A review of the underlying flow physics with some sample calculations.
Control measures, incubation, and the spread of a viral pandemic like COVID-19 form an interconnected set of many social, biological, and physico-chemical phenomena. The study of these processes spans numerous areas of social and physical science – all directed towards one unified goal of mitigating the toll it takes on the human life and economy. These physico-chemico-biological phenomena occur at a variety of length scales starting at the molecular level, followed by the length scales relevant to the common human experience, and finally culminating in the emergent effects at the societal level. At the molecular level, the most vital area of research for a viral disease is the study of the structure, classification, evolution of the virus, and its interaction with the physiology and immunity of the host organism. The cross-disciplinary merger of the areas like Tourism, Public Policy and Public Health is necessary to understand and mitigate the effects of a pandemic at the societal level. Somewhere between these extremes lies the range of scales that employs the host organism as a vehicle for the transmission of virus. In the case of respiratory viruses like SARS-CoV-2, this is where fluid dynamics plays an important role.
A respiratory virus like SARS-CoV-2 is primarily transmitted in three different ways: droplet transmission – the ejecta released into the ambient air by a typical expiratory event like coughing or sneezing consists of large droplets that have sufficient momentum so as to reach the recipient’s nasal/oral opening, contact transmission – physical contact with the contaminated surfaces subsequently followed by the transfer to the recipient’s oral/nasal opening, and airborne transmission – aerosolized viral particles that remain suspended in the air for longer periods of time by virtue of their small size, hence prone to inhalation. All these are mediated by complex flow phenomena and their study is important to understand the involved transmission mechanisms. Subsequently, a grasp of the underlying flow physics can thus lead to improved control measures and mitigation.
How are droplets formed?
Several fluid dynamic studies have attempted to explain the flow physics of viral transmission from the droplet generation stage to the latter stages of the aerosolization of the expiratory ejecta. In the droplet generation stage, the sites of interest are the upper parts of the respiratory tract where the fluid lining is affected by the air flow coming from the lungs. A study by Moriarty & Grotberg [1] describes how the airflow perturbs the mucus lining, destabilizes it by the action of shear and eventually fragments the mucus lining into droplets of varying radii. The hydrodynamic instabilities that play a role in destabilizing the thin mucus lining are Rayleigh-Taylor and Rayleigh-Plateau. Rayleigh-Taylor instability explains how the interface between the mucus and the air transforms while as the Rayleigh-Plateau instability explains how the blobs of mucus break up into smaller packets and ultimately detach from the respiratory tract walls. Other mechanisms responsible for droplet formation are related to the opening of the respiratory tract passages. For example, the opening and closing of vocal folds in the larynx during talking and coughing ruptures the mucus meniscus resulting in the generation of micrometer sized droplets. Movements of tongue and lips can also aid the process of droplet generation via this mechanism. The properties of mucus (which is a non-Newtonian fluid do not resemble to those of fluids like air or water which are dubbed as Newtonian fluids) and the moving and morphing walls of the respiratory tract make it difficult to predict the involved mechanisms to a greater degree of accuracy.
Droplets: how fast, how big, and how many?
Once the droplets detach from the walls of the upper regions of respiratory tract, airflow from the lungs convects them outwards. The fate of the virus and its eventual transmission depends on the velocity, size, and number density of these droplets. Several studies report the measurement of these factors. A good review of these studies was recently presented by Mittal et al. [2]. The results show that sneezing produces more droplets with higher velocities as compared to coughing. The droplet velocities and number density in the case of sneezing is almost two times and 10 – 100 times higher than in the case of coughing respectively. However, it is also reported that normal breathing releases about 50 particles a second which amounts to more bioaerosols than the intermittent events like sneezing and coughing over course of a day. This observation is important while considering the measures to mitigate transmission.
Droplet evolution and modes of transmission
Following from the discussion above, from a fluid dynamic perspective, an expiratory event or even normal breathing forces out a stream of droplets laden with viral particles, from the oral as well as the nasal passage of the affected person. The expiratory ejecta can be modeled as a turbulent jet that goes through several stages of spatial and temporal evolution. This jet is a two-phase buoyant jet – which means that two fluids of different density are released in a single stream. Buoyancy implies that the density difference between the jet and the surroundings plays a significant role in its evolution. The evolution of the droplets determines the mode of transmission that the virus adopts. Ligaments and sheets of saliva and mucus that are expelled out through the nasal and oral openings may eventually develop into droplets small enough to detach from the mouth/nasal walls and large enough to fall due to inertia under the action of gravity. These droplets may contaminate the nearby surfaces and thus contribute to the mode of contact transmission. The flow of these droplets is characterized by a large value of the quantity called Weber number, which measures the relative importance of fluid’s inertia compared to its surface tension.
In the case of droplet transmission, the distance over which large droplets can travel is of vital importance. Some earlier works by Wells (1934) (reviewed in [2]) indicate that the 6 feet social distancing guidelines might be sufficient to avoid viral transmission via the droplet transmission route. However, these guidelines are valid only under specific circumstances and some caveats must be kept in consideration in their implementation. It has been observed that the droplets ejected in normal breathing and coughing may lose their momentum within the 6 feet radius; however, a violent expiratory event like sneezing can produce droplets which can travel up to 20 feet or more. [3, 4] The strong recirculation zones (vortices) in the air induced by the ventilation systems and cooling devices like fans, air conditioning ducts etc. may carry the smaller droplets in a convection pattern and suspend the droplets for longer periods of time. Due to the complex nature of the flow by virtue of these devices, a location far away from the source of droplets may in fact be more prone to higher density of the droplets. This calls for extra precautions and makes it mandatory to wear a face mask while working indoors.
So far, the discussion mainly focused on the contact and droplet modes of transmission. These modes do not consider the microscopic droplet nuclei that can potentially linger in ambient air for significantly larger periods of time. To understand the dynamics of airborne transmission, the ambient temperature and atmospheric conditions play a crucial role. Factors like vapor pressure of the droplet and the surrounding air, humidity, relative velocity between the droplet and the surrounding air, and droplet surface-to-ambient temperature difference affect the evolution of droplets. Facilitated by the synergy of environmental factors mentioned above, droplet nuclei are produced when the droplets lose most of their moisture by evaporation. Studies have found it difficult to parameterize the complex effect of environmental conditions on the droplet size and evaporation. The droplet nuclei consist of the residual solid matter with the attached SARS-CoV-2 infection that remains after the complete evaporation of the moisture. The relative magnitude of the falling time compared to the evaporation timescale decides the route the virus adopts for transmission. If a droplet evaporates before it settles due to the action of gravity, all that remains are these nuclei which are convected by the surrounding air and can remain suspended for longer periods of time. On the other hand, if the droplet size is large, the falling time is small compared to the evaporation timescale. An estimate of the falling time of these droplets can be made by calculating the terminal velocity of the falling droplets, assuming that these droplets are subjected to gravitational and drag forces of comparable magnitudes. Of course, this is true for droplets of small radius \(R < 100 \ \mu m\); large droplets are heavy and are acted upon by higher gravitational forces compared to the drag [5]. The terminal velocity of a falling droplet is given by the balance of gravitational force and the drag acting on the droplet,
\[ F_g = Mg \ \ \ \ \ \ \ \ \ \ (Gravitational \ force \ acting \ on \ the \ droplet) \]
\[ F_d = 6\pi \eta V_{fall} R \ \ \ \ \ \ \ \ \ \ (Drag \ force \ acting \ on \ the \ droplet \ (Stokes' \ Law)) \]
When \( F_g = F_d \), the droplet attains its maximum velocity of fall, i.e. \(V_{fall} = V_{t} \), such that,
\[ \implies V_{t} = \frac{Mg}{6\pi \eta R} \ \ \ \ \ \ \ \ \ \ \]
where, \( \eta = 1.8 \times 10^{-5} \ kg/ms \) is the dynamic viscosity of air, \(V_{t}\) is the terminal velocity of fall, \( R \) is the radius of the droplet, \( M = \frac{4}{3}\pi R^3 \rho \) is the droplet mass with \( \rho = 997 \ kg/m^3\), \(g = 9.8 \ m/s^2\) is the standard acceleration due to gravity, and \(\pi = 22/7\) is a universal constant.
Suppose the droplets are released into the ambient air at a height of \(H=2 \ meters \) (which is a rough estimate for height of a human being), the falling time can be computed as,
\[t_{fall} = \frac{H}{V_t}\]
 |
| Figure 1: Comparison of falling time and evaporation time scale w.r.t droplet size, and critical droplet size for droplet-to-airborne transition. Click on the image for higher resolution. |
Table 1: Falling time and evaporation time scale with respect to droplet size.
\( R \ (\mu m)\)
|
\(M \ (kg)\)
|
\(t_{fall} \ (s)\)
|
\(t_{evap} \ (s)\)
|
|
50
|
5.22E-10
|
0.301562
|
6.632141
|
0.4
|
100
|
4.18E-09
|
1.206247
|
1.658035
|
1.7
|
200
|
3.34E-08
|
4.824988
|
0.414509
|
6.6
|
500
|
5.22E-07
|
30.15617
|
0.066321
|
41
|
1000
|
4.18E-06
|
120.6247
|
0.01658
|
165
|
2000
|
3.34E-05
|
482.4988
|
0.004145
|
660
|
From Table 1, a comparison of the falling time and evaporation time scales shows that the droplets with \(R < 100 \ \mu m\) have a larger falling time than the evaporation time scale. Droplets of such size will remain suspended in the air, contributing to the airborne mode of transmission of the virus. The droplets with \(R > 100 \ \mu m\), on the other hand, have larger evaporation time scale as compared to the falling time, which identifies such droplets responsible for droplet transmission mode. The transition from droplet-to-airborne routes is more clearly visible in Figure 1.
Droplet inhalation and susceptibility
The final stage in transmission of SARS-CoV-2 is the inhalation of droplets or the droplet nuclei by the host. Deposition of the virus laden droplets does not always result in an infection because of the protective properties of the mucus layer. Larger droplets get deposited in the upper parts of the respiratory tract where they are deactivated by the defensive layer of mucus. Smaller droplets and nuclei, despite possessing low impact speeds, can penetrate the mucus layer deeper into the respiratory system and thus can make the host more susceptible and prone to the infection.
Control measures and mitigation
Several control measures are available to disable the transmission of the virus between the infected individual and the recipient. These measures are applicable at different levels of transmission and are based on different approaches to tackle the problem. Modification of the physical properties of the mucus lining like its physical stability and surface tension enhancement decreases the rate of droplet generation, thereby reducing infection rates. Fogging indoor spaces with disinfectants is also an efficient method to counter the airborne route of transmission. The fogging machines release a mist of disinfectants with a droplet size less than \( 10 \ \mu m\), such that the droplets can stay suspended for extended periods of time to provide safety against the droplet and airborne modes of transmission. Hand-washing is very important when we are dealing with contact transmission mode, given that hands are the most common body parts prone to touching the contaminated surfaces as well as our nasal/oral openings. The hydrophobic end of the soap molecules binds with the lipid bilayer of the virus and emulsifies the lipid content present in it. The emulsion is then convected and washed away by the bulk flow of water.
The use of face masks to prevent the spread of COVID-19 has been a topic of great interest and controversy. Considering the scale at which the virus has affected us globally, the face masks are expected to remain an important part of our post-COVID life. The understanding of the underlying flow physics, thus, becomes important while trying to make decisions about their implementation as an effective control measure now and during future outbreaks. Face masks serve a dual purpose of protecting the wearer by filtering out the aerosolized viral particles and preventing the infection by trapping the virus laden droplets exhaled by an infected person. The former is termed as inward protection while as the latter is called outward protection. The effectiveness of a face mask depends on a variety of factors including the filtering properties of the mask material, fit of the mask and the perimeter leaks. During inhalation, a low pressure is created in the region covered by the mask. This decrease in pressure pulls the mask in momentarily during the inhalation and seals the perimeter. During exhalation, a region of higher pressure is developed in the region covered by the mask. This higher pressure pushes against the mask resulting in the perimeter leakage. The perimeter leakage jets disperse the respiratory ejecta in the transverse directions with large expulsion speeds, and therefore diminish the effectiveness of the mask in outward protection. Hopefully, the fluid-physics informed design changes may help to mitigate these shortcomings of face masks in the future.
Closing remarks
A comprehensive description of fluid dynamics of the transmission of respiratory viruses like SARS-CoV-2 is discussed. Some fundamental aspects of the droplet generation, their number density, impact velocity, size and their evolution into aerosols has been presented. In addition to educating the reader on important physical aspects of virus transmission, the goal is also to highlight the importance of fluid dynamics in solving the real-life problems. Whether we realize or not, Fluid dynamics has, and is always going to be with us, as they say – in health and in sickness!
Cite as
Bader, Shujaut H., “COVID-19 transmission: A fluid dynamic perspective.” Backscatter, June 19, 2020, https://backscatterblog.blogspot.com/2020/06/covid-19-transmission-fluid-dynamic.html
References
- Moriarty, J. A. & Grotberg, J. B. (1999). Flow-induced instabilities of a mucusserous bilayer. J. Fluid Mech. 397, 1–22.
- Mittal, R., Ni, R., & Seo, J. (2020). The flow physics of COVID-19. Journal of Fluid Mechanics, 894, F2. doi:10.1017/jfm.2020.330
- Bourouiba, L., Dehandschoewercker, E. & Bush, J. W. M. (2014). Violent expiratory events: on coughing and sneezing. J. Fluid Mech. 745, 537–563.
- Xie, X., Li, Y., Chwang, A. T. Y., Ho, P. L. & Seto, W. H. (2007). How far droplets can move in indoor environments – revisiting the Wells evaporation-falling curve. Indoor Air 17 (3), 211–225.
- Luis A. Anchordoqui, Eugene M. Chudnovsky (2020). A physicist view of the airborne infection. arxiv.org/abs/2003.13689
- W. F. Wells, On air-borne infection study II: droplets and droplet nuclei, Am. J. Hyg. 20, 611 (1934).
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