events before droplet splashing on a solid surface2 madhav mani, shreyas mandre and michael p....

Events Before Droplet Splashing on a Solid

Surface

Journal: Journal of Fluid Mechanics

Manuscript ID: draft

mss type: Standard

Date Submitted by the Author:

Complete List of Authors: Mani, Madhav; Harvard University, SEAS Mandre, Shreyas; Harvard University, SEAS Brenner, Michael; Harvard University, Engineering and Applied Sciences

Keyword: Collisions with walls/surfaces < Drops and Bubbles, Capillary flows < Interfacial Flows (free surface)

Under consideration for publication in J. Fluid Mech. 1

Events before droplet splashing on a solidsurface

Madhav Mani, Shreyas Mandre and Michael P. BrennerSchool of Engineering and Applied Sciences, Harvard University, Cambridge MA 02138

(Received 16 July 2009)

A high velocity impact between a liquid droplet and a solid surface produces a splash.Classical observations traced the origin of this splash to a thin sheet of fluid ejectednear the impact point, though the fluid mechanical mechanism leading to the sheetis not known. Mechanisms of sheet formation have heretofore relied on initial contactof the droplet and the surface. In this paper, we theoretically and numerically studythe events within 1µs of contact. The droplet initially tries to contact the substrateby either draining gas out of a thin layer or compressing it, with the local behaviourdescribed by a self similar solution of the governing equations. This similarity solution isnot asymptotically consistent: forces that were initially negligible become relevant anddramatically change the behaviour. Depending on the radius and impact velocity of thedroplet, we show that the solution is overtaken by either the surface tension of the liquid-gas interface or viscous forces in the liquid. At low impact velocities surface tension stopsthe droplet from impacting the surface, whereas at higher velocities viscous forces becomeimportant before surface tension. The ultimate dynamics of the interface in the viscousregime is not yet known.

1. IntroductionAt sufficiently high impact velocities, a droplet colliding with a solid surface produces

a splash. The commonality of splashing of droplets belies the fact that there is anotherpotential solution to the equations of fluid mechanics for the impact of droplets on asolid surface, that the droplet simply spreads smoothly outwards along the surface.

Why does splashing occur? In the 1960’s, Bowden and Field (1964); Jenkins and Booker(1960); Lesser (1981); Field, Dear, and Ogren (1989); Lesser and Field (1983) discoveredthat when a splash occurs, a thin fluid sheet is emitted very near the point where thedroplet contacts the solid surface, with the sheet subsequently breaking down into a sprayof droplets. Figure 1 shows a snapshot of a sheet (Xu, Zhang, and Nagel 2005), which hasbeen ejected into the gas before disintegrating into satellite drops; the sheet generationoccurs much earlier in the impact process. Although known for nearly half a century,thefluid mechanical origin of the thin sheet is not yet understood. On the experimental side,the events surrounding the sheet ejection itself occur so quickly and at such small lengthscales that they have thus far proven to be impossible to measure. On the theoreticalside, to our knowledge, there are not currently even credible estimates for even the basiclength scales and time scales at which the phenomena leading to sheet ejection occur.

Why are these sheets formed? It is often argued that thin sheet ejection is inevitablebecause of volume conservation: once the droplet impacts, the fluid that would havetraversed below the solid surface must be displaced. But fluid could just as easily bepushed upwards into the body of the droplet, instead of forming a thin fluid sheet.

Page 1 of 20

2 Madhav Mani, Shreyas Mandre and Michael P. Brenner

Figure 1. Sheet ejection and evolution thereof from a splash, taken from Xu et al. (2005),0.276ms after initial contact, for a 3.4mm diameter ethanol droplet hitting a solid surface atV0 = 3.74m/sec.

Furthermore, it does not give a mechanism for a sheet being propelled into the gas insteadof along the solid. Despite the lack of a mechanistic understanding, empirical studieshave well studied the conditions for sheet formation and splashing. There is a criticalvelocity , the “splash threshold”, above which splashing occurs; such thresholds (Reinand Delplanque 2008; Levin and Hobbs 1971; Stow and Hadfield 1981; Yarin and Weiss1995; Bussmann, Chandra, and Mostaghimi 2000; Deegan, Prunet, and Eggers 2008) havelong been known to depend on surface tension, density, liquid viscosity and the surfaceroughness. Surprisingly, recent experiments by Xu et al. (2005) demonstrated a regime inwhich the splash threshold also depends on the ambient gas pressure; below a thresholdgas pressure splashing is suppressed. Ultra high speed photography (Thoroddsen, Etoh,Takehara, Ootsuka, and Hatsuki 2005) gives a detailed picture of the dynamics of anentrapped air bubble between the liquid and solid surface, but is unable to resolve theearliest stages of the formation of the air bubble or sheet generation itself.

Current rationale for sheet formation heavily relies on the droplet physically contactingthe solid surface, and the associated contact singularity. The contact singularity arisesbecause of the following simple argument: after a time t, a droplet of radius R fallingat velocity V has penetrated the solid surface a distance V t. The radius of the wettedarea is rwet =

√2RV t, giving drwet/dt =

√RV/2t. As t → 0, the velocity at the edge

of the wetted area diverges. This is a real singularity due to the geometry of a parabolicsurface impacting a plane. One popular mechanism for sheet formation points out thatthe contact velocity is much larger than the liquid sound velocity; this means that thepressure disturbance from the collision of the droplet with the solid surface is initiallypinned to the contact line, so that there can be no (localized) distortion of the liquidsurface. Once drwet/dt slows below the liquid sound velocity, there is an abrupt increasein the pressure at the liquid gas interface; such an impulse could cause the generationof a liquid sheet (Lesser 1981; Haller, Poulikakos, Ventikos, and Monkewitz 2003; Rein1993).

The difficulty with this model is that observations show a liquid sheet launched intothe gas (Fig. 1), whereas the above arguments suggest that the sheet will be propelledalong the solid surface. Such a liquid film will experience enormous frictional forceswhich strongly resist deformation of the sheet off the solid surface. Viscous stresses inthe surrounding gas are necessarily smaller than those within the liquid, so it is hard tounderstand how the viscous response of the gas could counteract the friction with thesolid surface. This issue, coupled with the role of ambient gas pressure Xu et al. (2005),led us to the hypothesis that the liquid sheet might originate due to the interaction ofthe liquid with the intervening gas layer, before the droplet contacts the solid surfaceMandre et al. (2009).

Page 2 of 20

Events before droplet splashing on a solid surface 3

1 2 3

t = -0.1 µs

1 2 3

t = 0.0 µs

1 2 3

y (µ

m) t = 0.1 µs

1 2 3

t = 0.5 µs

0 1 2 3

-150 -100 -50 0 50 100 150

x (µm)

t = 0.6 µs

1

10t = -0.1 µs

1

10t = 0.0 µs

1

10

p (

atm

) t = 0.1 µs

1

10t = 0.5 µs

1

10

-150 -100 -50 0 50 100 150

x (µm)

t = 0.6 µs

Figure 2. Snapshots from simulations documenting the deformation of a droplet (R = 1.7×10−3

m; µg = 1.8 × 10−5 Pa s; V = 3.74 m/s; ρ` = 780 kg/m3) before contacting the solid surface.Time is measured relative to the moment of contact if no gaseous layer were present. Overtimescales of ∼ 0.1µs, left panel shows deformations in the interface of the droplet occurring onthe scales of µm. Depending on impact, fluid and gas parameters, we demonstrate that furtherphenomena set in when the gas gap is of order 10− 100nm. Right panel shows the build up ofthe gas pressure as the mnimum gap thickness decreases.

The goal of this investigation is therefore to study the approach of a droplet to asolid surface through an intervening gas layer, to search for potential fluid mechanicalmechanisms for sheet generation before contact. We demonstrate that a remarkably richcascade of events occurs well before contact with the solid surface occurs. Fig. 2 showssimulations, described in detail later in the paper, of a 1.7 mm radius ethanol droplethitting a solid surface with a speed of 3.74 m/s. On timescales of ∼ 0.1µsec and lengthscales of ∼ µm, deformations of the liquid interface occur. We will see that the behav-ior of the interface depends critically on both impact parameters (droplet radius andvelocity) as well as fluid and gas properties. Depending on parameters, important ef-fects include liquid inertia, liquid viscosity, interfacial surface tension, gas pressure, gasviscosity, compressibility of the gas, heat transfer in the gas, mass transfer between thegas and the liquid, and the mean free path of the gas. This work extends the results ofour recent Letter Mandre et al. (2009), which demonstrated that liquid surface tensionprohibits contact with the solid surface, causing the droplet to skate on a very thin gaslayer. We show that depending on the droplet radius and impact velocity, other physicaleffects–in particular, viscous forces in the liquid– can also stop the similarity solution,leading potentially to other qualitative behaviours, such as, hypothetically, sheets.

The organization of this paper is as follows. In the next section, we summarize ourmathematical model. We consider the full Navier Stokes equations for the flows in theliquid and the intervening gas layer; we then use the initial condition, in which the dropletis moving towards the solid at high velocity, to argue that the dominant forces initiallyare the inertia of the liquid, as well as compressible and viscous forces in the gas. Section3 discusses the very initial stages of the droplet’s interaction with the solid surface, whenthe liquid interface first deforms due to the pressure in the gas layer. We demonstratethat, depending on both the impact parameters of the droplet, and the properties ofthe liquid and the gas, the dynamics in the gas layer can either be compressible orincompressible; experiments (Xu et al. 2005) tend to occur in the regime where the gas isinitially compressible. Section 4 considers the subsequent evolution of the liquid interface

Page 3 of 20


after it initially deforms. The liquid layer attempts to contact the solid substrate byeither forcing the gas to drain out of the layer or compressing it, with the local behaviourdescribed by a self similar solution of the governing equations. The structure of thissimilarity solution is derived and compared with high resolution numerical simulations.Section 5 demonstrates that the similarity solution is not asymptotically consistent: thesolution causes forces that were initially deemed negligible to become important: thisincludes the surface tension of the liquid-gas interface, and viscous forces in the liquid.We report phase diagrams predicting the thickness of the gas film when each of theseforces becomes important, and indicate the dominant effect as a function of experimentalparameters. Surface tension dominates at low impact speeds (∼ 1m/sec) which we havepreviously demonstrated stops liquid-solid contact Mandre et al. (2009), causing thedroplet to skate along a thin film of gas; at higher impact speeds we show that viscousforces in the liquid become important before surface tension.

2. Mathematical modelWe consider the fluid mechanics of an initially spherical droplet of radius R approaching

a wall with speed V in a gaseous environment at pressure P0. As the droplet approachesthe wall, fluid forces build up in the gas film between the drop and the wall, and causethe droplet to deform. The principal goal of our analysis is to discover whether theseforces are capable of creating a thin fluid sheet. As emphasized in the introduction, thenumber of potentially relevant physical effects is quite large, and it is not clear a prioriwhich are the most relevant.

To discover the relevant effects we proceed in the following fashion: we first consider thefull equations of motion for both the droplet and the gas layer. Given the initial state ofthe droplet, some of the terms in the equation of motion are demonstrably unimportant.The major simplifications for a typical millimeter sized droplet with impact velocity oforder 1m/sec, are (i) slender geometry for the gas film, (ii) the neglect of viscous forcesin the liquid, (iii) the neglect of surface tension forces at the liquid-gas interface and (iv)the neglect of nonlinear inertia in the liquid. With respect to viscous forces, the liquidReynolds number is of order Re = ρ`RV/µ` = 1000, where µ`, ρ` are the liquid viscosityand density respectively; hence inertial forces in the fluid initially overwhelm viscousforces, and we therefore neglect viscosity. With respect to the relative sizes of inertialpressures to pressure variations caused by surface tension, the relevant dimensionlessnumber is the Weber number We = ρ`V

2/σ, where σ is the liquid surface tension. For atypical droplet in the splashing regime We ∼ 100 and hence surface tension is negligible.Finally, the nonlinear effect of inertia is initially small because before its interaction withthe wall, the velocity field inside the droplet is constant.

Using these approximations, we solve the equations of motion for the evolution of thedroplet as it impacts the wall. We then monitor the resulting solutions for the continuedvalidity of the approximations: for each of the major approximations made, we will seethat the approximation breaks down at some time before the droplet contacts the wall.The precise distance between the droplet and the wall at which this occurs dependson the parameter regime, as does the neglected force that first becomes important. Forexample at low impact velocities we will see that the neglect of surface tension is thefirst force to become important, whereas at higher impact velocities viscous stresses inthe liquid are the first. Once our computed solution is invalidated by the breakdownof an approximation, the continued evolution of the droplet impact requires a differentapproach.

It is worth remarking that in addition to the approximations listed above, there are a

Page 4 of 20


number of additional physical effects that we neglect in the present analysis that mightultimately prove to be important for splashing. These effects include: (i) heat transfer inthe gas, which affects the gas equation of state; (ii) non-continuum effects in the gas, thatbecome relevant when the mean free path of the gas molecule becomes a relevant lengthscale; (iii) mass diffusion from the gas into the liquid: we assume in our calculationsthat the gas layer cannot be absorbed into the liquid. The neglect of all of these effects isvalid in the regime we are calculating, though these approximations could well breakdownbefore contact, and hence be important for any putative sheet formation mechanism.

2.1. Dynamics in the Liquid

The aforementioned approximations imply that in the liquid, we need to study the incom-pressible Euler equations. At this point, we make one further approximation, this timefor computational convenience: we model the spherical droplet as a cylinder. Approxi-mating the dynamics as two dimensional allows us to employ complex analytic methods(e.g. see Smith et al. 2003) in solving for the inviscid dynamics in the liquid. The centralquantitative results of this paper are not affected by the two dimensional approximation:the similarity solution at the heart of this paper (dictating when the various neglectedfluid forces become important) applies quantitatively to both cylindrical and sphericalimpact. Subscripts ` and g refer to the variables in the drop and the gas respectively, andsubscripts x, y, ξ and t denote differentiation with respect to the respective coordinate.The two dimensional Euler equations for the velocity components u = (u`, v`)

ρ`ut + ∇p = −ρ`u ·∇u + µ`∇2u, ∇ · u = 0, (2.1)

where the terms on the right hand side of the first equation (fluid viscosity and thenonlinear effect of inertia) will be deemed negligible, as described above. Keeping accountof the terms assumed negligible allows us to later evaluate the validity of our assumptions.These equations can be rewritten in a computationally tractable form using complexanalytic methods: since ∇2p` ≈ 0, we can define its harmonic conjugate q` such that thefunction p` + iq` is holomorphic in the complex plane z = x+ iy. Dropping the subscriptfor now, Cauchy’s integral formula gives

d

dz(p` + iq`) =

12πi

∫ +∞

−∞

p`ξ(ξ, 0, t) + iq`ξ(ξ, 0, t)(ξ − z) dξ. (2.2)

Grouping the imaginary terms and evaluating them at the interface, y = 0 gives usan expression for the vertical pressure gradient at the liquid interface, in terms of thehorizontal pressure gradient evaluated at the interface. Namely, we have

p`y(x, 0, t) =1π−∫ +∞

−∞

pξ(ξ − x)

dξ (2.3)

where the principal value of the integral is considered on the right hand side. Recognizingthe right hand side of (2.3) as the Hilbert-transform of p`x and using equation (2.1) wearrive at an equation prescribing the acceleration of the interface subject to a horizontalpressure gradient at the interface

ρ`htt −H [p`x] = −ρ`(u`v`,x + v`v`,y) + µ`∇2v` − ρ`(u`hx)t, (2.4)

where we have made use of the kinematic boundary condition ∂h/∂t = v` − uhx andwritten in a form such that the terms on the right hand side will be deemed negligible.

Page 5 of 20


2.2. Dynamics in the gas and at the interfaceNow let us consider the dynamics in the gas. When the droplet is close to the wall, thevertical length scale (set by the distance from the droplet to the wall) is much smallerthan the horizontal length scale (set by the radius of curvature of the droplet). Massconservation then suggests that the horizontal flow velocities in the gas are much largerthan the vertical velocities. The gas follows its equations of state and can exhibit com-pressible behavior if the characteristic pressure scale exceeds the ambient gas pressure.The compressible lubrication equations are Taylor and Saffman (1957),

ρg,t + (ρgug)x + (ρgvg)y = 0, (2.5)−pg,x + µgug,yy = ρg(ug,t + ugug,x + vgug,y)− µgug,xx, (2.6)

where the right hand side of (2.6) is negligible for the dynamics under consideration.Since µl � µg, and also the scale for horizontal velocity is much larger in the gas thanin the drop, we employ a no-slip boundary condition on the interface,

ug = 0, vg = ht at y = h(x, t) and ug = vg = 0 at y = 0. (2.7)

The leading order solution to (2.6) is

ug = −pg,x2µg

y(h− y) + corrections, (2.8)

where the corrections are estimated by substituting the leading order solution back intothe right hand side of (2.6). Integrating (2.5) across the gap subject to (2.7) finally givesus the evolution equation for the gas

(ρgh)t − 112µg

(ρgh

3pg,x)x

=

−[ρg

(h5pg,xxx120µg

+h4pg,xhxx

48µg

)]x

−[ρ2g

(h5pg,xt120µ2

g

+h4pg,xht

48µ2g

)]x

,

(2.9)

where again the terms on the right hand side are the first order corrections to the com-pressible lubrication theory we employ. We do not use these terms except to later verifythe validity of our approximation. The first term on the right hand side comes from theviscous term in (2.6) and the second comes from the acceleration of the gas. Equations(2.4) and (2.9) are solved subject to the Laplace condition for pressure p` = pg+σhxx andthe equation of state for the gas pg = P0(ρg/ρ0)γ , where γ is a constant and models theheat transfer. We choose γ = 1 for an isothermal gas film, and γ = 1.4 for an adiabaticgas film.

3. Initial deformationTo gain intuition for the flows that are set up between the liquid droplet and the gas

layer, we first focus on the very initial stages of impact, when the droplet first deformsfrom its spherical initial shape, due to its interaction with the underlying gas layer. Figure2 shows subsequent snapshots in time of the interfacial and pressure profiles. The firstsnapshot is when the parabola first begins to deform away from its uniform curvaturestate, a unimodal profile for the pressure develops that signals the onset of lubricationforces in the gas. Later snapshots show clearly the development of a dimple at the originat a height H = H∗ from the solid surface which stays roughly constant.

What sets the thickness H∗ at which droplet deformation begins? This thickness is

Page 6 of 20


also what sets the height of the layer of gas trapped under the droplet. For the droplet todeform, the pressure in the gas must be sufficient to decelerate the falling liquid, locally,from velocity V to rest; that is, the gas pressure must balance the inertial pressure inthe droplet. Owing to its parabolic shape, the approaching droplet, in interaction withthe gas layer below, sets up a flow over a horizontal length scale L =

√RH in the gas.

We now carry out a simple scaling analysis of the equations of motions described above.The left hand side of the drop equation (2.4) gives the inertial pressure gradient in thedrop,

P`/L ∼ ρ`htt ∼ ρlV/τ (3.1)

where ht ∼ V is the velocity scale in the system and τ = H/V is the time scale overwhich the fluid is brought to rest. When the drop is sufficiently far from the solid surfacethe gas density is undisturbed, ρg ≈ ρ0, so that the gas pressure is set by incompressibleviscous drainage. Equation (2.9) then implies that

Pg/L ∼ µRV/LH2 (3.2)

The liquid interface distorts when the gas pressure is of order the inertial pressure,Pg ∼ P`. This balance gives an estimate for the height at which the dimple forms andthe pressure in the dimple

H∗Incomp = RSt2/3, P ∗Incomp =µV

RSt4/3, (3.3)

where St = µg/(ρ`V R) is the Stokes number, the ratio between the inertial forces in thedrop and viscous forces in the gas.

The observation that altering the ambient pressure alters the dynamics leads us toquestion whether compressible effects are important in the deformation of the dropletand its subsequent behaviour. Using the threshold values for splashing reported by Xuet al. (2005) (P0 = 30kPa) and the incompressible scalings derived above we can computethe dimple pressure, P ∗Incomp, that being, the value of the pressure at the dimple’s locationwhen the droplet first deviates from its circular state. We find P ∗Incomp = 700kPa. Thepressure required to produce a dimple is significantly greater than the typical thresholdambient pressure itself thereby making it apparent that the neglected compressible effectsmust be important. If the pressure in the gas, Pgas, is of order the ambient pressure, thenbefore the drop can deform, the underlying gas compresses. This transition happens whenPgas ∼ P0, or when

H∗ = R(µV/RP0)1/2 (3.4)

this critical height (indicated by a lower subscript star) marks the transition from in-compressible dynamics to a compressible one. Below this critical height viscous drainagecan be neglected and the gas equation (2.9) simplifies to

(ρgh)t ≈ 0 (3.5)

or ρgh = ρ0H∗ where ρ0 is the initial gas density. Balancing the gas pressure gradientwith the liquid deceleration as before we arrive at

H∗comp = RSt2/3ε2−γ2γ−1 where ε ≡ P0

(Rµ−1V 7ρ41)1/3

. (3.6)

This dimensionless parameter ε may be interpreted as the ratio of the ambient pressure(P0) to the pressure that would have built up in the film if the gas film were incompressible(P ∗Incomp from (3.3)); it determines the importance of compressible effects in the system

Page 7 of 20


compared to lubrication drainage. The height at which the dimple forms depends on thevalue of ambient pressure.

In order to verify the scaling predictions presented above we simulated the aboveequation for a range of St and ε, tracking the height H = H∗ at which the droplet’sinterface first deviates from a circular shape. To illustrate the numerical values of thedimple height in typical experiments, we carried out a series of simulations varying theambient pressure and the impact speed and tracking the height at which the dimple firstforms. Figure 3(a) shows H∗ in µm as a function of the ambient pressure P0 in kPa,for ethanol droplets identical to used in Xu et al. (2005). For sufficiently large P0, thedimple height is independent P0 in accordance with the incompressible scaling, whereas

for sufficiently small P0 the dimple height decreases scaling as P2−γ2γ−10 according to (3.6).

The value of pressure that demarcates this behaviour itself depends on the impact speed.Similarly figure 3(b) shows that the dimple height decreases with velocity as V −2/3 butis independent of the ambient pressure for impact speeds slower than a critical vlaue.Above this critical value, the power law changes to V −

16−7γ3(2γ−1) . The critical value of velocity

depends on the ambient pressure.All this data can be collapsed onto a single master curve by plotting H∗/RSt2/3 as

a function of ε as shown in figure 4 (reproduced from Mandre et al. (2009)). We showresults for γ = 1 and γ = 1.4 in this figure to demonstrate that the small and largeε asymptotes are in agreement with (3.3) and (3.6). Thus we conclude that ε capturesthe transition from an incompressible regime to a compressible one. The dynamics areincompressible for ε� 1 and compressible for ε� 1.

With the scalings derived and confirmed we rescale the governing equations for thedroplet (2.4) and gas (2.9) into dimensionless form. Barred variables refer to dimensionlessvariables and unbarred ones to their dimensional counterparts. Using

h = RSt2/3h, x = RSt1/3x, p` =µgV

RSt4/3p`, t =

RSt2/3

Vt, pg = P0p, ρg = ρ0ρ,

(3.7)

results in

htt − εH(px) = Re−1∇2v − St1/3(uvx + vvy) + εδH(hxxx), (3.8)

(ρh)t − ε

12(ρh3px)x = −εSt2/3

[ρ

(h5pxxx

120+h4hxxpx

48

)]x

− εRegSt2/3[ρ2

(h5pxt120

+h4htpx

48

)]x

,

(3.9)

p = ργ , (3.10)

where Re = ρ`V R/µl is the drop Reynolds number, Reg = ρ0RV/µg is the gas Reynoldsnumber, δ = σ/RP0 is the ratio of the Laplace pressure to gas pressure, and we havedropped the bars. For typical drop impacts St ≈ 10−5 , Re ≈ 103 and δ ≈ 10−4, hencethe terms on the right hand side are negligible, as we expected. Moreover, only onedimensionless parameters ε defined in (3.6) characterizes the behaviour of the solution,Hence, we begin by considering the evolution of the interface, and any dependence thatit might have on the ambient gas pressure, by considering the limit δ → 0, St → 0,RegSt2/3 → 0 and Re−1 → 0, and studying the ε dependence. Thus we initially focus

Page 8 of 20


10!2

100

102

104

106

10!2

10!1

100

101

P0 (kPa)

H* (

m)

Xu et al.

Xu et al. V=1.87 m/s

Xu et al. V=7.48 m/s

!

P0

(a)

101

100

101

102

100

102

V (m/s)

H* (

m)

Xu et al.: P0=30 kPa



(b)V

V

-2/3

-3

Figure 3. (a) Dimensional dimple height H∗ (in µm) as a function of the ambient pressureP0 (in kPa). Open circles correspond to Vimpact = 3.74 m/s, other data sets correspond toaltered values of impact velocity. The dimple height decreases as impact velocity is increasedand vice-versa. (b) Dimensional dimple height H∗ (in µm) as a function of the impact velocity V(in m/s). Open circles correspond to P0=30 kPa, other data sets correspond to altered values ofambient air-pressure. The dimple height increases with increasing ambient air-pressure, howeverare low velocities any dependence on air-pressure appears to be negligible. Other parametervalues are characteristic of experiments conducted by Xu et al. (2005) (Rdrop = 1.7 × 10−3 m;µgas = 1.8× 10−5 Pa s; ρdrop = 780 kg/m3).

10-1

10+0

10+1

10-2 10-1 10+0 10+1 10+2 10+3 10+4

H∗ /

RSt

2/3

ǫ−1

3.2ǫ1/3

22ǫ

4.3

St = 1/33

St = 1/53

St = 1/63

St = 1/103 Xu,et

.al,

regim

e1

Xu,et

.al,

regim

e2

Figure 4. Dimensionless dimple height: H = H∗/RSt2/3 versus ε for a range of St. Opensymbols denote γ = 1, while the corresponding filled symbols denote γ = 1.4. The data collapsesonto a single curve for a given value of γ, with asymptotes agreeing with the predictions. Thedata confirms that the dynamics are independent of the ambient pressure when the system isincompressible seen by the data asymptoting to a constant for large values of ε.

attention on the solution of (3.10), coupled with

htt = εH(px), (3.11)

(ρh)t − ε

12(ρh3px)x = 0, . (3.12)

4. Beyond the dimple: The Similarity SolutionWe now consider the evolution of the droplet interface after the formation of a dimple.

Figure 2 (ε = 5×10−3) shows that subsequent to dimple formation, the droplet interfaceand pressure profiles develop two symmetric kinks that move away from origin at aconstant speed, the location of the interface’s minima coincides with the maximum of the

Page 9 of 20


10-2

10-1

10+0

10+1

10+2

10+3

10+4

10+5

10-2 10-1 10+0 10+1 10+2

pm

ax

hmin

h−γmin

h−1/2min

5.1 × 10−2

10 −3

102

100

10-3

10-2

10-1

10+0

10-2 10-1 10+010-3

10-2

10-1

10+0

10-2 10-1 10+010-3

10-2

10-1

10+0

10-2 10-1 10+010-3

10-2

10-1

10+0

10-2 10-1 10+010-3

10-2

10-1

10+0

10-2 10-1 10+010-3

10-2

10-1

10+0

10-2 10-1 10+0

10-2

10-1

10+0

10+1

10-1 10+0 10+110-2

10-1

10+0

10+1

10-1 10+0 10+110-2

10-1

10+0

10+1

10-1 10+0 10+110-2

10-1

10+0

10+1

10-1 10+0 10+110-2

10-1

10+0

10+1

10-1 10+0 10+1

hm

inh

min

hm

inh

min

hm

in

hminhminhminhminhminhmin

h3/2minh3/2minh3/2minh3/2minh3/2minh3/2min

h1+γminh1+γminh1+γminh1+γminh1+γminh1+γmin

5.1× 10

−2

5.1× 10

−2

5.1× 10

−2

5.1× 10

−2

5.1× 10

−2

5.1× 10

−2

10−3

10−3

10−3

10−3

10−3

10−3

102

102

102

102

102

10210

010

010

010

010

010

0

ℓℓℓℓℓℓ

t0 − tt0 − tt0 − tt0 − tt0 − t

Figure 5. Maximum pressure pmax (left) and curvature lengthscale ` (right) as a function ofhmin shown in solid lines. Labels denote values of ε. The ambient pressure is subtracted frompmax for ε = 100. The dashed lines denote various power–law estimates (see text). The valueε = 5.1× 10−2 corresponds to figure 1 of Xu et al. (2005). Inset on the left shows hmin againstt0 − t, where t0 is the time to contact.

pressure profile. In an axisymmetric geometry, this suggests a circular rim moving awayfrom the origin. Accompanying this movement outwards, the minimum film thickness,hmin continues to decrease, and the pressure maximum, pmax, increases. The kink ischaracterized by a new horizontal length scale, `, which itself rapidly decreases as hmin →0, indicated by increasingly higher curvatures developing in figure 2. If we neglect forcesother than those considered in equations (3.10), we find that for every ε, the solutionforms a singularity in which the interfacial curvature and internal pressure diverges infinite time.

Figure 5 shows the evolutions of pmax and ` as a function of hmin, for a range ofdifferent values of 10−3 ≤ ε ≤ 102. Both the maximum pressure and ` eventually behaveas power laws, pmax ∼ h−βmin, ` ∼ hαmin, where the scaling exponents α, β show a strongdependence on ε. At large ε, the solution obeys β ≈ 0.5 and α ≈ 1.5; at the low εcharacteristic of experiments the behavior is much more complicated: for example at ε =0.001 the maximum pressure pmax does not even increase monotonically with decreasinghmin! Similarly at the ε of the Xu et al. (2005) experiment (ε = 0.051) the behavior iscomplicated with several apparent regimes. Below, we present an asymptotic descriptionof the solution that sorts out these different regimes.

We proceed by constructing a similarity solution for the behavior in the vicinity ofthe leftmost kink. In both regimes, the height and density fields are described by theself-similar ansatz

h(x, t) = hmin(t)H(η) ρ(x, t) = ρmax(t)R(η), p(x, t) = pmax(t)Π(η), (4.1)

where η = (x − x0(t))/`(t) is the similarity coordinate, with x0(t) being the time de-pendent location of the kink. Here ρmax is the time dependent maximum density. Beforewe plug this ansatz into the governing equations we can forsee that time derivatives ofvariables such as h(x, t) will have three contributions

∂

∂th(x, t) = hminH − η ˙(t)hmin

`(t)Hη − x0hmin

`Hη (4.2)

Corresponding to the decrease in the minimum height, a change in time of the charac-teristic lengthscale `(t) and an advective component corresponding to the movement inthe location of the kink away from the origin, respectively. We numerically evaluate therelative size of these terms in every regime we investigate and find that the advective

Page 10 of 20


term dominates in every case; i.e. it is at least as large as the other terms, if not larger ashmin → 0. When this term dominates, we hypothesize that the kink region is wave-likein character, that is, ∂t ≈ U∂x where we recognize that the appropriate scale for x is `(t)and that x0(t) = U , a constant for a given choice of parameters. The self-similar ansatzalong with the knowledge of the wave-like character of the solution can be plugged intothe governing equations (3.10), leading to the following

(x0)2hmin`2

Hηη = εhminρ

γmax

`H(Rγη) (4.3)

U

`ρmaxhmin(RH)η =

ε

12ρ1+γmaxh

3min

`2(RH3Rγη)η (4.4)

Motivated by the scalings derived above for the formation of the dimple, we expectthere to be two broad regimes: if the pressure is dominated by compression of the gas,then we expect that ρmaxhmin ∼ 1, so that pmax ∼ 1/hγmin. In the other regime, thepressure is dominated by viscous forces. These two regimes occur in taking differentlimits of the parameter of ε, with the ε� 1 corresponding to the incompressible/viscousdominated regime and ε� 1 corresponding to the compressible regime.

4.1. Compressible RegimeWe begin by considering the compressible regime where ε� 1. Motivated by the intuitionthat the density should increase because of the decreasing hmin (i.e. ρmaxhmin ∼ 1), weuse the ansatz

ρh = A(ε) + f(ε, t)F (η). (4.5)Here the first term represents the intuition, whereas the second term is a time depen-dent correction whose form we will determine. We require the correction boundary layerlike term to explain the non-monotonicity in the log-log plots of pmax vs hmin. The εdependence of the constant A is determined by the state of the gas when the similaritysolution sets in: as discussed above, the compressible behaviour sets in at the criticalheight H∗ = R(µV/RP0)1/2; this implies that A(ε) = Aε−1/2.

To proceed further, we first focus on the developing kink to the left of the origin inFig. 2. To the right of this kink, the density is high and the gas is compressible: to its leftof the kink the density relaxes to the ambient and the pressure is determined by viscousstresses. As mentioned previously the density maximum between these two regimes movesoutwards as it sharpens and it’s advection dominates the time derivatives in the system,that is, ∂t ≈ U∂x, where U = x0.

The advection approximation and the ansatz (4.5) in equation (3.10) gives

−Uf`Fη = A

ε1/2

12Pmaxh

2min

`2

[H2

(1Hγ

)η

]η

, (4.6)

which can be decomposed into a time–dependent scaling and the similarity profile as

Uf

`∼ ε1/2h2

min

Pmax`2

, Fη =A

12

[H2

(1Hγ

)η

]η

. (4.7)

Similarly the drop equation in (3.10) yields

Pmax`∼ ε−1U2hmin

`2, Hηη = H

[(1Hγ

)η

]. (4.8)

Solving the scalings from (4.5), (4.7) and (4.8) simultaneously for pmax, f and ` in terms

Page 11 of 20


0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

0.5

1

1.5

2

2.5

3x 10

7

X

PTime

!0.03 !0.02 !0.01 0 0.01 0.02 0.03 0.04

0.5

1

1.5

2

2.5

3

x 107

"

P/h

min

!!

(a1) (a2)

4.45 4.5 4.55 4.6 4.65 4.7 4.75

400

600

800

1000

1200

1400

X

P

Time

!2 !1 0 1 2

x 10!3

500

1000

1500

"

P/h

min

!1/2

(b1) (b2)

!0.08 !0.06 !0.04 !0.02 0 0.02 0.04 0.06 0.08

1

2

3

4

x 10!3

"

P/hmin

!1/2

3.5 4 4.5 5 5.5

1

2

3

4x 10

!3

X

P

Time

(c1) (c2)

Figure 6. Numerical results from the sub–compressible (a1-a2), super–compressible (b1-b2)and incompressible (c1-c2) regimes. (a1,b1,c1): Subsequent snapshots in time of the pressureprofile developing to the right of the origin. The maximum of the pressure profile can be seento be diverging while the characteristic lengthscale is tending to zero. (a2,b2,c2): Scaled plot ofthe profiles from the left panel. The pressures are scaled by the predicted scaling and expressed

as a function of η = X−X0(T )`(T )

instead of X. A single universal profile, P (η), can be seen giving

further evidence to the self-similar hypothesis and scalings predicted. The predictions for thevarious similarity laws are given in (4.9) for the top panel, (4.11) for the middle panel and (4.14)for the bottom panel.

of hmin leads to

Pmax ∼ ε−γ2 h−γmin, ` ∼ ε γ2−1U2h1+γ

min, f ∼ ε 32−γU−3h1−2γ

min . (4.9)

Surprisingly, this solution demonstrates that f grows in time; this implies that even-tually the intuited ρmaxhmin ∼ Aε−1/2 regime will give rise to another regime. Thecharacteristics of this regime can be found by repeating the analysis but neglecting theAε−1/2 term in Eqn. (4.5), so that ρh = f(ε, t)F (η). All three terms in the time derivativeexpression given in (4.2) are numerically observed to be equally dominant in this caseowing to ` ∝ t0 − t, where t0 is the time to contact. We then find that

Uρmaxhmin

`∼ εh3

minρmaxPmax`2

, Pmax ∼ ε−1U2hmin`

(4.10)

These balances give

Pmax ∼ ε−1U3/2h−1/2min ; ` ∼ U1/2h

3/2min. (4.11)

The similarity analysis thus predicts there are two distinct regimes of compressiblebehaviour: initially, the solution will follow the first regime, in which ρmaxhmin ∼ 1 and

Page 12 of 20


10!2

10!1

100

101

10!4

10!2

100

hmin

l/(!"/2!1!!0.3

hm

in

1+")

1

10!2

100

102

104

100

hmin

Pm

ax/(!!"/2h

min

!"

)

2

(a1) (a2)

10!1

100

101

102

10!1

100

hmin

l/h

min

3/2

!

1

10!1

100

101

10!1

100

hmin

Pm

ax/(!!1h

min

!1/2

)

1.4

(b1) (b2)

10!1

100

101

10!0.7

10!0.5

10!0.3

10!0.1

hmin

l/hm

in

3/2

0.8

10!1

100

101

10!0.8

10!0.5

10!0.2

100.1

hmin

(Pm

ax!

1)/

(!!

1h

min

!1/2

)

1.2

(c1) (c2)

Figure 7. Numerical results from the sub–compressible (a1-a2: γ = 1.4 ; ε ∈ [1×10−3, 1×10−6]),super-compressible (b1-b2: γ = 2 ; ε ∈ [1 × 10−2, 1 × 10−9]) and incompressible (c1-c2:ε ∈ [1×10, 1×104]) regimes. (a1,b1,c1): LogLog plot of the characteristic horizontal lengthscale,`, divided by its predicted scaling versus the minimum height, hmin for different ε. (a2,b2,c2):LogLog plot of the maximum pressure, pmax, divided by its predicted scaling versus the mini-mum height, hmin for different ε. For both dimensionless quantities the curves asymptote to auniversal constant indicating a verification of a universal self-similar law for both ` and pmax.

the scaling laws (4.9) are obeyed; we call this regime the sub–compressible regime. Belowa critical minimum film thickness this behaviour changes to that given by (4.11), thesuper–compressible regime. Indeed, evidence for this transition is shown clearly in Fig. 5:at the lowest ε, we see in the left panel that pmax ∼ h−γmin; near hmin = 0.002 the solutiontransitions to another behavior. At ε = 0.051, evidence is given for pmax ∼ 1/

√hmin.

Similarly, the right panel shows that ` ∼ h1+γmin, followed by a transition to ` ∼ h3/2

min.

4.2. Incompressible regime

We now consider the incompressible regime corresponding to large ambient pressures(ε >> 1). In this regime, the pressure in the gas is largely constant, close to its ambientvalue P0. Inhomogeneity in the gas’s pressure field arises from viscous forces. We thereforeuse the ansatz

pg = 1 + ε−1pg,1 (4.12)

Like the super–compressible case, all three terms in the time derivative expression in(4.2) are equally important. Substituting the pressure expansion into the scaled governing

Page 13 of 20


equations (2.4) and (2.9) gives us

htt = H[p1,x], ht = (h3p1,x)x (4.13)

which can be further simplified using the wave-like nature of the solutions leading to thefollowing scalings for the pressure, Pmax and lengthscale, `

Pmax ∼ U3/2h−1/2min , ` ∼ U1/2h

3/2min. (4.14)

These scalings are identical to those derived in the super–compressible regime, despitethe different physics. Figure 5 shows that at ε = 100, the maximum pressure pmax(defined here as the perturbation of the pressure from the ambient) obeys the law pmax ∼1/√hmin, whereas ` ∼ h3/2

min.

4.3. Numerical verification and transitionsThe prefactors for the aforementioned similarity solutions can be solved for explicitly byformulating and solving similarity equations, analogous to (4.8) for the sub-compressiblecase. The associated similarity equations are integro-differential equations, owing to theHilbert transform in the equation for the flow in the droplet; although such similarityequations have been solved before Cohen et al. (1991), we do not carry out such analysishere. There is a very real sense in which the most efficient way to solve the similarityequations is to conduct a careful study of the dynamics of the underlying nonlinear partialdifferential equations, as we have already done!

Figures 6 and 7 are verification of the scalings laws for the three aforementioned sim-ilarity regimes. Figure 6 demonstrates the collapse of the pressure profiles for differenttimes onto a single universal profile, depicting the validity of the self-similarity. Similarly,figure 7 show the collapse of Pmax and ` as a function of hmin for various ε, indicatingthat the leading order variation is ε is captured by the similarity solution. Moreover, theprefactors for the scaling laws (4.9), (4.11) and (4.14) can be read off of figure 7. Thecorresponding similarity scalings with prefactors read:

Sub–compressible: Pmax = 2.0ε−γ2 h−γmin ` = 1.0ε

γ2−1U2h1+γ

min (4.15)

Super-compressible: Pmax = 1.4ε−1U3/2h−1/2min ` = 1.0U1/2h

3/2min (4.16)

Incompressible: Pmax = P0 + 1.2ε−1U3/2h−1/2min ` = 0.8U1/2h

3/2min. (4.17)

Armed with the scaling laws and the prefactors, we are now in a position to theoreticallyaddress the transition between these similarity solutions. The transition between the sub–compressible and the super–compressible regimes is predicted to occur when Aε−1/2 ∼ f .We have tested this in a series of simulations with γ = 1.4: there, the transition shouldoccur when hsub−supermin ∼ ε0.6 and P sub−supermax ∼ ε−1.6 . Figure 8 verifies these predictedcrossovers from the sub–compressible regime over to the super–compressible regime andcompares the observed scaling laws with the ones predicted by the analysis. Figure 9shows the maximum pressure divided by its scaling in the super–compressible regimeto demonstrate the ultimate transition. The thinner curves are data from numericalsimulations of solutions that enter the super-compressible regime directly. The two thickcurves that stand out are results from simulations of solutions that initially start out inthe sub-compressible regime which then transition into the super-compressible regime.The plot illustrates that not only do the solutions transition to the super-compressibleregime, depicted by the thick curves asymptoting to a constant, but they asymptote tothe same constant as the solutions that start out in the super-compressible regime.

Once the transition to the ultimate asymptotic (super–compressible) regime occurs,

Page 14 of 20


10!4

10!3

105

!

Pm

ax

*

! -1.6

10!4

10!3

10!2

!

hm

in

*

0.6!

(a) (b)

Figure 8. Sub-compressible to super-compressible transition: Critical pressure (a) and minimumheight (b) at transition as a function of ε. Numerical data (solid) agrees with the predicted powerlaw (dashed line).

10!2

10!1

100

101

10!4

10!2

100

hmin

Pm

ax/(!!1h

min

!1/2

)

!=1" 10!2

; #=2.2

!=1" 10!3

; #=2.2

!=1" 10!4

; #=2.2

!=1" 10!6

; #=2.2

!=1" 10!7

; #=2.2

!=1" 10!9

; #=2.2

!=1"10!3

; #=1.4

!=1"10!4

; #=1.4

1.4

Figure 9. Numerical results of several super-compressible simulations and two sub-compressiblesolutions that transition and asymptote to the same constant as the ”pure” super-compressiblesolutions that start out in the super-compressible regime. The two sub-compressible solutionsstand out since the behaviour prior to transition is very different to all the other solutions inthe panel. Verification of the solutions asymptoting to the same constant lends further evidenceto the universality of the solutions found.

the solution loses all memory of the initial conditions from where they come. Figure9 shows numerical simulations of a variety of different situations reaching the super–compressible regime, with 10−9 ≤ ε ≤ 10−2 and different γ. In the figure, the thin curvesrepresent solutions that enter the super–compressible regime directly, whereas the thickcurves represent two simulations that start in the sub–compressible regime and thentransition into the super–compressible regime. Strikingly, all of the solutions asymptoteto the identical universal law, namely

Pmax =1.4

ε√hmin

; ` = h3/2min (4.18)

where the prefactor 1.4 and unity in the expression for the pressure maximum and criticallengthscale, respectively, are apparent universal constants.

5. The breakdown of the similarity solutionsThese similarity solutions appear to support the idea that liquid-solid contact occurs,

entraining a gaseous bubble underneath. However, the self consistency of the argumentrequires that we return and check that the series of approximations that we made at the

Page 15 of 20


beginning of this paper remain valid uniformly in time. A quick examination reveals thatit is impossible for the solution to be uniformly consistent: independent of the regime (ε)every solution eventually obeys ` ∼ h

3/2min, so the interfacial slopes diverge near contact.

But implicit in our derivation of the equation for the gas was lubrication theory, whichrequires that slopes remain small for all time. Additionally, we have neglected a numberof physical effects, whose neglect may be invalid as hmin → 0.

Here we use the universal scaling laws derived in the last section to quantitativelytrack the value of hmin the different effects can no longer be neglected and may causea breakdown of the similarity solution. In our previous Letter (Mandre et al. 2009) westudied the role of surface tension, which prohibits contact from occurring. Here we alsoinclude the departure from lubrication theory, viscous effects in the liquid and nonlinearinertia in the liquid. For each effect, we present contour plots for the critical heightwhere this effect becomes imoprtant as a function of P0, V,R. Underlying our analysisis a hypothesis that the parameter space can be divided into regions where particularphysical effects cause a breakdown of the similarity solution and result in a multitude ofqualitatively different dynamics.

5.1. Departure from lubrication theoryGoing back to (3.9), in the super–compressible regime the terms on the left hand side scaleas ρmaxhmin/` = ρmaxh

−1/2min . On the other hand, the first term on the right hand side

originating from the µgug,xx in (2.6) scales as εSt2/3ρmaxh5minpmax/`

3 = St2/3ρmaxh−3/2min .

Consequently as hmin → 0, separation of scales in the lubrication theory becomes invalidand the µgug,xx term may become as large as the µgug,yy term. This happens whenhmin ∼ O(St2/3), or dimensionally when hmin ∼ O(RSt4/3). This dimensional height, atwhich lubrication theory breaks down is shown in figure 10(a) as a function of the drop ra-dius and impact velocity. The other term on the right hand side of (3.9) is asymptoticallyconsistent.

5.2. Surface tensionThe terms on the left hand side of the drop equation (3.8) scale like hmin/`2 = h−2

min.The surface tension term εδH(hxxx) scales as εδh−7/2

min and can no longer be neglectedfor hmin ∼ O((εδ)2/3). Dimensionally, this amounts to a minimum film–thickness ofhmin ∼ O(RSt8/9We−2/3), which is plotted in figure 10(b) as a function of the drop radiusand impact velocity. Note that the film thickness where surface tension sets inbecomesimportant is independent of ambient pressure P0.

5.3. Viscous effect:

The first term on the right hand side of (3.8) scales as Re−1hmin/`3 = Re−1h

−7/2min . A

breakdown ensues when Re−1h−3/2min ∼ O(1), or when hmin ∼ O(Re−2/3). Dimensionally,

hmin = RSt2/3Re−2/3. This dimensional height is plotted in figure 10(c). Note that thefilm thickness where viscous forces becomes important is independent of ambient pressureP0.

5.4. Nonlinear inertia:The second term on the right hand side of (3.8) scales as St1/3h2

min/`3 = St1/3h−5/2

min .Breakdown occurs when hmin ∼ O(St2/3) or dimensionally when hmin ∼ O(RSt4/3).This is identical to the scale obtained for the breakdown of lubrication theory and theheight where this effect becomes important is shown in figure 10(a). Note that the film

Page 16 of 20


V (m/s)

R (

m)

5

1

0.25

0.1

0.05

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

x 10!3

V (m/s)

R (

m)

100

5

1

0.25

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

x 10!3

V (m/s)

R (

m)

50

10

2.5

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

x 10!3

Figure 10. Contour plot (in nm’s) of the height at which the neglect of (a) lubrication theory ornonlinear–inertial effects (b)surface tension, or (c) viscous effects in the liquid becomes invalid.The thick black curve partitions parameter space into an incompressible region to its left and acompressible one to its right.

thickness where nonlinear inertia becomes important is independent of ambient pressureP0.

6. Discussion and ConclusionThis paper has studied the dynamics of a droplet impacting a solid surface through

many different stages, all occurring before surface impact occurs. The air layer causesdeformation of the liquid surface, followed by the formation of a sharp kink on the surfacewhich propagates and attempts to contact the solid surface. However, before contact,other physical effects set in which change the dynamics. We have used the universalsimilarity solution for the solution of the kink to derive the critical height when threeimportant critical effects set in: surface tension, viscous forces in the liquid and nonlinearinertia in the liquid. We have demonstrated that each of these effects sets in at a criticalheight which depends on the droplet radius R and impact velocity V , but not the ambientpressure P0.

Figure 11 summarizes our results in a phase diagram of droplet velocity and radius. Wehave colored the phase diagram according to the physical effect that sets in at the largestgap thickness. At low impact velocities, surface tension becomes important first (shadedgrey), whereas at higher velocities viscous forces in the liquid dominate (shaded white).Our previous Letter has demonstrated that when surface tension becomes important,contact with the liquid surface is completely prohibited and instead the droplet skatesalong a thin air layer of thickness 2.54RSt8/9We−2/3. Here we see that this result isself–consistent at velocities below ∼ 1m/sec, with the precise value depending on dropletradius. Above the threshold depicted in Fig. 11, viscous forces in the liquid becomeimportant first and will modify the self similar dynamics before surface tension forces.The resulting dynamics of the liquid interface in this regime is not yet known. It isnoteworthy that the only role we have uncovered for the gas pressure P0 is its effect on thedimple height; neither the surface tension transition nor the viscous transition depends

Page 17 of 20


V (m/s)

R (

m)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

x 10!3

Viscous

Surface Tension

Figure 11. Phase diagram delineating which effect stops the similarity solution. In the shadedregion, surface tension becomes important first; we have already shown Mandre et al. (2009) thatthis causes the droplet to not impact the solid surface, skating along a thin gas layer. At highenough impact velocities, viscous forces in the liquid become important before surface tension(white region). What happens to the droplet in this region is under investigation.

on P0, and hence neither are sufficient to explain the pressure dependence observed inXu et al. (2005). There are a number of other physical effects that might also becomeimportant, including (i) heat transfer in the gas, which affects the gas equation of state;(ii) non-continuum effects in the gas, that become relevant when the mean free path ofthe gas molecule becomes a relevant length scale; and (iii) mass diffusion from the gasinto the liquid. Whether any of these effects conspire with surface tension and viscousforces in the liquid to give a mechanism of sheet formation remains to be seen.

7. AcknowledgmentsThis research was supported by the National Science Foundation through the Division

of Mathematical Sciences and the Harvard MRSEC.

Appendix. Numerical MethodThe numerical method of choice to solve these equations is dictated by the stiffness of

equations (3.11–refeqn:nondimgas). In the spirit of finite difference methods, we define adiscrete spatial grid defined by xj = x0 + j∆x, where x0 is the x–coordinate of the firstgrid point and ∆x is a constant grid spacing. The solution is approximated only at thesegrid points and at a times t0, t1, t2, · · · , tn, · · · and we use the notation φnj = φ(xj , tn)for arbitrary functions φ. For stability, an explicit finite–difference method for the dropequation (3.11) requires a time step ∆t ∝ ∆x3/2 for a given ∆x. Similarly, for thecompressible lubrication equation (3.12) the condition is ∆t ∝ ∆x2. Anticipating a smallspatial scale to develop as the solution evolves, a fine grid is required to resolve it. Thisimposes a very strict constraint on the allowable time–step and makes the computationprohibitively slow. Instead of explicit methods, we opt for semi–implicit schemes to tacklethe numerical instability and allow us to take arbitrary time steps. In particular, wesubstitute the equation of state (3.10) in the compressible lubrication equation (3.12)and at each time step solve for the change in the gas density ∆ρj = ρn+1

j − ρnj , given thechange ∆hj = hn+1

j −hnj in the film thickness. The lubrication equations is linearized for

Page 18 of 20


small ∆ρj to obtain a sparse linear system of equations,

∆ρj[hn+1j + (γ + 1)ρnγj (Rj+ 1

2+Rj− 1

2)]

−∆ρj−1(γ + 1)ρnγj−1Rj− 12−∆ρj+1(γ + 1)ρnγj+1Rj+ 1

2

= ρnj ∆hj + 2Rj+ 12

[ρn(γ+1)j+1 − ρn(γ+1)

j

]− 2Rj− 1

2

[ρn(γ+1)j − ρn(γ+1)

j−1

] (A 1)

where

Rj+ 12

=εγ∆t

12(γ + 1)∆x2

(1

hn3j+1

+1hn3j

)−1

(A 2)

At the boundary points ∆ρj is set to zero. These equations can be solved for ∆ρj inO(N) computation per time step.

Once ∆ρj is found, it is used to compute pn+1j to be substituted in the Hilbert transform

in (3.11) to advance h to the next time step. Although the Hilbert transform is linear,applying a semi–implicit method based on finite differences is not optimal because thelinear system generated to solve for ∆hj is not sparse and would require at least O(N2)computation. Instead, transforming to Fourier space as

hnj =x2

2+

N/2−1∑k=−N/2

hnk exp(

2πikxjL

), hnt,j =

N/2−1∑k=−N/2

vnk exp(

2πikxjL

), (A 3)

where L is the length of the domain, allows for the stiffness resulting from the surfacetension term to be integrated analytically. The Fourier coefficients satisfy

hn+1k = hnk cosωk∆t+

vnkωk

sinωk∆t+ pnk∆t2

2, k 6= 0

vn+1k = −hnkωk sinωk∆t+ vnk cosωk∆t+ pnk∆t, k 6= 0

hn+10 = hn0 + vn0 ∆t+ pn0

∆t2

2, vn+1

0 = vn0 + pn0 ∆t

(A 4)

where pnk denote the discrete Fourier transform of pnj as

pnj =N/2−1∑k=−N/2

png,k exp(

2πikxjL

)(A 5)

denotes the Fourier transform of the gas pressure and ωk =√

8π3σk3/ρlL3. The forwardand inverse Fourier transforms can be carried out in O(N logN) computation using fastFourier transform algorithms. Owing to the Fourier expansion, the interface perturbationsfrom a parabolic shape are strictly speaking periodic. However, we chose a large enoughsimulation domain to ensure that the perturbation decayed towards the boundary andthis periodicity does not affect the solution, as verified by doubling the domain length.

We implemented an adaptive time-stepping strategy to traverse any fast transientswithout losing accuracy. The error in each time step was estimated from a Richardsonextrapolation; i.e. by comparing the solution obtained by a time step ∆t with that ob-tained by taking two time steps of duration ∆t/2 each. If the maximum value of therelative error was larger than a preset tolerance (usually set to 10−3), the time step wasreduced by a factor of 2.

The spatial grid was also adapted depending on the current spatial scale of the solution.The spatial scale was estimated as the distance between two landmarks of the solution;examples of landmarks we used are the minimum of h, the maximum of p and the

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extrema of ht, etc. The choice of landmarks depended of the particular solution underconsideration. The number of points in the grid were doubled if the distance betweenthe chosen landmarks spanned less than a preset number (usually 40) of grid points. Therefined solution is found by interpolating the unrefined solution using cubic splines.

Each simulation is started with the initial condition

h0j = H0 +

x2j

2, htj = −1, ρj = 0, (A 6)

for sufficiently large H0. The particular H0 we used depended on the parameters andthe solution was accepted only if doubling the value of H0 yielded the same solution to3 significant digits.

REFERENCES

F.P. Bowden and J.E. Field. The deformation of solids by liquid impact, by solid impact andby shock. Proc. R. Soc. London Ser. A, 282:331–352, 1964.

M. Bussmann, S. Chandra, and J. Mostaghimi. Modeling the splash of a droplet impacting asolid surface. Phys. Fluids, 12:3121, 2000.

I. Cohen, M. P. Brenner, J. Eggers, and S. R. Nagel. Two fluid drop snap-off problem: Experi-ments and theory. J. Fluid Mech Phys Rev Lett, 83:1147, 1991.

R.D. Deegan, P. Prunet, and J. Eggers. Complexities of splashing. Nonlinearity, 21:C1–C11,2008.

J. E. Field, J. P. Dear, and J. E Ogren. The effects of target complance on liquid drop impact.J. Appl. Phys., 65:533–540, 1989.

K.K. Haller, D. Poulikakos, Y. Ventikos, and P. Monkewitz. Shock wave formation in dropletimpact on a rigid surface. J. Fluid. Mech., 490:1–14, 2003.

D. C. Jenkins and J. D. Booker. In Aerodynamic Capture Particles, pages 97–103. Ox-ford:Permagon, 1960.

M. B. Lesser. Analytic solutions of liquid-drop impact problems. Proc. R. Soc. Lond. A, 377:289–308, 1981.

M. B. Lesser and J. E. Field. The impact of compressible liquids. Ann. Rev. Fluid Mech., 15:97–122, 1983.

Z. Levin and P. V. Hobbs. Splashing of water drops on solid and wetted surfaces. Philos. Trans.R. Soc. London A, 269:555, 1971.

S. Mandre, M. Mani, and M. P. Brenner. Precursors to splashing of liquid droplets on a solidsurface. Phys. Rev. Lett., 102, 2009.

M. Rein. Phenomena of liquid drop impact on solid and liquid surfaces. Fluid Dyn. Res., 12:6193, 1993.

Martin Rein and J.P. Delplanque. The role of air entrainment on the outcome of drop impacton a solid surface. Acta. Mech., 201:105–118, 2008.

F. T. Smith, L. Li, and G. X. Wu. Air cushioning with a lubrication inviscid balance. J. FluidMech, pages 291–318, 2003.

C. D. Stow and M. G. Hadfield. An experimental investigation of fluid flow resulting from theimpact of a water drop with an unyielding dry surface. Proc. R. Soc. London A, 373:419,1981.

G. I. Taylor and P. G. Saffman. Effects of compressibility at low reynolds number. J. Aeronaut.Sci., 24:553, 1957.

S.T. Thoroddsen, T.G. Etoh, K. Takehara, N. Ootsuka, and Y. Hatsuki. The air bubble en-trapped under a drop impacting on a solid surface. J. Fluid Mech., 545:203–212, 2005.

L. Xu, W. W. Zhang, and S. R. Nagel. Drop splashing on a dry smooth surface. Phys. Rev.Lett., 94:184505, 2005.

A. L. Yarin and D. A. Weiss. Impact of drops on solid surfaces: self similar capillary waves andsplashing as a new type of kinetic discontinuity. J. Fluid Mech., 283:141, 1995.

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events before droplet splashing on a solid surface2 madhav mani, shreyas mandre and michael p....

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