Drops in the wind: their dispersion and COVID-19 implications
DDrops in the wind: their dispersion and COVID-19 implications
Mario Sandoval a) and Omar Vergara Department of Physics, Universidad Autonoma Metropolitana-Iztapalapa, Mexico City 09340,Mexico. (Dated: 17 February 2021)
Most of the works on the dispersion of droplets and their COVID-19 (Coronavirus disease) implicationsaddress droplets’ dynamics in quiescent environments. As most droplets in a common situation are immersedin external flows (such as ambient flows), we consider the effect of canonical flow profiles namely, shear flow,Poiseuille flow, and unsteady shear flow on the transport of spherical droplets of radius ranging from 5 µ mto 100 µ m, which are characteristic lengths in human talking, coughing or sneezing processes. The dynamicswe employ satisfies the Maxey-Riley (M-R) equation. An order-of-magnitude estimate allows us to solvethe M-R equation to leading order analytically, and to higher order (accounting for the Boussinesq-Bassetmemory term) numerically. Discarding evaporation, our results to leading order indicate that the maximumtravelled distance for small droplets (5 µm radius) under a shear/Poiseuille external flow with a maximumflow speed of 1 m/s may easily reach more than 250 meters, since those droplets remain in the air for around600 seconds. The maximum travelled distance was also calculated to leading and higher orders, and it isobserved that there is a small difference between the leading and higher order results, and that it dependson the strength of the flow. For example, this difference for droplets of radius 5 µm in a shear flow, and witha maximum wind speed of 5 m/s , is seen to be around 2 m . In general, higher order terms are observed toslightly enhance droplets’ dispersion and their flying time. I. INTRODUCTION
So far, COVID pandemic is still growing, as of 18:34pmCentral European Time (CET), February 6 2021, therehave been 104 , ,
439 corroborated cases of COVID-19,encompassing 2 , ,
488 deaths, reported to the WorldHealth Organization (WHO). It is believed that air-borne transmission is one of the main mechanisms forCOVID spreading and that potential sources of in-fected droplets are breathing, sneezing, coughing or sim-ply talking. Unfortunately, most of the reported liter-ature neglects the effect of ambient flows on dropletstransport. These flows are often present in daily activi-ties such a wind flows, ventilation generated currents inoffices, homes, malls, among other public places.Recent studies suggest that talking may be amongthe most dangerous mechanisms for generating infecteddroplets . According to Tan and Bourouiba , speechinduced plumes can travel 1 . m in 2 s or even 8 m , whichis a distance way longer than the 2 m social distancing.Abkarian and Stone using high-speed imaging showedthat pronouncing consonants (typical to many languages)such as ’Pa’,’Ba’, and ’Ka’ are potent aerolization mech-anisms. Abkarian et al using theory, experiments, andsimulations documented the flow generated after speak-ing and breathing, which is in fact the responsible fordroplets’ transport. Notice that the 2m social distanc-ing, only considered quiescent flows . This situationis not always satisfied in daily conditions, where wind isfrequently present either naturally or induced by air con-dition in buildings or even by simple motion of people .Further evidence of airborne transmission of COVID-19 a) Electronic mail: [email protected] disease possible enhanced by air currents are: A reportedinfection of 96 people out of 216 working in a eleventhfloor in a call center of South Korea ; a singing rehearsalin Washington, where 53 singers were infected even theywere located in a volleyball court but ventilation air cur-rents were present . A study precisely on the effective-ness of air ventilation and COVID implications suggeststhat only certain type of ventilation called ’displacementventilation’ properly designed to extract contaminatedhot air, could be the most effective air condition mecha-nism to reduce the risk of infection. Related works wheredroplets produced by breathing, sneezing, coughing, ortalking, and immersed in external flows are few. Someexamples are Cummins et al who studied micrometricspherical droplets in the presence of a source-sink flow,which simulated a scenario where droplets are producedand subjected to an extraction mechanism (air condi-tion). Incorporating evaporation, humidity, and an uni-form external wind, Dbouk and Drikakis presented aconjugated heat and flow transfer problem (occurring ina saliva droplet) and coupled to a CFD analysis. Theyproposed a transient Nusselt number and varied relativehumidity (RH), temperature, and speed of flow. Con-trary to past studies, they conclude that evaporationof droplets is enhanced at low RH and high tempera-tures. As an example, they results indicate that in acloud of droplets in an environment at RH=50%, temper-ature T < o C , and under an external flow of 4 km/h ,there will not be evaporation. Their simulation how-ever, only reached five seconds, hence the full disper-sion of droplets could not be reached. B. Blocken et al also performed a CFD analysis for people exhaling whilewalking or running, and emanating from them possibleinfected droplets. However, their simulation only con-sidered droplets of radius 20 µm and beyond. As it has a r X i v : . [ phy s i c s . f l u - dyn ] F e b yz v ,N = v e ,N < < U max u ( r , t ) h FIG. 1. Schematic of the problem studied. A person talk-ing either normally (exit speed of droplets around 1 m/s ) orstrongly (exit speed of droplets around 5 m/s ) , and underthe presence of an external flow u ( r ,t). recently been observed, the smallest the size of a droplet,the more dangerous it may be . Feng et al locatedtwo virtual humans 3 . m apart and let one human toeject droplets while coughing or sneezing. Using a com-putational particle fluid dynamics model that consideredevaporation, external wind and condensation, and evenBrownian motion, they found that at this distance, po-tentially infected droplets easily reach hair and face ofthe other human. They also performed a study on thefiltration efficiency of several masks. In conclusion, mostof the previous works suggest that the safe distancing isnot enough under wind conditions.In this work, we consider micrometrical sphericaldroplets emanating from a person while talking ei-ther normally (exit speed of droplets around 1 m/s ) orstrongly (exit speed of droplets around 5 m/s ) , andunder the presence of external flow currents, and calcu-late, following the Maxey-Riley (M-R) equation , theireffect on the maximum spreading of droplets of constantradius ranging from 5 µm to 100 µm . The wind currentsprofiles are modeled as a shear flow, a Poiseuille flow,and an unsteady shear flow that considers the typicaltime-dependent situation of wind blowing and ceasing.The present study is organized as follows: Section II de-scribes the model, and the order-of-magnitude of eachterm in the M-R equation. Section III presents analyticalresults for the dispersion of micrometric droplets subjectto three external flow profiles namely, shear, Poiseuille,and unsteady shear flow. Here, the Boussinesq-Bassetmemory force is neglected. Section IV is intended tostudy, by performing numerical simulations, the effect ofthe Boussinesq-Basset memory force (and the same ex-ternal flows as in Sec. III) on the dispersion of droplets.Discussions and conclusions are offered in Sec. V. II. PHYSICAL MODEL
Let us analyze the motion of noninteracting sphericaldroplets of mass m , radius a , in an environment with air density ρ a and air viscosity µ a , under gravity g , and sub-ject to an external flow of the form u ( r , t ), with a charac-teristic speed U max , where r ( t ) = ( x, y, z ) is the droplet’sposition, here t represents time. In this study we will beconsidering three flows namely shear, Pouiseuille, andunsteady shear flow. Droplets immersed in these profileswill have a characteristic speed v ( t ) which decreases asthe droplets fall due to gravity. With the latter phys-ical quantities a Reynolds number ( Re ) can be definedas Re = ρ a ( U max − v ) a/µ a which satisfies Re < v ( t ) = ( v x , v y , v z ), followsthe Maxey-Riley equation m d v dt = ( m − m a ) g + m a D u Dt (cid:12)(cid:12)(cid:12)(cid:12) r − m a ddt (cid:18) v T + a ∇ u (cid:12)(cid:12) r (cid:19) − R T v T − R T a v T (0) √ πυ a t − R T a (cid:90) t d v T dτ dτ (cid:112) πυ a ( t − τ ) , (1)where m a is the mass of air displaced by the sphere, υ a = µ a /ρ a represents the kinematic viscosity, R T =6 πaµ a is the resistance coefficient; while ρ indicates thedroplet’s density. In addition, the following definitionsare included: v T = v − u | r − a ∇ u (cid:12)(cid:12) r (2) D u Dt = ∂ u ∂t + u · ∇ u , (3) d u dt = ∂ u ∂t + v · ∇ u . (4)The forces on the right hand side of Eq. (1) are re-spectively, the droplet’s weight; bouyancy force; forces inthe undisturbed flow due to local pressure gradients; theadded or virtual mass force; Stokes drag force; and thelast two terms represent the Boussinesq-Basset memoryforce. To find the order-of-magnitude of each term in theMaxey-Riley Eq. (1), one can introduce the followingdimensionless quantities r = r /a, v = v /U, t = ( U/a ) t, where the characteristic speed is defined as U = mg/R T , which is the terminal velocity of a falling droplet. Afterapplying those dimensionless variables to Eq. (1), onearrives at Re d v dt = − λ − λ ) k + λRe D u Dt (cid:12)(cid:12)(cid:12)(cid:12) r − λRe ddt (cid:18) v T + 115 ∇ u (cid:12)(cid:12)(cid:12) r (cid:19) − λ v T − λ √ Re √ π v T (0) √ t − λ √ Re √ π (cid:90) t d v T dτ dτ (cid:113)(cid:0) t − τ (cid:1) . (5) < < z ( m ) y ( m ) z ( m ) y ( m ) ( a ) ( b ) < < µm < a < µm µm < a < µm µm < a < µm µm < a < µm Shear
Poiseuille ( c ) y ( m ) Unsteady shear < < { U max , v } = 1 m/s { U max , v } = 5 m/s . s . s . s . s . s . s . s . s . s . s . s . s FIG. 2. A person talking and spreading micrometric droplets of radius a ranging from 5 µm to 100 µm . The spreading is underthe presence of an external flow u , and it is shown for three different times after the droplets were ejected from a cone shape ofvelocities modeling a person’s mouth namely, t = 0 . s, t = 0 . s , and t = 1 . s . (a), Drops’ distribution during a normal andstrong talk with exit speed of v = 1 m/s and v = 5 m/s , respectively; and under a shear flow with U max = { m/s, m/s } (seecolor code). (b) The same as in (a) but under the presence of a Poiseuille flow. (c) The same as in (a) but under the presenceof an unsteady shear flow. The safe distancing is indicated as a vertical dotted-black line. The light blue arrows represent theflow profile.TABLE I. Order-of-magnitude for different terms appearingin the M-R Eq. (5) as a function of the radius a . a ( µm ) Re λ √ Re λRe . × − . × − × −
10 0.0067 1 × − . × −
20 0.05 2 . × − . × −
30 0.18 5 . × − . × −
50 0.83 0.0011 0.001100 6.7 0.0032 0.0082 where the Reynolds number Re = ρ a U a/µ a has beenintroduced, as well as the ratio λ = m a /m = ρ a /ρ =1 . × − . Here g = g a/U . III. DYNAMICS WITHOUT BOUSSINESQ-BASSETMEMORY
Let us start analyzing the order of magnitude estimateof the terms in Eq. (1) based on typical droplets ejectedafter talking. According to recent studies , the ra-dius of those droplets range between 5 µm to 100 µm ,hence Table I shows their respective Reynolds numberand the order-of-magnitude involved in Eq. (5). Fromthis table and for droplets of radius a ∈ [5 µm, µm ],the limit Re → v T = − k , (6)which implies that the dynamics of droplets of this sizeis v x = 0; v y = u y , v z = − . (7) The latter result belongs to the so-called overdampedapproximation, where inertia does not play a role, andparticles immediately reach the external flow speed. Al-ternatively, by keeping inertia and neglecting terms oforder √ Re and higher, the Maxey-Riley equation can berewritten as d v dt = − g ( k + v T ) . (8)This equation allows us to observe the dynamics ofdroplets at short times. A. Shear and Poiseuille flows as external wind
Let us solve Eq. (8) under the presence of shear andPoiseuille flows modeling the effect of wind on the spher-ical droplets. These profiles are given by u = (cid:16) U max zh + U z ( z − h ) (cid:17) j . (9)Notice that the shear flow profile correspond to the case U = 0 . Given Eq. (9), we solve Eq. (8) whose solutionincluding dimensions and subject to r (0) = (0 , , h ) and v (0) = ( v x , v y , v z ) can be shown to be v x ( t ) = v x e − Kt , v z ( t ) = γ z e − Kt − gK , (10) v y ( t ) = γ y e − Kt − U (cid:16) γ z K (cid:17) e − Kt + U γ z gK t e − Kt − Ate − Kt + B − Ct + U g K t , (11)where K = R T /m and β = Ka U /
3. Integrating the − − U max = 1 m/sU max = 5 m/s U max = 1 m/s Shear
Poise.
Unst. ( a ) ( b ) a ( µm )
30 40 50 −
60 70 80 90 ( c ) a ( µm ) a ( µm ) y M ( m ) y M ( m ) y M ( m )
125 6 7 8 9 10 a ( µm ) − − − t F ( s ) t F ( s ) t F ( s ) a ( µm ) a ( µm ) a ( µm )
05 10 15 20
20 40 60
60 80 100 ( d ) ( e ) ( f ) FIG. 3. (a)-(c) Maximum dispersion ( y M ) along the y − direction of spherical droplets as a function of the radius a , as well asa function of different external flows namely, shear, Poiseuille, and unsteady shear flow. The color code in (b) is valid for allfigures. (a) Maximum dispersion for a ranging from 5 µm to 10 µm . (b) Maximum dispersion for a ranging from 12 µm to 60 µm .(c) Maximum dispersion for a ranging from 60 µm to 100 µm . (d)-(f) Flying time ( t F ) of spherical droplets as a function of theradius a . Here, v z (0) = 0. Red circles represent the whole z − component in Eq. (12), while the black-solid line represents theapproximated equation t F ≈ µ a h/ ρga . (d) a ranging from 5 µm to 20 µm . (f) a ranging from 20 µm to 60 µm . (e) a rangingfrom 60 µm to 100 µm . velocities, we get x ( t ) = − v x K e − Kt + s x , z ( t ) = − γ z K e − Kt − gK t + s z , (12) y ( t ) = 1 K (cid:18) AK − γ y − U γ z gK (cid:19) e − Kt + 1 K (cid:18) A − U γ z gK (cid:19) te − Kt − U γ z gK t e − Kt + U γ z K e − Kt + s y + Bt − C t + U g K t , (13)where constants A, B, C, γ y , γ z , s x , s y , s z are defined inAppendix IX. B. Unsteady shear flow as external wind
A more realistic situation, is to consider the fact thatair flows for a while, and then stops, and then flows again.The simplest model is to assume a time-dependent shearflow scenario. This unsteady vector flow field u ( r , t )should satisfy the time-dependent Navier-Stokes equa-tions, which after assuming u ( r , t ) = (0 , u y ( z, t ) , ∂u y ∂t = υ a ∂ u y ∂z . (14) This parabolic equation must satisfy u y ( z,
0) =( U max /h ) z, u y (0 , t ) = 0 , and u y ( h, t ) = U max / ωt ). Its solution can be veryfied tobe u y ( z, t ) = ∞ (cid:88) m =1 D m g ( t ) sin mπzh + U max h (1 + cos ωt ) z, (15)where g ( t ) = e − d m t + d m ω sin ωt − cos ωt, (16) D m = ωf m d m + ω , (17) f m = ( − m +1 ωU max /mπ, (18) d m = υ a (cid:16) mπh (cid:17) . (19)Once Eq. (15) is available, it is plugged into Eq. (8)and its solution subject to r (0) = (0 , , h ) and v (0) =( v x , v y , v z ) although very lengthy, can be analyticallyfound. The dominant terms of the solution along the y − component for long times read − − − a ( µm )
20 22 24 26 28 40 60 8030 100 a ( µm ) a ( µm ) ( a ) ( b ) ( c ) y ( m ) y ( m ) y ( m )
13 15 17 190 0 0 U max = 1 m/sU max = 5 m/s U max = 1 m/s Shear − − a ( µm ) t F ( s )
20 40 60 80 100 − t ( s ) t ( s ) y ( m ) y ( m ) ( d ) ( e ) ( f ) FIG. 4. (a)-(c) Effect of higher order terms on the dispersion along the y − direction of spherical droplets as a function of radius a and an external flow. Here ∆ y = y MB − y M , where y MB is the maximum travelled distance by a droplet along the y -directioncontaining first order terms ( a ≤ µm ) or the full terms in the M-R equation ( a > µm ), whereas y M represents the solutiondirectly obtained from Eqs. (12) and (13). (d) Effect of higher order terms on the flying time of spherical droplets as a functionof radius a . Here ∆ t F = t FB − t F , where t FB is the flying time by a droplet containing first order terms ( a ≤ µm ) or the fullterms in the M-R equation ( a > µm ); whereas t F represents the the solution directly obtained from Eqs. (12). (e)-(f) Linearconvergence of ∆ y = y MB − y M as the time-step in the simulations decreases. (e) Convergence for a droplet of radius 30 µm after discretizing the full M-R equation (5). (f) Convergence for a droplet of radius 19 µm after discretizing the overdampedM-R equation (22). v y ( t ) = − U max g hK t − U max g [ K cos( ωt ) + ω sin( ωt )]2 h ( K + ω ) t, (20) y ( t ) = U max h (cid:110) gK t − g K t − g ( K + ω ) (cid:20) Kω t sin( ωt ) − t cos( ωt ) (cid:21)(cid:27) . (21)Once Eq. (12), Eq. (13), and Eq. (21) are available, wecan now exemplify a typical droplets’ dispersion (cloud)after a person talks and generates micrometric droplets(radius between 5 µm to 100 µm ), which are sub-ject to either a shear flow (Fig. 3(a)), a Pouiseuille flow(Fig. 3(b)), or an unsteady shear flow (Fig. 3(c)). Wesimulate the situation of a normal and a strong talk byimposing an exit speed of droplets (from a cone shape ofvelocities representing a person’s mouth) of v = 1 m/s and v = 5 m/s , respectively. The cloud is made of1000 droplets of random size between 5 µm to 100 µm .Additionally, we impose a moderate ( U max = 5 m/s )and a calm ( U max = 1 m/s ) wind scenario. The coneshape as the initial velocity distribution, and as ob-served from experiments , is implemented by imposing v (0) = ( v cos ϕ sin θ , v sin ϕ sin θ , v cos θ ), whoseangular polar and azimuthal initial extremum are set to θ max = 110 o , θ min = 81 o , ϕ max = 110 o , and ϕ min = 81 o . The case of a Poiseuille flow profile considers for acalm flow { U max , U } = { m/s, − . m − s − } whereadfor a moderate flow { U max , U } = { m/s, − m − s − } :The simulations for the unsteady flow profile consider ω = 0 . s − . Finally, ( ϕ, θ ) will randomly vary betweentheir extremum. The results can be visualized in Fig. 3which shows the distribution of droplets at three differenttimes namely, t = 0 . s , t = 0 . s , and t = 1 . s . A colorcode indicating the droplets’ sizes is also introduced. Thesafe distancing is indicated as a vertical dotted-black line.Clearly, droplets move beyond the safe distancing. As itcan be seen, droplets under a moderate wind and af-ter only 1 . s , are already 6 m away from the person’smouth. Droplets under a calm wind and less than 50 µm in radius will overpass the safe distancing in the next sec-ond. These results indicate a potential danger for peo-ple in a public space since the safe distancing has beeneasily surpassed. We finally take Eq. (12), Eq. (13),and Eq. (21) to find the maximum travelled distance( y M ) along the y − direction as a function of droplets’ size,as well as a function of different external flows namely,shear, Poiseuille, and unsteady shear flow. The resultsare shown in Fig. Figs. 3(a)-(c). Figures 3(a)-(c) showthat droplets under a Pouiseuille profile reached the thelongest distance compared to the other analyzed profiles.This figure also indicates that for U max = 5 m/s , dropletsof radius 5 µm , and under a Poiseuille flow, can travel un-til 2000 m , while around 1500 m under a shear flow. Asexpected, the condition of having a wind blowing andceasing (unsteady flow) reduces the droplets maximumdispersion to around 750 m for U max = 5 m/s . On theother hand, our results to leading order indicate that themaximum travelled distance for small droplets (5 µm ra-dius) under a shear/Poiseuille external flow with a maxi-mum speed of U max = 1 m/s may easily reach more than250 meters. The long distance achieved by these smalldroplets is because they remain in the air for around 600seconds, see Figs. 3(d)-(e). In these figures, the flyingtime obtained from Eq. (12), is shown as a solid-blackline; while the red circles represent the approximatedequation t F ≈ µ a h/ ρga . These figures also indicatethat the largest (100 µm radius) droplets can only stay inthe air for about 1 . s . It is worth mentioning that all thestudied droplets ( a ∈ [5 µm, µm ]) reached a constantvertical terminal velocity U ≈ mg/R T . IV. DYNAMICS WITH BOUSSINESQ-BASSETMEMORY
In the literature, the Boussinesq-Basset (B-B) mem-ory force has been less frequently considered, probablybecause of its order-of-magnitude and because of the re-quired computational effort. However, there exist sometheoretical , numerical and experimental works dealing with this force. In this section we analyzethe effect of the memory force term on the dispersion anflying time of spherical droplets.Consider first small droplets ranging between a ∈ [5 µm, µm ]. According to Table I, their motion canbe modeled by the M-R equation under the overdampedapproximation. However, by keeping terms of order √ Re to see the effect of the B-B memory force, Eq. (5) reads v T = − (1 − λ ) k − (cid:114) Reπ (cid:90) t d v T dτ dτ (cid:113)(cid:0) t − τ (cid:1) . (22)To solve this integro-differential equation, a first orderEuler method is chosen. Under this method, a compo-nent of the B-B force term can be shown to be: (cid:90) t dvdτ dτ (cid:113)(cid:0) t − τ (cid:1) = k − (cid:88) i =1 v i +1 − v i √ ∆ t α ( k, i ) , (23)where we have assumed that ∆ t = ∆ τ and defined α ( k, i ) = 2 √ k − i − √ k − − i . After certain steps,one can prove that the overdamped M-R Eq. (22), alongthe z − direction and in dimensional variables, acquiresthe following discrete form for k = 3 , ...N : v z,k = −√ ∆ t c c + c c v z,k − − c c k − (cid:88) i =1 ( v z,i +1 − v z,i ) α ( k, i ) , (24) − − − − − − − l n ( v z ( t ) + U ) ln ( t ) ⇠ t / simulationsimulation ( a = 30 µm )( a = 98 µm ) ⇠ e Kt FIG. 5. Effect of the Boussinesq-Basset memory force termon the sedimentation velocity ( v z (0) = 0), for two sphericaldroplets reaching their terminal velocity U . The black-solidlines indicate an exponential decay towards U when the B-Bterm is absent. The red-dashed lines indicate a t − / decaywhen the B-B term is present. where constants c , c , c , c are defined in Appendix X.A similar expression as Eq. (24) is obtained for the otherspatial components. In the case of larger droplets andfrom Table I, one notices that the order-of-magnitude ofall terms in Eq. (5) is the same. Therefore, one has tosolve for the full M-R Eq. (5). Using the Euler methodtogether with Eq. (23), one can show that the discretized z − component of Eq. (5) in dimensional variables reads v z,k = − ∆ tb Rg + b b v z,k − − b b k − (cid:88) i =1 ( v z,i +1 − v z,i ) α ( k, i ) , (25)where constants R, b , b , b , b are defined in AppendixX. The discretization of the other components in Eq. (5)is similar to Eq. (25). After posing Eq. (24) and Eq.(25),we are ready to find the droplets’ dynamics under higherorder terms.Because of the external flows we have chosen and fromthe simulations in Sec. III, we observe that the dynam-ics mostly occurs along the z − y plane, and that initialconditions are not relevant for long times (at least for v T (0) = 0); thus from now on, a 2D problem with initialconditions v z (0) = 0 and v y (0) = U max , will be consid-ered. Equations (5) and (22) are then numerically solvedusing the discretized scheme in Eq. (24) and Eq. (25),under the presence of a shear flow (the other flows basi-cally share the same features), and for two typical exitinitial speeds while talking namely, 1 m/s and 5 m/s .The time-step used for solving Eq. (5) and Eq. (22)was 4 × − s and 5 × − s , respectively. The resultsof the simulations are shown in Figs. 4(a)-(c). In thosefigures, ∆ y = y MB − y M , where y MB is the maximumtravelled distance along the y -direction by droplets of size a ≤ µm and obtained from Eq. (22). For droplets ofsize a > µm , y MB represents the maximum travelleddistance along the y -direction obtained after solving thefull M-R equation (5); whereas y M represents the solutiondirectly obtained from Eqs. (12) and (13). One can ob-serve that for a low wind speed, the effect of higher orderterms barely enhance the droplets’ dispersion. However,for a wind speed of 5 m/s and for the smallest considereddroplet, higher order terms can increase its dispersionaround 2 m . Figures 4(a)-(c) also indicate that as thesize of the droplets increases, higher order terms effectsbecome smaller until they finally disappear.The effect of first order terms and the full terms in theM-R equation, on the flying time of spherical dropletsas a function of the radius a is also analyzed. UsingEq. (12), the numerical solutions from Eq. (24) and Eq.(25), and defining ∆ t F = t F B − t F ; where for a ≤ µm , t F B represents the flying time of a droplet calculatedusing first order terms (Eq. (22)). For a > µm , t F B represents the flying time calculated using the full termsin the M-R equation. On the other hand, t F is the flyingtime directly obtained from Eqs. (12). This analysis isshown in Fig. 3(d). It can be seen that ∆ t F increases asthe droplets’ sizes decrease. In fact, for a = 5 µm there isa 0 . s flying time difference between the dynamics of Eq.(5) and Eq. (8). This extra time also contributes to theobserved enhancement of dispersion of small droplets. Anumerical analysis on the convergence of ∆ y = y MB − y M , as the time-step ∆ t in the simulations decreases, isalso performed. Fig. 4(f) shows this convergence fora droplet of radius 30 µm after discretizing the full M-R equation (5). As expected, a linear convergence canbe appreciated. The convergence for a droplet of radius19 µm and after using the overdamped M-R equation (22)is shown in Fig. 3(b). Once again a linear convergenceis achieved. Therefore, the latter analysis validates ouremployed first order numerical algorithm.Finally, the implications of the Boussinesq-Bassetmemory force term on the sedimentation velocity compo-nent v z ( t ) with v z (0) = 0 is also studied. The numericalresults are shown in Fig. 5, which illustrates the dynam-ics of v z ( t ) + U versus time in a log-log plot, and fortwo different spherical droplets reaching their terminalvelocity U . The black-solid lines belong to an exponen-tial decay of v z ( t ) towards U ; whereas the red-dashedlines indicate a t − / decay. It can be observed that forshort times, v z ( t ) exponentially decays towards U ; how-ever, v z ( t ) decays according to the scaling t − / for longtimes. This is a rather surprising result, since the order-of-magnitude of the B-B term is really small. This t − / decay of the sedimenting velocity has been also recentlyreported . Further consequences of the B-B term onthe motion of particles at low Reynolds numbers may besearch for in future investigations. < < z ( m ) y ( m ) UNS SS UNS y ( m ) t ( s ) < ( a ) ( b ) ( c ) y ( m ) z ( m ) This work
Dbouk and Drikakis
FIG. 6. (a) Droplets’ distribution at t = 5 s under a shear(S) and an unsteady shear flow (UNS) at calm (1 m/s ) andmoderate winds (5 m/s ). The rest of the parameters are thesame as in Fig. 2. Under these parameters, together with RH = 50% and temperature les than 30 o C , evaporation doesnot play a role . (b) Comparison of droplets’ position underan uniform flow ( { h, U max , v } = { h = 1 . m, . m/s, m/s } )either using single particle dynamics (this work), or using amore elaborated CFD analysis . (c) Droplets’ distributionat t = { s, s, s, s, s } , and linear paths followed by somedroplets of different sizes under an uniform flow. See Fig. 2for color code. V. DISCUSSIONS AND CONCLUSIONS
According to Dbouk and Drikakis and others, evap-oration and relative humidity play a role on droplets’dispersion. Those factors may reduce or increase the sizeof droplets and their cloud shape as it travels. Therefore,our results could be improved to account for a rocket-likedynamics (drops varying mass). Following Dbouk andDrikakis , who considered a conjugated flow-heat-masstransfer problem and CFD dynamics, it can be concluedthat evaporation of droplets takes place at high relativehumidities, and high temperatures. Thus for the caseof countries with an annual average of relative humidity, RH = 50%, a wind speed of 4 km/h = 1 . m/s , a temper-ature less than 30 C o , and five seconds later since a cloudof droplets originated, there will be a null evaporation .Other works also supporting a long time survival of in-fected droplets is Stadnytskyi et al. Based on this information, Fig. 6(a) shows fourdroplets’ distributions (cloud) five seconds later since thecloud originated under a shear ( S ) or an unsteady shear( U N S ) flow with { U max , v } = { m/s, m/s } . Theother numerical parameters employed and the initial ve-locity distribution of droplets are the same as in Fig.2. This figure indicates that droplets with a > µm have already reached the floor, and that droplets of size50 µm < a < µm are about to reach the floor. How-ever, the smallest droplets under a calm wind (1 m/s ) forboth U N S and S , have surpassed the social distancing(vertical black-dashed lines). The same droplets undera moderate wind (5 m/s ) for both U N S and S , reached15 m and 25 m , respectively. Thus from this figure, onecan explicitly visualize a more real scale to which peo-ple may be in risk of contagion. Fig. 6(b) compares theposition of droplets’ distribution (cloud) under an uni-form flow ( { h, U max , v } = { h = 1 . m, . m/s, m/s } ),either using single particle dynamics (this work) or em-ploying a more elaborated CFD analysis . Both meth-ods result in a similar cloud’s position. Finally, usingthe latter parameters, the droplets’ distribution at t = { s, s, s, s, s } and some paths followed by dropletsof different sizes are shown in Fig.6(c). These paths havea linear behavior since the cloud dynamics is practicallyoverdamped, implying y ( t ) ≈ U max t and z ( t ) ≈ h − U t ,thus particles will follow the function z ≈ h − ( U/U max ) y .In summary, using single particle dynamics, which hasthe advantage of requiring a minimum computationalcost, this paper provided an estimate of the maximumdispersion of micrometric droplets generated after talk-ing. Briefly, under conditions of a null evaporation andonly five seconds later since droplets were originated, itwas found that an unsteady shear calm wind (1 m/s ) candisperse droplets beyond the social distancing, and untilmore than 15 m when droplets are subject to a moderateunsteady shear flow (5 m/s ). As expected, an unsteadyshear profile is less efficient to disperse droplets than aconstant Poiseuille or shear profile. These constants pro-files modeling a calm wind (1 m/s ) were found to pro-vide a maximum dispersion beyond 250 m for the smallestdroples (5 µm ). The effect of the Boussinesq–Basset forceterm was also analyzed. Although its order-of magnitudeis small, it was found to be enough to change the be-havior of the sedimentation velocity (from exponentiallydecaying towards its terminal velocity, to proportionallydecaying as t − / ) of a micrometric particle and slightlyincrease droplets’ dispersion and their flying time.Future research would be to consider external flows inthe presence of buildings and to find the complex stream-lines generated and their effects on droplets’ distribution.It may be inferred for example that the presence of acorner on a common street, could generate stagnationpoints, that may risk areas of infection, since those pointscould storage for a while infected droplets. Furthermore,walls may induce three-dimensional flows that may dragparticles away from the floor, thus increasing its flyingtime and hence their capability of traveling longer. Wehope this study helps people to be more aware about theeffect of daily wind currents on the propagation of poten-tially infected droplets. Based on our findings of dropletseasily dispersing beyond the social distancing when sub-ject to wind currents, we recommend the use of masks able to contain the virus, as well as flex seal googles,since droplets dragged by wind may reach eyes. We alsorecommend to wash all your wearing clothes and shoes,and to shower after being out from home, since infected droplets may be attached to clothes or hair . Direct ex-posure to wind currents, mainly in crowded cities, shouldalso be avoided. VI. AUTHOR CONTRIBUTIOS
All authors contributed equally to this research.
VII. ACKNOWLEDGEMENTS
M. S. thanks Consejo Nacional de Ciencia y Tecnolo-gia, CONACyT for support. M. S. dedicates this paperto the memory of his relatives Arturo Sandoval and Mar-cos Fernandez reached by this pandemic.
VIII. DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
IX. APPENDIX 1: CONSTANTS ADDED
The constants
A, B, C, γ y , γ z , s x , s y , s z appearing inEqs. (10)-(13) are defined as A = γ z (cid:18) U max h + U (2 s z − h ) (cid:19) , (26) B = U s z + βK + (cid:18) U (cid:20) gK − h (cid:21) + U max h (cid:19) (cid:16) gK + s z (cid:17) , (27) C = gK (cid:20) U (cid:16) (cid:16) gK + s z (cid:17) − h (cid:17) + U max h (cid:21) , (28) γ y = v y + U (cid:18)(cid:16) γ z K (cid:17) − s z (cid:19) − (cid:18) U (cid:18) gK − h (cid:19) + U max h (cid:19) (cid:16) gK + s z (cid:17) − βK , (29) γ z = v z + gK , s x = v x K , s z = h + γ z K , (30) s y = − K (cid:20) AK − γ y + U γ z K (cid:18) γ z − gK (cid:19)(cid:21) . (31) X. APPENDIX 2: NUMERICAL PART FOR THE B-BEQUATION
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