outbreak of COVID-19. The aerosols concentrations of SARS-CoV-2 were much higher in the toilet used by patients with COVID-19 and in areas where crowds that included infected individuals gathered. They proposed that SARS-CoV-2 may be transmitted through aerosols.

Coughing, which can generate droplets and droplet nuclei, is a typical symptom of COVID-19. These virion-laden respiratory fluid droplets are expelled into the environment by powerful air flow during coughing. Measuring the air velocity, droplet number density, velocity and size distribution are critical to understanding and modeling virus transmission. Zhu, Kato, and Yang (2006) investigated the transport characteristics of saliva droplets produced by coughing in a calm indoor environment via particle image velocimetry (PIV) experiments and numerical simulation.

The PIV results indicated that the initial velocity of cough airflow is as high as 22 m/s with an average velocity of 11.2 m/s. Numerical analyses that did not consider evaporation indicated that droplets smaller than 30 µm in diameter are mostly transported with the airflow, droplets with diameters of 50-200 µm are significantly affected by gravity, and droplets of 300 µm or larger are mostly affected by inertia. Gupta, Lin, and Chen (2009) measured the flow rate, flow direction and mouth opening area for 25 human subjects and found that the flow rate of a cough as a function of time can be defined as a combination of gamma-probability-distribution functions. Xie et al. (2009) carried out a series of experiments to measure the number and size of respiratory droplets produced by talking and coughing. The distribution of the droplet diameter at the origin was given in their paper. To predict the trajectory and settling speed of droplets, a discrete fallout model and a continuous fallout model were developed based on the theory of multiphase turbulent buoyant clouds by Bourouiba, Dehandschoewercker, and Bush (2014) . These models can be used with clinical data to yield better estimations of the range of airborne respiratory disease transmission. The influence of environmental conditions on droplet transmission was investigated using a three-dimensional numerical simulation by Dbouk and Drikakis (2020a) . It was found that saliva droplets can travel up to 6 m, which is much further than the 2 m social distance metric, when the wind speed varies from 4 km/h to 15 km/h. Wearing face masks is a fundamental and efficient protection against virus transmission. The Schlieren optical method was applied to visualize cough airflow with and without standard surgical and N95 masks (Tang et al., 2009) . The results indicated that human coughing produces a rapid turbulent jet into the surrounding air and that a mask can block the formation of this jet. David et al. (2012) used flow visualization to investigate the expelled air dispersion distances during coughing with a human patient simulator with and without a surgical mask or N95 mask in a negative pressure isolation room. The turbulent jet caused by coughing without a mask can propagate close to 0.7 m, while the air dispersion distance is approximately 0.15 m with an N95 mask. To clarify whether masks can offer effective protection against droplet infection, Kähler and Hain (2020) conducted three experiments of fluid mechanics to analyze the blockage caused by masks, to determine the effectiveness of different filter materials and masks, and to verify the effects of leakage flows at the edges of masks. The effectiveness and leakage of face masks were qualitatively visualized by Verma, Dhanak, and Frankenfield (2020) using smoke. Wearing masks can significantly curtail the speed and range of respiratory jets. Dbouk and Drikakis (2020b) used multiphase computational fluid dynamics in a fully coupled Eulerian-Lagrangian framework to investigate the droplet dynamics of mild coughing. This numerical results showed that a mask can reduce airborne droplet transmission. However, some droplets still spread around and away from the mask.

As stated by Asadi et al. (2020) , virologists and epidemiologists are racing to understand COVID-19 and how best to treat it. Many unknowns remain, but one thing is eminently clear: COVID-19 is both deadly and highly transmissible. In addition to direct large droplet transmission and contact transmission, airborne transmission is a potential important pathway, especially in an enclosed small space (Asadi et al., 2020; Liu et al., 2020) . Therefore, investigation of the generation and evolution of expiratory activities such as breathing, talking, coughing and sneezing, which can produce virion-laden droplets and aerosols traveling with the airflow, is very significant (Mittal, Ni, and Seo, 2020) . Regarding coughing, which is the main symptom of COVID-19, most studies include qualitative visualization without dynamic analysis. In the present work, we use a high-speed PIV system to capture the spatial-temporal evolution of coughing (He, Rubinstein, and Wang, 2002; Zhao and He, 2009; He, Jin, and Yang, 2017) . We carried out three optics-based fluid experiments. First, we analyzed large-scale visualization images generated using cigarette smoke Gupta, Lin, and Chen (2009) . Second, we used the PIV technique to calculate the airflow velocity near the mouth and analyzed the time-resolved flow fields. Third, we used particle shadow velocimetry (Estevadeordal and Goss; Hessenkemper and Ziegenhein (2018) , PSV) to track the trajectories of large droplets and analyzed the velocity distribution of the droplets. Finally, we constructed a model to consider the evaporation and motion of a droplet and analyzed the influence of weather conditions on droplet transmission. The rest of the paper is organized as follows.

In Sec. II, we first introduce the experimental facility, setup and data processing methods used in this work. In Sec. III, we present the results and discussion in accordance with these experiments.

Finally, we offer conclusions in Sec. IV.

A. Experimental setup shared opening of 60 mm × 60 mm was mounted at the end of the tube to block the laser and protect the volunteers. The length, width and height of the tube were 1000 mm, 300 mm and 300 mm, respectively. When conducting the experiment, volunteers were asked to sit in front of the opaque plate and place their mouths as close as possible to the opening, and the exhaled airflow or droplets passed into the testing room through this opening. Additionally, volunteers were asked to wear safety goggles to protect their eyes. As stated in the introduction section, three kinds of experiments for different objectives were performed in this work. Each experiment was carried out by four healthy male volunteers, and each volunteer repeated three times. The detailed experimental configurations are listed in Tab. I and are introduced in the next sections.

The coughing was visualized using cigarette smoke recorded by a high-speed Photron camera (FastcamSA2/86K-M3). The volunteer drew heavily on his cigarette and coughed out a cloud of smoke as naturally as possible. The diameters of cigarette smoke particles are mostly distributed in the submicron size range, i.e., 0.01-1 µm, as reported by Sahu et al. (2013) . A continuous laser sheet with a thickness of approximately 1 mm at a power of 9 W was used to illuminate the smoke.

Images with a resolution of 1280×1024 pixels 2 were recorded at 1000 frames per second (fps). A high-speed camera with a 50 mm F1.4 lens was placed approximately 1 m from the measurement domain, and this configuration yielded a field of view (FOV) of approximately 0.32×0.26 m 2 .

The velocity field near the mouth was estimated using the PIV technique. We still used a continuous laser sheet at a power of 9 W to illuminate the cigarette smoke exhaled during coughing.

Different from the coughing visualization, the smoke images were acquired at a very high sampling frequency of 10000 Hz due to the high velocity of the flow near the mouth. Limited by the transfer bandwidth of the camera, the image resolution was reduced to 320×256 pixels 2 . The distance of the camera from the measurement domain was approximately 1.0 m, and a 50 mm F1.4 camera lens was used in this experiment. The size of the FOV was approximately 0.08×0.06 m 2 .

The velocity fields were estimated using in-house PIV software with a three-pass window deformation iterative multigrid scheme (WIDIM) (Scarano, 2002) . Before the velocity estimation, the particle images were preprocessed using a Gaussian filter for image denoising. The interrogation window (IW) size used for the first pass was 48×48 pixels 2 , and the IW size of the final pass was set to 32×32 pixels 2 . The interval of the adjacent vectors was set to 2 pixels to increase the number of vectors. The cross-correlation map was calculated using a fast Fourier transform (FFT)-based approach, and the subpixel displacement was obtained by three-point Gaussian fitting. Outliers in the PIV fields were detected using the normalized median test proposed by Westerweel and Scarano (2005) , and the missing vectors were filled using linear interpolation. The final velocity fields were smoothed by average filtering with dimensions of 3×3 to further reduce the measurement noise.

This experiment was performed to investigate the velocity of saliva droplets during coughing.

To capture a clear droplet shape and track the trajectories, the PSV method was adopted. The flow region was illuminated using a high-power LED backlight, as shown in Fig. 1 , instead of the laser sheet used in the smoke visualization and velocimetry. Additionally, a small depth of focus (DoF) corresponding to a small value of lens aperture was used to generate a thin volume where the droplets can be clearly identified. There were no droplets captured by our camera for dry coughing; therefore, the volunteer drank some water to moisten their throat and kept little water in their mouth. The remaining water in their mouth was driven by the pressure of coughing and generated many droplets due to hydrodynamic instability (Scharfman et al., 2016; Mittal, Ni, and Seo, 2020) . A camera with a 105 mm F2.4 lens was placed 0.4 m from the measurement domain.

Images with a resolution of 800×400 pixels 2 were recorded at a frequency of 5000 Hz, and the exposure time was 1/8000 seconds. The FOV size was 0.028×0.014 m 2 .

Many droplets of different sizes were ejected from the volunteer's mouth during coughing, and only the droplets located in the region of the DoF were in focus in the images. To estimate a droplet's velocity, the location of the droplet had to be detected first. Segmentation methods based on a gray-level or gradient threshold are not appropriate for the droplets because of the intensity variation with the size of the droplets and the refraction of irregular shapes. Additionally, the presence of blurred droplets can locally modify the background around the in-focus droplets (Castanet et al., 2013) . In the present work, the processes for droplet detection are given in Fig good agreement with the edges of droplets, and the red dots represent the locations of the droplets.

Note that a few droplets present a crescent shape due to light refraction.

After droplet detection, the velocity of the droplets is determined using the noniterative doubleframe particle tracking velocimetry (PTV) proposed by Fuchs, Hain, and Kähler (2017) . The combination of particle shadow images and PTV is referred to as particle shadow tracking velocimetry (PSTV) (Hessenkemper and Ziegenhein, 2018) . We did not track the trajectories of the droplets because the measurement domain in the x direction is as short as 400 pixels. Figure 3 gives an example of the velocity vectors of the droplets obtained using PSTV; the minimum and maximum droplet velocities are 5.0 m/s and 10.0 m/s, respectively.

The coughing process is visualized by the expelling of smoke. Figure 4 shows the evolution of a cough at a time interval of 0.02 s. The onset of coughing is determined by visually inspecting the image sequence. The red rectangles indicate regions with high gray levels, corresponding to concentrated smoke. The distance s is defined as the distance from the left origin to the right border of the rectangle, and w denotes the width of the rectangles. At the beginning of coughing, the ejected smoke presents a cone-like shape and a high concentration and then rapidly transitions into turbulence. With the entraining of the ambient quiescent air, the cough smoke is gradually 

This result is different from that of Bourouiba, Dehandschoewercker, and Bush (2014) , who stated that s is related to t by s ∼ t 0.5 in the first phase, which is dominated by jet-like dynamics. We believe that s varies as t 0.3 because a cough is a pluse-like jet and not a continuous jet. In accordance with the fitting functions, the relationship between U c and s can be expressed as: imately 0.08 s) are considered because the smoke gradually diffuses to a low gray level. Figure 6 shows x, and the radial position is normalized by the jet half-width b 1/2 , which is defined as the radial distance corresponding to half of u max (Krishnan and Mohseni, 2010; Xu, Feng, and Wang, 2013; Xu et al., 2018) . In this figure, a hyperbolic cosine function u/u max = cosh −2 (y/b 1/2 ) that is used to describe the velocity profiles of synthetic jets (Zhang and Wang, 2007; Xu, Feng, and Wang, 2013; Xu et al., 2018) is given for comparison. The normalized velocity profiles agree well with the blue curve when y/b 1/2 ≤ 0, while velocity when y/b 1/2 > 0 is higher than the blue curve.

This asymmetric axial velocity profile may be due to two possible reasons. First, the temperature of the expelled airflow is higher than that of the outside environment, which can cause the airflow to be driven by buoyancy (Bourouiba, Dehandschoewercker, and Bush, 2014) . In our opinion, this is the primary reason. Second, the number of samples is still not high enough to obtain converged results.

The cough jet interacts with the surrounding air via flow entrainment, which can be viewed as a combination of small-scale nibbling plus a large-scale engulfment and induced inflow (Philip and Marusic, 2012) . The entrainment of a cough is qualitatively stronger than that of a continuous turbulent jet or synthetic jet because of the variation in the mouth position and flow direction. 

where we assume that the cough jet is axisymmetric and |r| is the absolute value of the radial s is presented in Fig. 8 (b) . It is obvious that the flow rate Q increases with the axial position.

The entrainment can be estimated from the axial derivative of Q as dQ/dx, and the increase in Q implies that the jet can entrain the ambient air. However, it is difficult to quantize the entrainment effect of coughing in the limited measurement domain of the data. 

As introduced in Sec. II D, the velocity of saliva droplets is estimated using PSTV. There are 12 cases overall. The onset of coughing is visually determined as when the droplets just begin to enter the measurement domain. Because the domain is as small as approximately 1.4 cm × 2.8 cm, the velocity of the droplets is averaged over the domain and cases, and the result is presented in Fig.   9 (a). At the beginning (t = 0 s), the velocity of the droplets is approximately 9.0 m/s; the velocity has reduced to 6.2 m/s at t = 0.005 s. After t = 0.005 s, the velocity of the droplets presents large fluctuations around the mean value of 6.2 m/s. This large deviation may be caused by an insufficient number of particles, as shown in Fig. 9 (b) . Because the camera used cannot resolve smaller droplets, the maximum number of detected droplets in the domain is approximately 15 and decreases to approximately 3 droplets after t = 0.01 s.

We also consider the joint probability density function (PDF) of the droplet velocity and diameter, as shown in Fig. 10 . This joint PDF is counted over t = 0 s to t = 0.02 s for all cases. 

We consider a saliva droplet as a pure water sphere with diameter D. According to Newton's second law of motion, the evolution of the saliva droplet's velocity can be calculated as:

where m d and u d are the mass and velocity vectors of the droplet, respectively. The mass m d is estimated as 1/6πρ d D 3 , where ρ d is the density of the droplet. F g denotes the gravity and F a denotes the air drag force. In the CFD analysis of Zhu, Kato, and Yang (2006) , the pressure force is also considered. However, this term is ignored in Eq. 3 because an accurate pressure field is difficult to obtain in our experiments. The forces F g and F a are given as:

Here, g is the gravitational acceleration and A d is the windward area, given as π( D 2 ) 2 . The parameter ρ a is the density of the air. White and Corfield (2006) offered the following curve-fit function for the drag coefficient C a .

The Reynolds number of the droplet is computed by

where µ a is the dynamic viscosity of the air. If the location of the droplet is in the region of the cough airflow, the velocity u a is given as Eq. 1. Otherwise, the velocity u a is set to zero for still ambient air. Eqs. 4, 5, 6 and 1 are substituted into Eq. 3 to calculate the velocity and trajectory of the droplets.

When a respiratory droplet is expelled into the air, the physical processes of mass transfer and heat transfer are simultaneously generated at the droplet surface (Kukkonen, Vesala, and Kulmala, 1989; Xie et al., 2007) . Mass reduction due to evaporation has a significant influence on the velocity and trajectory of small droplets. The rate of decrease in the diameter D of a spherical drop in air due to evaporation is expressed as (Kukkonen, Vesala, and Kulmala, 1989; Holterman, 2003) :

Here, M L is the molecular weight of vapor, which is given as 0.018 kg/mole; D ∞ is the binary diffusion coefficient far from the droplet; P t denotes the atmospheric pressure of air (101 kPa in this work); R is the universal gas constant (R = 8.3144 J mol −1 K −1 ); T ∞ is the ambient air temperature far from the droplet; RH is the relative humidity of the ambient air; T w is the wet-bulb temperature and p sat denotes the saturated vapor pressure. The Schmidt number Sc is a dimensionless quantity relating the momentum diffusivity (kinematic viscosity) to the mass diffusivity and is calculated by: The estimation of D ∞ , T w and p sat and the validation of this model are given in detail in Sec. A of the appendix. In the present work, the temperature and relative humidity of the cough airflow are T cough = 34 • C and RH cough = 85%, and these values for ambient air are T air = 23 • C and RH air = 50% (Bourouiba, Dehandschoewercker, and Bush, 2014) . The height of mouth from the ground is set to 1.8 m. The calculation is performed until the diameter is less than 5 µm. The temperature variation of the droplet is neglected in this work. Figure 11 displays the trajectories of droplets at the five different diameters of D = 30 µm, 80 µm, 200 µm, 400 µm, 800 µm. The simulated results considering the cough airflow and evaporation are shown in Fig. 11 (b) . For comparison, only gravity and the drag force are considered in Fig. 11 (a) . The initial horizontal velocity of the droplets is 6 m/s. It can be seen from Fig. 11 (a) that the horizontal velocity rapidly decreases to zero for the small droplet due to its relatively high viscosity. All the droplets fall to the ground, and the distance from the cougher at which this occurs x increases with increasing diameter. However, considering the real physical process, there are three different types of trajectories, as shown in Fig. 11 (b) . First, the trajectory for a droplet of D = 30 µm stops in the region of the cough airflow, which is denoted region A. The diameter of this droplet deceases to 5 µm due to evaporation. Second, the trajectory for a droplet of D = 80 µm stops in the region of ambient air, which is denoted region B. Third, droplets with larger diameters of D = 200 µm, 400 µm and 800 µm fall to the ground before evaporating to droplet nuclei. Moreover, the droplets in Fig. 11 (b) travel longer distances than those in Fig. 11 (a) .

Taking D = 200 µm as an example, the cough airflow can increase the distance from 0.25 m to 0.7 m.

The evaporation time and falling time as a function of the droplet diameter and initial velocity are presented in Fig. 12 (a) . There are two dashed lines in this figure: the black line represents the dividing line between droplets that fall to the ground and those that evaporate to droplet nuclei, and the red line represents the dividing line between the droplets that evaporate in region A and those that evaporate in region B. With our parameter settings, droplets larger than the critical droplet size D ≈ 100 µm are deposited on the ground before evaporating to a small droplet of D = 5 µm.

This critical droplet size is consistent with the results given by Wells (1955) and Xie et al. (2007) .

The second critical size indicated by the red dashed line is approximately 50 µm, where droplets smaller than this size evaporate to droplet nuclei in region A (cough airflow). The droplets with diameters of 50 ∼ 100 µm evaporate in region B (ambient air). These droplet nuclei are suspended in the air and travel with the movement of the air. In particular, the droplets in region A are easier for people of the same height to inhale; therefore, these droplets pose a higher probability of infection. Moreover, the initial velocity of the droplets u 0 has a negligible influence on the time.

The horizontal distance covered as a droplet evaporates to a small droplet or falls to the ground is displayed in Fig. 12 (b) . The contour line of x = 2 m is indicated by a black arrow. The contour map above the black dashed line represents the distance at which the droplet falls to the ground, and the contour map lower the back dashed line represents the distance traveled before evaporation. The practical distance of the droplets with D ≤ 100 µm is hard to predict because of the long-range airborne transmission. In addition to the small droplets, the droplets with D ≥ 500 µm and u 0 ≥ 5 m/s can travel further than 2 m, as shown in Fig. 12 Distance traveled from the cougher (to evaporate to a small droplet with D = 5 µm or to fall to the ground) as a function of the droplet diameter and initial velocity. The distinction between the droplets that fall to the ground and those that evaporate to droplet nuclei is represented by a black dashed line, and the distinction between the droplets that evaporate in region A and those that evaporate in region B is denoted by a red dashed line. winter-outdoor conditions, droplets with initial diameters larger than 50 µm evaporate slowly and deposit on the ground before drying. Additionally, the settling time of droplets with diameters of 50 ∼ 100 µm is much longer than those of the other three cases. If we consider the effect of condensation in winter, more large droplets are slowly settling to the ground in these conditions. This finding indicates that low temperature and high relative humidity may significantly increase the possibility of large droplet transmission and contact transmission, which may be a potential reason for a second pandemic wave in the autumn and winter seasons (Dbouk and Drikakis, 2020c). 

In the present work, the motion of saliva droplets produced by coughing was investigated through experiments and theoretical analysis.

Three different types of experiments, including flow visualization, PIV and PSTV, were performed to experimentally investigate the transport characteristics of coughing. For the flow visualization and PIV, image sequences of cigarette smoke expelled by healthy adult male volunteers were recorded using a high-speed camera. For the PSTV, large droplets were recorded using a shadow imaging technique, and the velocity and diameter of these droplets were extracted from the images by tracking the motion of the droplets. Our key findings from the experiments are as follows. First, the visualization shows that the cough jet interacts with the ambient air via flow entrainment, which leads to increasing size and decreasing velocity with increasing distance from the cougher (Bourouiba, Dehandschoewercker, and Bush, 2014) . Further quantitative analysis indicates that the convection velocity of the cough airflow presents the relationship t −0.7 with time;

hence, the distance from the cougher increases as t 0.3 in the range of our measurement domain.

The dependence of the distance s on time t is different from the relationship s ∼ t 0.5 that was obtained by considering a cough as a continuous jet by Bourouiba, Dehandschoewercker, and Bush (2014) . The width of the cough airflow evolves linearly as the distance increases. Second, the normalized mean velocity profiles agrees well with that of a synthetic jet in the region y/b 1/2 ≤ 0 and is higher than that of a synthetic jet in the region y/b 1/2 > 0, mainly due to the buoyancy driven by the relatively high temperature of a cough. The maximum airflow can reach 15.0 m/s. Third, the measured minimum diameter is approximately 250 µm due to the limitations of our experimental hardware. The mean velocity of the droplets is approximately 6.0 m/s, and smaller droplets are more susceptible to cough airflow.

With these experimental results, a physical model considering the evaporation effect was built to predict the movement of droplets under different environmental conditions (temperature and relative humidity). Our findings indicate that there are two critical sizes for the expelled droplets.

The first critical droplet size, where droplets larger than this size fall to the ground or otherwise evaporate to droplet nuclei, is approximately 100 µm. The second critical droplet size, which distinguishes droplet evaporation in the region of cough airflow from that in the region of ambient air, is approximately 50 µm. Droplets smaller than 50 µm are easier for people of the same height to inhale, resulting in a higher probability of infection. Additionally, both the small droplets (initial diameter D ≤ 100 µm) evaporating to droplet nuclei and the large droplets with D ≥ 500 µm and initial velocity u 0 ≥ 5 m/s can travel more than the social distance of 2 m. With this model, we also investigated the effects of weather conditions on the evaporation of cough droplets. Winter conditions, with low temperature and high relative humidity, can result in more droplets settling to the ground, which may be a possible driver of a second pandemic wave in the autumn and winter seasons.

Our study visualizes and analyzes the cough process using techniques of experimental fluid mechanics, and the theoretical results reinforce the importance of maintaining social distance and wearing masks to stem the spread of viruses. We should take precautions against a second COVID-19 wave due to the winter conditions of low temperature and high relative humidity. A potential future study is to build a relationship between pandemic evolution and the motion of respiratory droplets to accurately predict the infection rate and range for the population. 

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

A schematic representation of droplet evaporation is shown in Fig. 14 . In this figure, the open and solid circles represent air and vapor, respectively. The ambient air temperature and relative humidity are denoted by T ∞ and RH, respectively. The molecules in the droplet are bound to their neighbors by intermolecular forces; however, the liquid molecules at the interface are more weakly bounded to their neighbors than those inside the liquid due to having fewer adjacent molecules (Holterman, 2003) . Therefore, some molecules with relatively high kinetic energy escape from the droplet into the air, and a very thin evaporating film with a saturated vapor is generated at the interface between the droplet and air. Because escaped molecules can absorb heat, the droplet's temperature T w is lower than the temperature of the ambient air T ∞ . In this work, we ignore the variation in the temperature of the droplet and assume that the final droplet temperature is the wet-bulb temperature, which can be described by a linear relation with T ∞ and quadratic relation with relative humidity RH as (Holterman, 2003) : 

At 100% relative humidity, the wet-bulb temperature T w is equal to the air temperature T ∞ (drybulb temperature).

If the air is saturated with the water vapor, the partial vapor pressure is defined as the saturated vapor pressure p sat . p sat is strongly dependent on the temperature, and an empirical equation is given as (Holterman, 2003) :

p sat = 610.7 × 10 7.5T /(T +237.3) .

The above equation is valid for temperatures of 0 • C up to approximately 100 • C.

The dependence of the diffusion coefficient D ∞ of water in air on the temperature is given by Holterman (2003) as:

D ∞ = 21.2 × 10 −6 (1 + 0.0071T ∞ ).

The validity of the present model is first examined by comparing the diameter evolution of motionless droplets with the experimental results given by Chaudhuri et al. (2020) (Fig. 15 (a) ).

A motionless droplet is allowed to evaporate under ambient conditions at T ∞ = 30 • C and RH = 50%. The diameter variations as a function of time for two different initial diameters D 0 are shown in Fig. 15 (a) . The solid curves estimated by the present model qualitatively agree with the experimental results. We also compare the evaporation-falling time of the present model with the result extracted from the paper by Xie et al. (2007) . The droplet is released at a height of 2 m, and the ambient temperature T ∞ and relative humidity RH are 18 • C and 0%, respectively. As shown in Fig. 15 (b) , the curve estimated by the present model agrees well with the result of Xie et al. (2007) . Droplets smaller than 125 µm totally evaporate before reaching the ground. With these comparisons, we can conclude that the model used in this work can describe the evaporation and motion of a droplet. Xie et al. (2007) . The ambient temperature T ∞ is set to 18 • C, and the relative humidity RH is set to 0%. The droplet is released at a height of 2 m.

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