On the airborne aspect of COVID-19 coronovirus
aa r X i v : . [ phy s i c s . pop - ph ] A p r On the airborne aspect of COVID-19coronovirus N avinder S ingh , ∗ and M anpreet kaur † [email protected]; [email protected] April 22, 2020
Abstract
It is a widely accepted view that COVID 19 is either transmitted via surface contamination or viaclose contact of an un-infected person with an infected person. Surface contamination usually happenswhen infected water droplets from exhalation/sneeze/cough of COVID sick person settle on nearby surfaces.To curb this, social distancing and good hand hygiene advise is advocated by World health Organization(WHO). We argue that COVID 19 coronovirus can also be airborne in a puff cloud loaded with infecteddroplets generated by COVID sick person. An elementary calculation shows that a µ m respiratory infecteddroplet can remain suspended for about 9.0 minutes and a µ m droplet can remain suspended for about anhour! And social distancing advise of 3 feet by WHO and 6 feet by CDC (Centers for Disease Control andPrevention) may not be sufficient in some circumstances as discussed in the text. COVID-19 is a pandemic. It has infectedover 2.2 million people worldwide, and over1.5 lakh people have died due to it, as of 20thApril 2020. Most of the information related toCOVID-19 can be found at WHO website[1],and at CDC website[2]. In the next section wepresent a calculation with the aim to calculatetimescale over which the virus in infected wa-ter droplets exhaled by a COVID sick personstays in air. ∗ Theoretical Physics Division, Physical Research Lab-oratory, Ahmedabad, India. Cell Phone: +919662680605. † Department of Pediatric and preventive dentistry,Ahmedabad Dental College, Ranchodpura, Ahmedabad.India. Cell Phone: +919409290203
I. A N ewtonian -S tokescalculation (N eglecting B rownian motion ) Let h be the typical height of a human be-ing ( ∼ meter ). Human exhalation gener-ates mucosalivary droplets of varied size from0.5 µ m to 12 µ m [3]. Consider the trajectoryof a droplet. Time taken to fall from height h using Newtonian mechanics neglecting air drag is given by p ( h / g ) , where g is accelerationdue to gravity. For h = meters this timeis roughly 0.6 sec (less than a second). In thiscase we say that droplets are not airborne and1irborne aspect of COVID-19settle to nearby surfaces in a time less than asecond.But this picture is radically modified whenair drag is taken into account. The equation ofmotion including drag force ( f d ) for the verti-cal downward direction is given by m dv ( t ) dt = mg − f d . (1)Here f d Stokes’ drag force given by 6 πη rv ( t ) . η is the viscosity of air, r is the radius of waterdroplet, and v ( t ) is its downward velocity attime t . For the time being consider that initial( t =
0) downward velocity is zero. The solu-tion of the above equation immediately gives mg h f d Figure 1:
Various forces acting on the water droplet. v ( t ) = g τ ( − e − t / τ ) , (2)where τ = m πη r . And the instantaneous height(measured from the top end) v ( t ) = dh ( t ) dt isgiven by h ( t ) = g τ ( t + τ ( e − t / τ − )) . (3)Let time taken to fall down be t and h ( t ) = h the typical height of a person. Let us take radius of a typical water dropletto be 5 micro − meters . Its mass will roughlybe 5.25 × − kg . Let us say ambient temper-ature is 25 degree centigrade. Air viscosity atthis temperature is η = × Kg / ( m − sec ) . Plugging these numbers we get τ = m πη r ≃ × − or 0.3 milli − secs . For theshortest (drag free) time scale (0.6 seconds) theexponentials e − t / τ in the expressions of v ( t ) and h ( t ) are extremely small. Thus v ( t ) = v terminal ≃ g τ ≃ mm / sec (from equation(2)) which is very low terminal speed! Andthat the terminal speed is reached even withinmillisecond time scale (as τ is in sub-milli-second timescale).The total time taken t to fall down fromheight h is t ≃ hg τ ≃ µ m can take 9 minutes to settle down! Smallerdroplets can take even more time! This timescale is inversely proportional to the square ofthe radius of droplet: t = (cid:18) η hg ρ r (cid:19) , (4)where ρ is the density of water droplet. 2 µ mdroplet can remain suspended for about an hour! If we consider initial downward componentof velocity (as exhalation imparts some initialspeed to droplets i ) then the solution of theequation of motion leads to i Nasal breathing typically leads initial speeds of about1.4 m / sec . v ( t ) = v ( ) e − t / τ + g τ ( − e − t / τ ) . (5)Whatever be the initial speed it is going todecay in millisecond timescales as τ is verysmall in sub milli-second time scales. Alsoany transverse component of the water dropletwill be dissipated away in such small timescales. It does not form a parabolic projectiletrajectory as commonly visualized. II. I mportant physical situationrelated to social distancing
As an implication of the above calculation,consider a situation like this: Consider thatpeople are standing in a queue in front of amilk parlor or in front of an ATM for cashwithdrawal waiting for their turn. Also con-sider that all are following social distancingnorm and they stand 6 feet away from eachother. Let us suppose that a person is infectedbut he/she is asymptomatic. ii The person isstanding at his/her place say for more thanfive minutes. Also consider a nil wind con-dition. This person will create an invisiblecloud loaded with respiratory droplets of var-ied size. These droplets are all infected! Afterfive minutes this person moves ahead to takehis/her turn, and a person behind him/hertakes his/her position. It will take coupleof seconds to move 6 feet ahead. But the ii It has been seen that a person can be infectious evenif she/he has no symptoms. cloud loaded with infectious droplets is go-ing to stay there! (a two micrometer dropletwill remain suspended for about an hour, andsmaller droplets take more time to settle). Sothis well person will enter this infectious cloudand can get infection, even if social distancingnorms are obeyed!Now, if the wind is flowing transverse to thepeople’s queue, it will take away that infec-tious cloud, and it will disperse in the ambi-ent air. At sufficiently large distances concen-tration of infectious droplet nuclei will be ex-tremely low, and it will be no more dangerous.But if wind is flowing along the queue, thenwind can transfer those infectious droplets tounprotected well persons.
III. L ife - time of the waterdroplets In the above discussion it is assumed thatwater droplets preserve their size. It is nottrue. They evaporate in short time depend-ing upon the size of the droplet. However, arecent research by Lydia Bourouiba and col-laborators show that life-times are radicallymodified as the droplets are in a very specialenvironment of respiratory puff cloud whichcontains lots of water vapors[4]. A thoroughinvestigation of droplet life times and trajec-tories taking into account in this special en-vironment is much needed (in equation (1)mass will then be a time depended param-eter). But, when an infected droplet evapo-3irborne aspect of COVID-19rates, it ends up in an infected droplet nucleior an infected aerosol. These are sub-micronsized and take even more time to settle. Al-though the above mechanical-hydrodynamicalapproach cannot be applied, as aerosols dy-namics will be stochastic and Brownian mo-tion will be important in its transfer from oneplace to another, but it will take much moretime to settle to ground! In addition, a tur-bulent wind can take it away. This points to-wards airborne nature of the disease. A dis-cussion about the Brownian motion is givenin the next section.
IV. C onsideration of the B rownian motion In equation (1) we considered the system-atic component (drag component f d ) of theforce due to bombardment of droplet by airmolecules and we neglected a random compo-nent of this force which is also due to bom-bardment of droplet by air molecules iii . Thusthe total force has two components: F ( t ) = f d + ξ ( t ) . ξ ( t ) is the random force whose av-erage is zero h ξ ( t ) i = h ξ ( t ) ξ ( t ′ ) i = Γ δ ( t − t ′ ) [5].This random force can be neglected for iii Just imagine slow motion movie in which the dropletis being bombarded from all sides by air molecules. Atone instant of time more molecules will bombard fromleft and the droplet will jerk towards left, and at the otherinstant of time this imbalance topples to some other direc-tion. This leads to a random force. bigger droplets. The criterion is based onthe value of Reynolds number. The fluctuat-ing force ξ ( t ) and the drag force f d are in-timately related to each other through fluc-tuation dissipation theorem (as both origi-nate from the common origin via molecularbombardment)[5]. Thus we need to comparedrag force with another important force in theproblem that is called the inertial force. Theinertial force is the force required to changedroplets momentum over a distance of its size f inertial ∼ mv ( v / r ) (average rate of change ofdroplet’s momentum over its size). The otherforce is the viscous force f drag = η rv (we ne-glect prefactors, as we are interested in orderof magnitudes). Their ratio is called Reynoldsnumber: Re = f inertial f drag = ρ rv η . (6)For 5 µ m droplet drifting down with speed3 mm / sec , Re ∼ ∼ nm )moving at this drift speed the Reynolds num-ber turns out to be of the order of 10 − . Thisis completely a fluctuation dissipation domi-nated regime, and the virus will perform aBrownian motion.Thus, the timescales obtained above byNewton-Stokes analysis are the minimum pos-sible timescales (these are the lower bounds).That is, the 5 µ m droplet will take more than9 minutes to settle. Detailed investigation us-ing Langevin or more sophisticated model isbeyond the scope of this investigation.4irborne aspect of COVID-19 V. A dvise to P ublic In view of the airborne nature of the disease tosome extent it is advised that when someoneis out of one’s house to get essential goods ata frequently visited place (like a grocery store)one should not only wear a good mask for oraland nasal protection, but a protective gear foreyes (eye guard or eye shield) is also highlyrecommended to cover-up the airborne aspectof the disease discussed in this work.
VI. S ummary