Natural Convection Above Circular Disks of Evaporating Liquids

Natural convection above circular disks of evaporating liquids

Benjamin Dollet1 and Fran¸coisBoulogne2 1Institut de Physique de Rennes, UMR 6251 CNRS and UniversitéRennes 1, Campus Beaulieu, Bâtiment 11A, 35042 Rennes Cedex, France 2Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, UniversitéParis-Saclay, Orsay 91405, France

May 16, 2017

Abstract We investigate theoretically and experimentally the evaporation of liquid disks in the presence of natural convection due to a density difference between the vapor and the surrounding gas. From the analogy between thermal convection above a heated disk and our system, we derive scaling laws to describe the evaporation rate. The local evaporation rate depends on the presence of a boundary layer in the gas phase such that the total evaporation rate is given by a combination of different scaling contributions, which reflect the structure of the boundary layer. We compare our theoretical predictions to experiments performed with water in an environment controlled in humidity, which validate our approach.

1 Introduction where ρs and ρ are respectively the vapor density at saturation and at∞ infinity, g the gravity constant, R the The evaporation of small liquid disks takes its roots in radius of the drop and ν the kinematic viscosity of air. botany, in particular with some studies on the transpi- For small Grashof numbers, evaporation is limited by dif- ration of plants published in the early twentieth century fusion while for large Grashof numbers, buoyancy creates [1, 2, 3, 4]. Beyond this original inspiration, evaporation a convective flow. is an ubiquitous phenomenon in nature as for the evapo- Additional experimental questioning [11] and evi- ration of liquids from water drops [5] to lakes or oceans [6] dences [12, 13, 14, 15, 16, 17] of the importance of con- and in the industry, in particular for coating processes. vective effects have been reported more recently in the Indeed, the transport of solutes is often driven by evap- literature. Direct visualizations of evaporating drops oration and dictates the self-organization of particles at have been achieved by X-ray imaging method [12], IR micro and macroscopic lengthscales [7]. Therefore, a cor- absorption [14], Schlieren technique [18] or by interfer- rect modeling of the evaporation dynamics is crucial for ometric measurements [19] and these studies concluded understanding evaporation kinetics and colloidal deposi- that evaporation can be enhanced by convection. The tion. transition between diffusive and convective evaporation When the evaporation is limited by the diffusion in the regimes has been reported and the convective evapo- β vapor phase [8], the derivation of the evaporative flux ration rate can be captured as Gr with β 0.20 ≈ shows that the total evaporation rate is linear with the [13, 14, 15]. radius of the liquid surface [9]. In 2006, Shahidzadeh- As we expect from the definition of the Grashof num- Bonn et al. reported experimental observations of evap- ber (Eq. (1)), the threshold for convective evaporation orating drops of water and hexane [10]. They observed is a function of the radial lengthscale of the evaporating arXiv:1704.03243v2 [physics.flu-dyn] 14 May 2017 that while a drop of hexane evaporates as predicted by a surface. Recently, Carrier et al. investigated the mutual diffusive model, a drop of water has an anomalous evap- influence of closely deposited drops on the evaporation oration rate above a certain radius. They attributed this rate [20] as it can be encountered for a sprayed liquid. If different behavior to the natural convection that takes the drops are separated by a distance comparable to their place, as water vapor is less dense than air. The di- radius, they found that a cooperative effect induces con- mensionless number representing the balance between vective evaporation. In addition, they studied the evap- the buoyant forces in favor of convection and the vis- oration of circular evaporating surfaces of different radii cous forces in favor of diffusion, is the Grashof number and they concluded that for large Grashof numbers, con- defined as vection dominates with an evaporation rate that scales 3 as R2. ρs ρ gR Gr = − ∞ , (1) ρ ν2 From the point of view of the governing equations, the ∞

1 evaporation of a liquid is similar to the dissolution of a z liquid into another. Recent attention has been devoted to the sessile drop dissolution [21, 22, 23] and bubble ~g Water growth in supersaturated solutions [24] with a combination of experimental, numerical and theoretical investiga- R tions. These studies show that natural convection is also r observed above dissolving drops. The main difference Ring between evaporation and dissolution is the magnitude of the Schmidt number Sc = ν/ defined as the ratio of the kinematic viscosity ν of theD surrounding phase and Figure 1: Sketch presenting the notations for the convec- the diffusivity of the molecules. For evaporation, the tive evaporation above a circular disk of volatile liquid. Schmidt numberD is close to unity as diffusivity and kinematic viscosity are similar for gases. In contrast, the constant, except in the terms where it acts as a driving Schmidt number is large for dissolution [22]. This differ- force for the flow; in particular, the flow is considered ence of Schmidt number is important for the structure of incompressible. the boundary layer of the convective flow [25]. Under these conditions, the velocity field of the vapor In this paper, we propose to derive scaling laws for a above the drop writes u = u e + w e and depends only flat circular evaporating surface in a regime dominated r z on the cylindrical coordinates r and z, with origin at by convection. We base our analysis on an analogy be- the center of the drop interface (Fig. 1). The continuity tween convective evaporation of flat circular surfaces and equation is the thermal convection that takes place above a heated 1 ∂(ru) ∂w disk. We show that our prediction is in agreement with + = 0. (2) r ∂r ∂z our measurements and with some empirical predictions available in the literature. With p the pressure field, c the mass concentration field, ρ(c) the gas density field and µ its dynamic viscosity, the radial and vertical components of the Navier-Stokes 2 Model equation are respectively ∂u ∂u ∂p As stated in the introduction, the evaporation of the liq- ρ u + w = + uid changes the composition of the surrounding atmo- ∞ ∂r ∂z − ∂r sphere and thus, its local density. Therefore, we con- ∂2u 1 ∂u u ∂2u µ + + , sider the natural convection that can take place above ∂r2 r ∂r − r2 ∂z2 a circular disk of liquid. In this Section, we establish (3a) a scaling law between the evaporation flux and the ra- ∂w ∂w ∂p dius of the flat drop. By analogy between heat and mass ρ u + w = ρ(c)g + ∞ ∂r ∂z − − ∂z transfer, this derivation closely follows that of the heat ∂2w 1 ∂w ∂2w transfer above a horizontal heated surface, which is a µ + + . (3b) classical problem [26, 27, 28, 29]. Dehaeck et al. have ∂r2 r ∂r ∂z2 also performed a detailed study of the role of natural In these equations, according to the Boussinesq approx- convection on an evaporating pendant droplet, see the imation, the gas density is taken as constant and equal Supporting Information of [19]. In particular, they dis- to its value at infinity ρ , except in the driving force cussed the role of the varying slope of the droplet, and of term of natural convection,∞ ρ(c)g in the equation (3b) of thermal Marangoni flows induced in the droplet by the vertical motion. From the ideal gas law, the gas density latent heat released during evaporation. However, these varies linearly with the vapor concentration: papers either considered a plate geometry instead of a disk [26, 27], or started from dimensionless or simplified c ρ(c) = ρ0 ∆ρ , (4) equations [28, 29, 19], which is why we prefer to present − cs the derivation in details. where ρ0 is the density of dry air, ∆ρ = ρ0 ρs the density difference between pure air and air saturated− with 2.1 Diffusion-convection equations vapor, and cs the saturation concentration of vapor in We establish the equations to describe the flow in the air. Finally, the diffusion-convection equation for the gas phase and our analysis requires the following as- concentration field is sumptions. (i) The liquid-vapor interface is horizontal. ∂c ∂c 1 ∂ ∂c ∂2c u + w = r + , (5) (ii) The flow of vapor is axisymmetric and in a steady ∂r ∂z D r ∂r ∂r ∂z2 state. (iii) Thermal effects are negligible. (iv) We use the Boussinesq approximation: the air density is assumed where is the diffusion coefficient of vapor in air. D

2 Let p = pstat + $ with pstat the hydrostatic pressure, 2.0 1.0 such that ρ g ∂p /∂z = 0. The Navier-Stokes stat 0.8 equations (3)− ∞ become:− 1.5 0.6 ˜ ˜

c ∂u ∂u ∂$ z 1.0 ρ u + w = + 0.4 ∞ ∂r ∂z − ∂r 0.5 ∂2u 1 ∂u u ∂2u 0.2 µ + + , ∂r2 r ∂r − r2 ∂z2 0.0 0.0 (6a) 2 1 0 1 2 − − r˜ ∂w ∂w ∂$ ρ u + w = [ρ(c) ρ ] g + ∞ ∂r ∂z − − ∞ − ∂z Figure 2: Dimensionless vapor concentration map above ∂2w 1 ∂w ∂2w an evaporating disk in the dimensionless space (˜r, z˜) ob- µ + + . (6b) ∂r2 r ∂r ∂z2 tained from equation (12). White and cyan lines are the oblate spheroidal coordinates (κ, σ), for respectively con- We now nondimensionalize equations (2), (5) and (6) stant κ and σ values. by the characteristic length scale R, the characteristic velocity ν/R and the characteristic pressure ρ ν2/R2. We thus introduce the following dimensionless∞ quanti- 2.2 Diffusion-limited evaporation ties, denoted by a tilde:r ˜ = r/R,z ˜ = z/R,u ˜ = Ru/ν, For small Grashof numbers, the unique driving term of w˜ = Rw/ν andp ˜ = R2p/ρ ν2. We also define a di- ∞ the gas flow, namely Grc ˜ in equation (7c), vanishes. mensionless concentration fieldc ˜ = (c c )/(cs c ), From the definition of the Grashof number given by equa- where c is the vapor concentration far− from∞ the− drop.∞ ∞ tion (1), and from (4), the condition Gr < 1 is equivalent We then obtain: to a drop radius smaller than R? defined as 1 ∂(˜ru˜) ∂w˜ + = 0, (7a) 2 1/3 ? cs c ν r˜ ∂r˜ ∂z˜ R = − ∞ . (10) ∂u˜ ∂u˜ ∂$˜ ∂2u˜ 1 ∂u˜ u˜ ∂2u˜ c g u˜ +w ˜ = + + + , ∞ ∂r˜ ∂z˜ − ∂r˜ ∂r˜2 r˜ ∂r˜ − r˜2 ∂z˜2 Hence, in this condition, there is no gas flow i.e. u˜ = (7b) w˜ = 0, and the set of equations (7) reduces to the Laplace ∂w˜ ∂w˜ ∂$˜ ∂2w˜ 1 ∂w˜ ∂2w˜ equation for the concentration field. From equation (7d), u˜ +w ˜ = + Grc ˜ + + + , ∂r˜ ∂z˜ − ∂z˜ ∂r˜2 r˜ ∂r˜ ∂z˜2 the Laplace equation writes (7c) 2 2 1 ∂ ∂c˜ ∂ c˜ ∂c˜ ∂c˜ 1 1 ∂ ∂c˜ ∂ c˜ r˜ + = 0. (11) u˜ +w ˜ = r˜ + , (7d) 2 ∂r˜ ∂z˜ Sc r˜ ∂r˜ ∂r˜ ∂z˜2 r˜ ∂r˜ ∂r˜ ∂z˜ The boundary conditions are a saturated vapor concen- with two dimensionless numbers, the Grashof number tration at the surface of the disk c(r < R, z = 0) = cs defined by equation (1), and the Schmidt number Sc = and a vapor concentration c far from the disk. Intro- ν/ , which is of order one because all diffusivities have ∞ D ducing the oblate spheroidal coordinates (κ, σ) defined as the same order of magnitude in gases. r˜2 = (1 κ2)(1+σ2) andz ˜ = κσ, the vapor concentration Since the liquid is out of equilibrium with the atmo- can be− written in the simple form [30, 31] sphere, a mass transfer occurs at the liquid-vapor interface. From Fick’s law, the local flux normal at the 2 c˜(κ, σ) = 1 arctan(σ). (12) liquid-vapor interface is given by − π

∂c(r, z) This solution is represented in Fig. 2. j(r) = , (8) −D ∂z Thus, for a diffusion-limited evaporation, the local flux z=0 calculated from equation (8), is where the concentration gradient is taken at the liquid- vapor interface, i.e. z = 0. The total evaporating flux is 2 (cs c ) jdiff (r) = D − ∞ , (13) given by π √R2 r2 Z − Q = j(r) dS, (9) which is the well-known flux for a drop of a small contact S angle. This flux presents a divergence at the edge that where S is the surface of the liquid-vapor interface. can be interpreted as a tip effect by analogy with electro- To determine these local and total fluxes, we must es- static problems. Substituting equation (13) in equation tablish the vapor concentration gradient at the interface. (9), we derive the total flux Therefore, we analyze in the next paragraphs the set of equations (7) to derive this concentration gradient. Qdiff = 4 (cs c )R. (14) D − ∞

3 Plume Due to this convective flow, unsaturated air is brought (a) toward the edge of the evaporating film. At some distance from the edge of the film, a slender boundary layer is established and progressively thickens. At the center of the disk, the flow field converges, and thus moves up- Center Intermediate zone Edge wards to form a rising plume of wet air. zone zone Consequently, we distinguish three zones as depicted 3/5 in Fig. 3(a): the edge and the center zones, that do Gr− R δ(r) R? not satisfy the conditions for a slender boundary layer, 0 R r in contrast to the intermediate zone, which is slender in the sense that vertical variations are much sharper than (b) R (mm) horizontal ones. We first focus on this intermediate zone 10 100 where the approximation of a thin horizontal boundary 1 layer can be applied to determine its thickness variation. Then, from this first analysis, we precise the horizontal 0.8 extension of the intermediate zone and, by consequence,

) we also deﬁne the sizes of the edge and center zones. 2 0.6 Center

πR Edge (

/ 0.4 Intermediate 2.3.2 Intermediate zone S We assume that the flow is almost horizontal and that the 0.2 vertical variations occur over lengthscales mush smaller than the horizontal ones. To retain the necessary inertial, 0 101 102 103 104 105 106 viscous and pressure terms describing such a flow, the following rescaling is introduced [26]:z ˆ = Gr1/5z˜,u ˆ = Gr 2/5 1/5 4/5 Gr− u˜,w ˆ = Gr− w˜ and$ ˆ = Gr− $˜ . Substituting these new variables in equations (7), we obtain: Figure 3: (a) For a large Grashof number, sketch representing the three characteristic zones of the boundary 1 ∂(˜ruˆ) ∂wˆ + =0, (15a) layer depicted in red above an evaporating disk of liq- r˜ ∂r˜ ∂zˆ uid. The thickness of the boundary layer is given by ∂uˆ ∂uˆ ∂$ˆ uˆ +w ˆ = + Eq. (17). Blue arrows indicate the direction of the gas ∂r˜ ∂zˆ − ∂r˜ flow for a light vapor, ∆ρ > 0. (b) Relative surface area 2 2/5 ∂ uˆ 1 ∂uˆ uˆ of the center, edge and intermediate zone as a function Gr− + ∂r˜2 r˜ ∂r˜ − r˜2 of the Grashof number. Each area is calculated from the ∂2uˆ scaling presented in (a). The corresponding radius R is + , (15b) indicated for water. ∂zˆ2 2/5 ∂wˆ ∂wˆ ∂$ˆ Gr− uˆ +w ˆ = +c ˜+ ∂r˜ ∂zˆ − ∂zˆ Equations (13) and (14) have been largely commented ∂2wˆ 1 ∂wˆ in the literature, particularly in the frames of evaporat- 4/5 Gr− 2 + ing droplet lifetime [32, 33] and the so-called coffee stain ∂r˜ r˜ ∂r˜ 2 effect [9, 34, 35, 36, 17]. Therefore, we do not develop 2/5 ∂ wˆ + Gr− , (15c) further the diffusion-limited case and we analyze in the ∂zˆ2 2 next paragraph the convective evaporation. ∂c˜ ∂c˜ 1 2/5 1 ∂ ∂c˜ ∂ c˜ uˆ +w ˆ = Gr− r˜ + . ∂r˜ ∂zˆ Sc r˜ ∂r˜ ∂r˜ ∂zˆ2 2.3 Convective evaporation (15d) 2/5 2.3.1 Spatial structure of the boundary layer Hence, within corrections of relative order Gr− , 1 ∂(˜ruˆ) ∂wˆ For large Grashof numbers, a flow is established in the + = 0, (16a) gas phase. The direction and the spatial structure of this r˜ ∂r˜ ∂zˆ ∂uˆ ∂uˆ ∂$ˆ ∂2uˆ flow depends on the sign of the density difference ∆ρ, as uˆ +w ˆ = + , (16b) well as the spatial structure of the flow, especially at the ∂r˜ ∂zˆ − ∂r˜ ∂zˆ2 ∂$ˆ edge and at the center of the liquid disk. As the exper- 0 = +c, ˜ (16c) iments conducted in Section 3 are for a positive density − ∂zˆ difference, we henceforth assume that ∆ρ > 0. Under ∂c˜ ∂c˜ 1 ∂2c˜ uˆ +w ˆ = , (16d) this assumption, the flow is mostly horizontal inwards. ∂r˜ ∂zˆ Sc ∂zˆ2

4 which are the analogue equations that describe natural significant. The inner layer has a dimensionless thickness convection above a heated disk [29]. r˜2/3 [29]. Similarly to equation (18), the slenderness of 1/5 1/3 Although these equations can only be solved numer- the inner layer scales as Gr− (R/r) . Thus, the inner 3/5 ically, the full solution is not necessary to obtain the layer cannot be considered as slender for r . Gr− R, scaling of the evaporative flux. Starting from the edge of which defines the extent of the central zone where the the drop (˜r = 1), a boundary-layer solution emerges. To boundary-layer approximation underlying equation (16) see this, we setr ˜ = 1 x˜ in equations (16), and consider breaks down (Fig. 3(a)). their behavior at small− x ˜. We then obtain equations similar to those describing natural convection above a 2.3.4 Evaporative flux horizontal heated plate with a straight edge. As shown by Stewartson [26], such equations admit a self-similar A proper estimate of the evaporative flux must consider solution depending on the rescaled variable1 ηˆ =z/ ˆ x˜2/5. in principle the three zones (Fig. 3(a)). In the edge ? Merkin [29] showed that, although such a self-similarity zone x < R , we showed that the boundary layer has ? is lost when considering equations (16), it is still possible both horizontal and vertical extensions of lengthscale R . to useη ˆ as a rescaling variable forz ˆ. Hence, the concen- Thus, we can estimate the scaling of the local flux defined tration gradient is localized within a boundary layer of by equation (8) as 2/5 dimensionless thicknessx ˜ . In dimensional units, from cs c j − ∞ , (19) the definitions ofz ˆ,z ˜ andx ˜, the scaling of the boundary edge ≈ D R? layer thickness is thus where R? is the characteristic lengthscale of the vertical vapor concentration gradient. The domain x < R? has (R r)2/5R3/5 a surface 2πRR? in the limit of large Grashof numbers, δ(r) − 1/5 . (17) ≈ Gr i.e. R? R. Hence, the evaporative flux in this zone has the following order of magnitude: 2.3.3 Spatial extension of the three zones ? (cs c ) Qedge 2πRR D − ∞ = 2π R(cs c ). (20) Now, we estimate the spatial extension of the different ≈ R? D − ∞ zones. From equation (17), the slenderness of the bound- Interestingly, this is exactly the same scaling as the evap- ary layer scales as oration flux of the whole drop in the purely diffusive 3/5 regime as shown in Eq. (14). δ 1 R In the intermediate zone, for r < R R?, the flux = 1/5 . (18) x Gr x scales as − cs c j (r) − ∞ , (21) As a consequence, starting from the drop edge, the int ≈ D δ(r) boundary layer becomes slender only for distances larger where δ(r) is the boundary layer thickness that corre- than x = R/Gr1/3 = R? from Eqs. (1) and (10), which sponds to the lengthscale of the vapor concentration gra- defines the extent of the edge zone; in this domain, hori- dient. From the scaling of δ(r) given by Eq. (17), we zontal and vertical variations occur over a similar length- have scale R?. This is similar to the breakdown of the bound- 1/5 (cs c )Gr ary layer approximation, for the flow of a fluid at high jint(r) D − ∞ , (22) ≈ R3/5(R r)2/5 Reynolds number, at the immediate vicinity of the lead- − ing edge of a solid, which is well known in fluid mechanics and using equation (9) with dS = 2πrdr, we obtain [37]. This effect is often negligible in the estimation of 6/5 1/5 (cs c ) 8/5 the friction force on a solid. However, as we discuss later, Qint 2π R(cs c )Gr = 2π D − ∞ R . ≈ D − ∞ c1/5ν2/5 the contribution of the edge on the total evaporation rate ∞ (23) is significant in our situation. The main prediction of this analysis is that the evapora- Close to the center of the drop, the solution given by tion flux must be proportional to R8/5. Eq. (22) also breaks down, and must match the rising A corrective term must be established for the center plume. As discussed by Merkin [29], the situation is com- zone. However, the ratio of the central zone area to the plex close to the center, where a strong pressure builds 6/5 total drop area is Gr− . As shown in Fig. 3(b), the up in response to the converging horizontal flow. Indeed, center zone represents a small portion of the drop. Thus, the flow structure is still described by Eqs. (16) but, ap- we neglect the contribution of the center zone on the total proaching the center, the boundary layer splits into two evaporative flux. layers: an inner layer in contact with the drop where Consequently, from Eqs. (23) and (20), our analysis viscous and inertial effects balance the pressure buildup yields the following law for the evaporation flux: while concentration gradients are not significant, and an Q a Q + a Q (24a) outer inviscid layer where the concentration gradients are conv ≈ 1 int 2 edge 1/5 1Notice that there is a typo in the definition (9) of η in [26]. 2π R(cs c )(a1Gr + a2), (24b) ≈ D − ∞

5 Desiccant (a) 0.2 0 0.2 −0.4 Humidity sensor −0.6 −0.8 − Air pump (g) 1 Sample − 4 mm

m 1.2 PID controller 5 mm ∆ −1.4 10 mm Water −1.6 15 mm −1.8 27 mm − 2 45 mm − 63 mm Figure 4: Sketch of the experimental setup with a scale 2.2 86 mm in a box for which the atmosphere is controlled in hu- −2.4 − midity. 0 3000 6000 9000 t (s) Gr (b) where a1 and a2 are two constant numbers. By inspec- 100 101 102 103 104 105 tion of Eq. (24b), we notice that in a convective regime, 1000 the total flux is a combination of a term reminiscent of a diffusive evaporation regime and a second term depend- 100 ing on the Grashof number, a signature of the convective flow. In the following, we perform experiments to mea- g/s) sure the evaporation rate for large Grashof numbers to µ 10 ( validate our prediction given by equation (24). Q 1 Qconv, fitted 3 Experiments Qdiff 0.1 1 10 100 Our experiments of controlled evaporation are performed R (mm) in a box made in polycarbonate (50 50 50 cm3). A × × precision scale (Ohaus Pioneer 210 g) with a precision of (c)101 0.1 mg is placed at the center of the box and is interfaced Qconv, fitted Q with a Python code using the pyserial library to record diff the time evolution of the weight. The humidity is reg- ulated with a PID controller based on an Arduino Uno and a humidity sensor (Honeywell HIH-4021-003) posi- 100 tioned far from the evaporating surface (Fig. 4). Dry air Sh 1 5 10 / is produced by circulating ambient air with an air pump 1 100 (Tetra APS 300) in a container filled with desiccant made 10 1 of anhydrous calcium sulfate (Drierite). Moist air is ob- − 2 Sh /10 Gr − tained by bubbling air in water. The relative humidity 0 1 2 3 4 5 6 1 10 10 10 10 10 10 10 is set to RH = 50% in all of our experiments. 10− 100 101 102 103 104 105 106 Circular troughs of different radii are filled with pure water right to the brim and a particular attention is paid Gr to get a flat surface. The diffusion coefficient of water 5 2 vapor in air is 2 10− m /s, and the kinematic D ≈ × 5 2 Figure 5: (a) Time evolution of the weight of water for viscosity of the gas is ν 1.5 10− m /s in our experimental conditions [17].≈ The time× variation of the weight different radii of circular troughs. The solid lines cor- of these containers is reported in Fig. 5(a) over a time responds to linear fit of the experimental data points. duration between 6000 to 9000 s. Data points are fitted (b) Measured evaporation rate as a function of the sur- with a linear function to get the mass evaporation rate face radius and the Grashof number. The dotted blue Q such that ∆m(t) = Qt. In Fig. 5(b), we report the line is a fit with equation (24) with a Grashof number − evaporation rates Q as a function of the radius of the liq- to the power 0.18, and with the coefficients a1 = 0.31 uid patch and the equivalent Grashof number as defined and a2 = 0.48. The solid line is the diffusive prediction by equation (1). Qdiff . (c) Sherwood number Sh defined by equation (26) To estimate the Grashof number, we evaluate the den- as a function of the Grashof number. Lines are equiva- sity gradient in the atmosphere due to the difference of lent to the plot (b). The inset shows the compensated 1/5 vapor concentration. The air density ρ at a pressure P0, plot Sh/Gr vs Gr.

6 a temperature T and a relative humidity RH is given by In their study, Kelly-Zion et al. [13] also considered [38] that the total evaporation flux is the result of two significant contributions, one based on diffusion, and the other 1 P M P ρ = 0 M 1 fR 1 w s , (25) on convection. In our theoretical analysis in Section 2, z T d − H − M P m R d 0 we saw that the evaporative flux is always a diffusive flux evaluated right at the interface of the evaporating where zm and f are the compressibility and enhancement drop. However, it turns out that the analysis of the spa- factors, the ideal gas coefficient, Md and Mw the mo- R tial structure of the flow of vapor above the drop, and lar density of dry and saturated air, and Ps the saturated pressure. At room temperature, we have f = 1, its separation into two convective-diffusive zones: one slender far from the edge, and one more isotropic close zm = 1 [38] and a saturated pressure Ps = 2.3 kPa [39]. Therefore, the density variation at a relative humidity to the edge, justifies that the evaporative flux separates 3 into two contributions, one scaling as a purely diffusive RH = 0.5 is ρs ρ /ρ 5 10− . In the next section, we discuss| the− ∞ evaporation| ∞ ≈ × model that we proposed flux, the other one as a convective one in a regime of large in Section 2 in comparison with our experimental results. Grashof number. Kelly-Zion et al. [13] studied volatile compounds with vapor heavier than air, contrary to water vapor. Although this leads to a different flow struc- 4 Discussion ture, starting from the center outwards, this does not affect the scaling derived in Sec. 2. Therefore, our model In Fig. 5(b), we show a fit from equation (24b) that cor- justifies a posteriori the hypothesis made by Kelly-Zion rectly captures our data. From our experimental results, et al. [13] and later by Carle et al. [15], who have added we determine the values of a1 and a2 by fitting equa- a diffusive and a convective flux to model their data. tion (24b) with a variable Grashof exponent. The result- ing exponent is 0.18 and coefficient values are a1 = 0.31 and a2 = 0.48 (Fig. 5b). The two dimensionless coefficients a1 and a2 are close to unity and the power of the Grashof number is close to 1/5, in agreement with Carrier et al. investigated the evaporation of water the theoretical prediction. The fact that a1 a2 and with beakers filled to the rim and they reported two 1/5 ' Gr is between 1 and 10, shows that the contribution regimes [20]. For radii smaller than 30 mm, the evap- of the edge (a2 Qedge), although smaller, remains signifi- oration rate is proportional to the radius R. For larger 2 cant compared to the intermediate zone (a1 Qint) in our radii, they suggest that the evaporation rate scales as R , experimental conditions. for radii up to 300 mm and they concluded that this sec- To non-dimensionalize the convective flux, we intro- ond regime can be attributed to convection. This power duce the Sherwood number Sh defined as the ratio of the exponent is significantly larger than the value found by convective and diffusive fluxes, Kelly-Zion et al. [13] and ours. Carrier et al. attributed the difference to measurements by Kelly-Zion et al. to a Qconv π 1/5 Sh = a1Gr + a2 , (26) crossover between the diffusive and a regime character- Q ≈ 2 diff ized by the development of convection cells. In the spirit from equations (14) for Qdiff and (24b) for Qconv. In of the classical Rayleigh-Bénardinstability, these authors Fig. 5c, we plot the Sherwood number as a function of proposed a direct transition from a quiescent state for the the Grashof number. Thus, we can estimate for instance gas with diffusive exchanges solely, to an unstable con- that convection becomes significant for Sh = 1.5, i.e. the vection state where the gas would be set in motion along convective flux is 50% higher than the predicted diffusive convection cells. However, such a transition appears in flux, which corresponds to Gr 20. configurations where the gas would be confined between The evaporation regime in the≈ experiments is convec- horizontal surfaces, which is far from our experimental tive. To follow the first data analysis performed by Kelly- configuration (Fig. 4). Unfortunately, the precise con- Zion et al. [13], we fit our data with a power law against figuration and the confinement are not reported for the the disk radius. We obtain that Q R1.39, in agree- experiments performed by Carrier et al., so we cannot ment with their experimental results∝ on heptane and 3- conclude as to whether the difference in exponent can be methylpentane where the radius exponents are 1.37 and ascribed to a difference in confinement. In our case, the 1.43, respectively. Nevertheless, we must underline that steady single-plume situation becomes unstable above a the vapors of heptane and 3-methylpentane are denser certain Grashof number, with the appearance of multiple than air, contrary to water vapor. This difference in va- plumes and even of a large-scale circulation [40]. How- por density can change the total evaporation rate, espe- ever, to the best of our knowledge, the onset of such an cially at the edge of the disk. The similar values for the instability is unknown. Nevertheless, our measurements exponent can be attributed to the fact that this scaling suggest that, even if such an instability appears in our mainly probes the effect of the radius due to the inter- range of Grashof numbers, it does not lead to a signifi- mediate zone, which is insensitive to the sign of ∆ρ. cant deviation from the scaling (24).

7 5 Conclusion downwards gravity current starting from the edge can evacuate the vapor collected from the disk, playing the In this paper, we used an analogy between thermal con- same role as the rising plume in the case of light vapor. vection that occurs above a uniformly heated disk and In contrast, if the disk is placed on an infinite plane or in the convective evaporation of a circular patch of liq- a trough, vapor cannot be evacuated, and accumulates, uid. The dimensionless form of the flow equations in the probably leading to a reduction of the evaporation rate. gas phase reveals a dimensionless parameter called the From a general perspective, the convective flow strongly Grashof number Gr, which relates the opposite effects depends on the geometrical aspects, which can lead to of buoyant and viscous forces. This number increases complex expressions of the evaporative flux with effects with the radius R of the disk as R3. Therefore, natural of boundaries. convection is expected for large Grashof numbers. In our theoretical analysis, we recalled the vapor concentration field surrounding an evaporating disk and the 6 Appendix related evaporative flux for small Grashof numbers, i.e. a diffusion-limited evaporation. In addition, we inves- In Section 2.1, we assume that the flow in the gas phase is tigated the evaporation dynamics under natural rising steady in order to neglect the time derivatives that would convection. The evaporating surface must be split in appear in equations (3). This assumption can be justified three domains. Near the edge of the disk, over a distance as follows. For large Grashof numbers, the characteristic R?, the boundary layer is not slender and the evapora- velocity in the gas phase is tive flux is diffusive with a concentration gradient across ν 2/5 ? = Gr , (27) a distance R . In an intermediate zone, the boundary U R layer can be considered as slender with a thickness that 2/5 2/5 scales as (R r) and the evaporative flux scales as as suggested by the rescalingu ˆ = Gr− u˜ withu ˜ = − Gr1/5. Near the center of the disk, the flow is converging Ru/ν. Therefore, the characteristic timescale associated and is directed upward with a more complex boundary to the natural convection is layer structure. The characteristic area ratio of this zone 2 6/5 R R 2/5 is Gr− , such that the precise contribution of the cen- τ = = Gr− . (28) conv ν ter zone may be neglected in a first approach, although U further experimental and theoretical investigation of this For a disk radius of water of R = 20 mm, the Grashof zone could be performed to clarify its fine structure. 3 number is Gr 1.5 10 and the timescale is τconv 1.4 Therefore, the total evaporative flux under natural con- s, which is much≈ smaller× than the observation timescales.≈ vection results in a combination of a diffusive-like flux Equivalently, we can estimate the timescale for small at the edge and a flux across the boundary layer in the Grashof numbers. For a diffusion-limited evaporation, intermediate zone, and is correctly described by equation the timescale is simply [41] (24). The Sherwood number, which compares the con- 2 vective flux Qconv to the diffusive flux Qdiff shows that R the convective flow has a significant effect for Gr > 20. τdiff = . (29) With our experimental measurements, we successfully D 2 For a millimeter water drop radius, τ 5 10− verify our prediction under natural convection. The scal- diff ≈ × ing laws of our theoretical developments provide the dif- s, which is also much smaller than the lifetime of such ferent evaporating mechanisms that occur above a cir- droplet [11]. Consequently, this supports the steady state cular liquid disk. In particular, we explain the origin assumption made in Section 2.1. of the convective evaporation dynamics observed experimentally [13, 15], where a combination of diffusion and convection-like contributions has been noticed. Acknowledgments Further refinements would require numerical calcula- We thank F. Ingremeau and H.A. Stone for stimulating tions to provide a full resolution of the gas flow and, discussions. F.B. thanks M. Decraene for her assistance. therefore, a prediction of the evaporation rate without This study has been carried out with a funding support unknown prefactors. In addition, it would be interesting by the ANR (ANR-11-BS04-0030-WAFPI project). to investigate also the case of drops with a non-zero contact angle to evaluate how the interface slope can modify the spatial extension of the intermediate zone. Further- References more, it would be interesting to revisit the case of vapor heavier than air, for which the flow is expected to be [1] B. E. Livingston and W. H. Brown. Relation of outwards above the evaporating disk. In such a case, the daily march of transpiration to variations in the we expect a significant effect of the geometrical config- water content of foliage leaves. Botanical Gazette, uration at the edge. If the disk is placed at height, a 53(4):309–330, 1912.

8 [2] N. Thomas and A. Ferguson. On evaporation from [16] S. Somasundaram, T. N. C. Anand, and S. Bakshi. a circular water surface. Philosophical Magazine Se- Evaporation-induced flow around a pendant droplet ries 6, 34(202):308–321, 1917. and its influence on evaporation. Physics of Fluids, 27(11), 2015. [3] N. Thomas and A. Ferguson. On the reduction of transpiration observations. Annals of Botany, [17] F. Boulogne, F. Ingremeau, and H. A. Stone. 31(122):241–255, 1917. Coffee-stain growth dynamics on dry and wet surfaces. Journal of Physics: Condensed Matter, The Lon- [4] H. Jeffreys. Some problems of evaporation. 29(7):074001, 2017. don, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 35(207):270–280, 1918. [18] P.L. Kelly-Zion, J. Batra, and C.J. Pursell. Corre- lation for the convective and diffusive evaporation [5] A.-M. Cazabat and G. Guéna. Evaporation of a sessile drop. International Journal of Heat and of macroscopic sessile droplets. Soft Matter, Mass Transfer, 64:278 – 285, 2013. 6(12):2591–2612, 2010.

[6] E. Bodenschatz, W. Pesch, and G. Ahlers. Re- [19] S. Dehaeck, A. Rednikov, and P. Colinet. Vapor- cent developments in Rayleigh-B´enardconvection. based interferometric measurement of local evapora- Annual Review of Fluid Mechanics, 32(1):709–778, tion rate and interfacial temperature of evaporating 2000. droplets. Langmuir, 30(8):2002–2008, 2014.

[7] A. F. Routh. Drying of thin colloidal ﬁlms. Reports [20] O. Carrier, N. Shahidzadeh-Bonn, R. Zargar, on Progress in Physics, 76(4):046603, 2013. M. Aytouna, M. Habibi, J. Eggers, and D. Bonn. Evaporation of water: evaporation rate and collec- [8] I. Langmuir. The evaporation of small spheres. tive eﬀects. Journal of Fluid Mechanics, 798:774– Phys. Rev., 12(5):368–370, 1918. 786, 2016.

[9] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, [21] X. Zhang, J. Wang, L. Bao, E. Dietrich, R. C. A. S. R. Nagel, and T. A. Witten. Capillary flow as the van der Veen, S. Peng, J. Friend, H. J. W. Zandvliet, cause of ring stains from dried liquid drops. Nature, L. Yeo, and D. Lohse. Mixed mode of dissolving 389(6653):827–829, 1997. immersed nanodroplets at a solid-water interface. Soft Matter, 11:1889–1900, 2015. [10] N. Shahidzadeh-Bonn, S. Rafa¨ı, A. Azouni, and D. Bonn. Evaporating droplets. Journal of Fluid [22] E. Dietrich, S. Wildeman, C.W. Visser, K. Hofhuis, Mechanics, 549:307–313, 2006. S. Kooij, H.J.W. Zandvliet, and D. Lohse. Role of natural convection in the dissolution of sessile [11] G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and droplets. Journal of Fluid Mechanics, 794:45–67, K. Sefiane. The strong influence of substrate con- 2016. ductivity on droplet evaporation. Journal of Fluid Mechanics, 623:329–351, 2009. [23] G. Laghezza, E. Dietrich, J.M. Yeomans, R.A. [12] B. M. Weon, J. H. Je, and C. Poulard. Convection- Ledesma-Aguilar, S. Kooij, H. Zandvliet, and enhanced water evaporation. AIP Advances, D. Lohse. Collective and convective effects compete 1(1):012102, 2011. in patterns of dissolving surface droplets. Soft Mat- ter, 12:5787–5796, 2016. [13] P.L. Kelly-Zion, C.J. Pursell, S. Vaidya, and J. Ba- tra. Evaporation of sessile drops under combined [24] O. R. Enr´ıquez,C. Sun, D. Lohse, A. Prosperetti, diffusion and natural convection. Colloids and Sur- and D. van der Meer. The quasi-static growth of faces A, 381(13):31 – 36, 2011. CO2 bubbles. Journal of Fluid Mechanics, 741, 2014. [14] P.L. Kelly-Zion, C.J. Pursell, N. Hasbamrer, B. Car- dozo, K. Gaughan, and K. Nickels. Vapor distribu- [25] A. Bejan. Heat Transfer. John Wiley & Sons, 1993. tion above an evaporating sessile drop. International Journal of Heat and Mass Transfer, 65:165 – 172, [26] K. Stewartson. On the free convection from a hor- 2013. izontal plate. Journal of Applied Mathematics and Physics, 9:276–282, 1958. [15] F. Carle, B. Sobac, and D. Brutin. Experimen- tal evidence of the atmospheric convective transport [27] Z. Rotem and L. Claassen. Natural convection above contribution to sessile droplet evaporation. Applied unconfined horizontal surfaces. Journal of Fluid Me- Physics Letters, 102(6):061603, 2013. chanics, 39:173–192, 10 1969.

9 [28] M. Zakerullah and J.A.D. Ackroyd. Laminar natural convection boundary layers on horizontal circular discs. Journal of Applied Mathematics and Physics, 30(3):427–435, 1979. [29] J.H. Merkin. Free convection above a heated horizontal circular disk. Zeitschrift fürangewandte Mathematik und Physik, 34(5):596–608, 1983. [30] N. Lebedev. Special Functions and their Applica- tions. Prentice-Hall, 1965. [31] B.M. Marder and N.R. Keltner. Heat flow from a disk by separation of variables. Numerical Heat Transfer, 4:485–497, 1981. [32] J.M. Stauber, S.K. Wilson, B.R. Duffy, and K. Sefi- ane. On the lifetimes of evaporating droplets. Jour- nal of Fluid Mechanics, 744, 2014.

[33] J. M. Stauber, S. K. Wilson, B. R. Duﬀy, and K. Se- ﬁane. On the lifetimes of evaporating droplets with related initial and receding contact angles. Physics of Fluids, 27(12), 2015. [34] R. Deegan, O. Bakajin, T. Dupont, G. Huber, S. Nagel, and T. Witten. Contact line deposits in an evaporating drop. Phys. Rev. E, 62:756–765, 2000. [35] Y. Popov. Evaporative deposition patterns: Spatial dimensions of the deposit. Phys. Rev. E, 71:036313, 2005.

[36] A. G. Mar´ın, H. Gelderblom, D. Lohse, and J. H. Snoeijer. Order-to-disorder transition in ring-shaped colloidal stains. Phys. Rev. Lett., 107:085502, 2011.

[37] L. G. Leal. Advanced Transport Phenomena. Cam- bridge University Press, 2007. [38] P.T. Tsilingiris. Thermophysical and transport properties of humid air at temperature range between 0 and 100 ◦C. Energy Conversion and Man- agement, 49(5):1098 – 1110, 2008. [39] R.M. Tennent. Science Data Book. Oliver and Boyd, 1971. [40] S. Grossmann and D. Lohse. Scaling in thermal convection: a unifying theory. Journal of Fluid Me- chanics, 407:27–56, 2000. [41] J. Crank. The Mathematics of Diﬀusion. Oxford, 1975.