PHYSICAL REVIEW E 87, 063015 (2013)

Thermally driven flows between a Leidenfrost solid and a ratchet surface

Steffen Hardt,1 Sudarshan Tiwari,2 and Tobias Baier1

1Center of Smart Interfaces, TU Darmstadt, Petersenstrasse 17, D-64287 Darmstadt, Germany2Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrodinger-Strasse, D-67663 Kaiserslautern, Germany

(Received 1 March 2013; published 24 June 2013)

The significance of thermally driven flows for the propulsion of Leidenfrost solids on a ratchet surface isstudied based on a numerical solution of the Boltzmann equation. The resulting flow patterns are dominated byvortices developing at the edges of the ratchet teeth. In a previous analysis it had been claimed that thermallydriven flows could cause the propulsion of Leidenfrost objects. In contrast to that analysis, it is found that suchflows make an insignificant contribution to the thrust of Leidenfrost solids on ratchet surfaces, which is dominatedby the pressure-driven flow due to the sublimating solid.

DOI: 10.1103/PhysRevE.87.063015 PACS number(s): 47.61.Cb, 47.45.−n, 47.15.G−

I. INTRODUCTION

Droplets deposited on a ratchet surface heated to temper-atures above the Leidenfrost point [1,2] exhibit a directedmotion, as was discovered by Linke et al. [3]. In correspondingexperiments a droplet hovers above a microstructured surfaceequipped with parallel, asymmetric grooves, resting on acushion of its own vapor. A droplet motion normal tothe orientation of the grooves is observed. Pinpointing themechanism of droplet propulsion is a quite challenging tasksince a variety of different effects come into consideration.The most prominent ones are viscous shear, net pressureforces due to the drop surface following the ratchet’s contour,thermocapillary flows, gradients in Laplace pressure, couplingof surface waves and droplet oscillations to the translationalmotion, recoil pressure due to phase change, and thermalcreep flow [3–6]. In that context, a major step forward wasthe observation that sublimating Leidenfrost solids such asdisks of dry ice are propelled on ratchet surfaces in a similarmanner as Leidenfrost droplets [4]. This excludes many ofthe effects listed above (such as thermocapillary flows) butstill does not allow us to clearly pinpoint the propulsionmechanism. In that context there are at least two differentpictures that have been suggested to explain the propulsion ofLeidenfrost solids. First of all, there are experimental hints thatviscous stresses originating from the flow of sublimating vaporguided over the topography of the microstructured surfaceresult in a significant net force on the solid hovering abovethe surface [5]. Alternatively, it was suggested that thermalcreep flow developing inside the narrow gap between theLeidenfrost solid and the surface could be the origin of thepropulsion mechanism [6]. These are two completely differentscenarios since the former picture suggests that the productionof vapor is the cause of the net force, whereas in the lattera motion could be induced even without any vapor influxinto the gap between the surfaces in situations where theirseparation is not maintained by a cushion of vapor due tosublimation. The purpose of this article is to examine thenature of thermally driven flows in that context and to quantifytheir contribution to the propulsion of Leidenfrost solids. Forthis reason the flux of vapor due to sublimation is initiallyneglected.

II. COMPUTATIONAL MODEL

The corresponding model geometry is shown in Fig. 1. Thegap between a ratchet surface and a planar surface is filledwith a gas. The ratchet surface is periodic in the x directionand translationally invariant in the y direction. Therefore, itis sufficient to consider a unit cell of extension L in the x

direction, as indicated by the dashed rectangle. Each of the twosurfaces is assumed to be isothermal, with the upper surfacerepresenting the bottom of the Leidenfrost solid and having atemperature of Tsubl and the lower one having a temperature ofT0. The model geometry of Fig. 1 closely resembles the surfacetopographies that have been used to experimentally study thepropulsion of Leidenfrost objects on ratchet surfaces.

Thermally driven gas flows are phenomena that cannotbe captured from first principles using the Navier-Stokesequations. Therefore, the Boltzmann equation

∂

∂tf + u · ∇rf = JHS [f ] (1)

is employed to model the transport processes occurring insidethe gas, where it is assumed that no external forces are actingon the gas molecules. The phase-space distribution function f

depends on (r,u), the position and velocity of a gas molecule.Collisions between the gas molecules are modeled by the hard-sphere collision integral JHS (see [7]). In [8] it was shown thatthe problem considered is insensitive to the specific choiceof the collision integral, with the variable hard-sphere termgiving very similar results.

The Boltzmann equation is solved using a Monte Carloscheme, a variant of the Direct Simulation Monte Carlo(DSMC) method [9,10]. For that purpose, the unit cell denotedby the rectangle in Fig. 1 is considered. Periodic boundaryconditions are assigned to the boundaries between two unitcells. At all solid surfaces the diffuse reflection boundarycondition with complete thermal accommodation is used[7]. The lower surface has a temperature of T0 = 723.15 K.The temperature of the upper surface corresponds to thesublimation temperature of dry ice at a pressure of 1 bar,i.e., Tsubl = 194.55 K. The key parameter determining the flowregime is the Knudsen number Kn, being the ratio of the meanfree path of the gas molecules and a characteristic length scale.The latter is chosen as B + H/2, and the former is of the order

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STEFFEN HARDT, SUDARSHAN TIWARI, AND TOBIAS BAIER PHYSICAL REVIEW E 87, 063015 (2013)

FIG. 1. (Color online) Model geometry considered for studyingthermally driven gas flows between a Leidenfrost solid and a ratchetsurface. The blue upper surface represents the bottom of the Leiden-frost solid, and the red lower surface represents the ratchet surface.The simulation domain is represented by a dashed green square.

of 65 nm for carbon dioxide at a mean temperature of 450 K anda pressure of 1 bar. In order to vary Kn, different initial densitiesare assigned in the simulations. While the model geometry isessentially two-dimensional, all three velocity components ofthe molecules are computed. A uniform Cartesian grid withquadratic cells is used for tracking the molecules. For Kn � 1a fixed cell size is chosen, whereas for Kn < 1 the cell sizeis proportional to the molecular mean free path. Simulationsare carried out initializing between 20 and 50 molecules percomputational cell using a constant time step. To obtain thevelocity field and the shear stress at the upper wall (theboundary of the Leidenfrost solid), averaging over at least105 time steps is performed. As far as the geometry of thecomputational domain is concerned, a situation characteristicfor experimental studies of Leidenfrost solids is considered.Specifically, L = 1500 μm, H = 200 μm, and B = 40 μm,which is referred to as “geometry 1.” The latter corresponds toa disc of dry ice with a diameter of 10 mm and a thickness ofabout 7 mm. As an alternative scenario, a ratchet geometrywith much shallower fins is considered. Specifically, L =1500 μm, H = 10 μm, and B = 40 μm, which is referred to as“geometry 2.” This serves the purpose of making a connectionwith publications in which small ratios of H/B are studiedtheoretically [6] and experimentally [11].

III. RESULTS AND DISCUSSION

In Fig. 2 the velocity vectors of the computed flow field in aregion around the tips of the ratchet are shown for geometry 1and two different Knudsen numbers, Kn = 0.06 and Kn = 1.Since the velocity scales are quite different for these twoKnudsen numbers, the vectors were rescaled appropriately forbetter visualization. There are two counterrotating vortices,a large one forming along the vertical wall and a small onealong the incline. When the Knudsen number decreases, thevortices shrink and move closer to the tip of the fin. Bycontrast, for larger Knudsen numbers the vortices expandand fill an increasing portion of the space between the twosurfaces. The CPU time requirements for simulations at smallKnudsen numbers are considerable since very small (comparedto the molecular velocity scale) velocities have to be extractedby averaging over a large ensemble of particles and a largenumber of time steps. Therefore, the smallest Knudsen numberthat could be considered with the available computational

FIG. 2. (Color online) Velocity vectors for geometry 1 in a regionaround the tips of the ratchet for (top) Kn = 0.06 and (bottom)Kn = 1. For better visualization, the vectors are scaled differently inboth cases.

resources while still keeping the statistical fluctuations at atolerable level was 0.015.

Apparently, the computed flow field is different from thethermally induced flow, as suggested by Wurger [6]. Wurger,referring to a seminal paper by Maxwell [12], suggested thatalong the inclined faces of the ratchet a thermal creep flowestablishes, having a velocity of

uc = 3

4ν∇||TT

, (2)

where ν is the kinematic viscosity of the gas, T is itstemperature, and the gradient is to be taken along the wall. Inhis analysis he reinterpreted the thermal creep flow boundarycondition, usually applied to nonisothermal surfaces, byevaluating the temperature gradient one mean free path awayfrom the wall. He concluded that this flow should drag theLeidenfrost solid along with it, leading to the experimentallyobserved direction of propulsion. The flow field displayed inFig. 2 does not correspond to such a picture. It is apparent thatthe flow along the incline is in a direction opposite to what isexpected from the arguments presented in [6].

The discrepancies between the results presented here andthose obtained in [6] can be explained as follows. Qualitatively,the thermally induced flow arises because the isotherms insidethe gas are not parallel to the wall boundaries. Moleculesimpinging onto the walls originating from a distance smallerthan their mean free path rarely experience intermediatecollisions with other gas molecules. Therefore, moleculescoming from hotter regions within the Knudsen layer transfera larger tangential momentum to the wall than those comingfrom colder regions, giving rise to a net tangential momentumtransfer. As a result, via Newton’s third law, the oppositemomentum is transferred to the gas, propelling it. Because ofthe orientation of these wall segments and their proximity tothe upper boundary (the Leidenfrost solid), the correspondingthermal driving force is strongest along the vertical segmentsof the ratchet geometry close to the edges. As a consequence,comparatively strong flows develop at the left of the ratchet

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FIG. 3. (Color online) Dimensionless average shear stress actingon the Leidenfrost solid as a function of Knudsen number in geometry1. The inset shows a blow up of the data points at small Knudsennumbers together with a power-law fit to the data.

teeth close to the edges, as visible in the top panel of Fig. 2.From Fig. 2 it is apparent that the corresponding recirculationzone will induce a shear force onto the Leidenfrost solid. Forthe situation considered here and at Kn = 0.06, the flowsalong the vertical wall segments are much stronger thanthose along the inclines of the ratchet, which are at thelevel of the numerical noise of the simulation. However, gaspropulsion along the vertical wall segments was neglectedin the mathematical model of Ref. [6], which explains thequalitative difference from the results presented here. Similarthermally induced flow patterns close to sharp edges have beenidentified before [13,14], referred to as “edge flow.”

To decide which effect this quite complex flow field has withrespect to propelling the Leidenfrost solid, the shear stress atthe upper surface needs to be evaluated. The vortices at the leftof the ratchet teeth produce a shear force onto the Leidenfrostsolid in the positive x direction, and those at the right producethe opposite. From the flow pattern of Fig. 2 it is expected thata net positive shear force is produced since the vortices at theleft are significantly stronger.

Figure 3 shows the nondimensionalized average shearstress τzx at the upper boundary as obtained from theMonte Carlo simulations. Nondimensionalization was donevia dividing by the average pressure evaluated at the upperboundary. Specifically,

τzx =∑

i

�p(i)x

/ ∑i

�p(i)z , (3)

where �p(i)x (�p(i)

z ) denotes the x momentum (z momentum)transferred by a particle onto the upper boundary in collision i.Typically, the summation runs over an ensemble of the orderof 109 collisions.

The shear stress shows the characteristic behavior oftransport processes predominantly occurring in the transitionflow regime, with a maximum around Kn = 1. For the chosengeometry at a pressure of 1 bar we have Kn ≈ 4.7 × 10−4,a value too small to consider in the framework of the

Monte Carlo method. However, taking into account thatlimKn→0 τzx = 0 [15], the average shear stress at very smallKnudsen numbers can be determined approximately byinterpolation. For that purpose, the data points for the sevensmallest Knudsen numbers shown in Fig. 3 were used to com-pute an interpolation curve going through (0,0), as displayedin the inset. From this curve a scaling of the shear stressτzx ∝ Kn1.71 is obtained. The scaling law allows us to computean approximate value of the thermally induced shear stresscorresponding to realistic situations studied in experiments.

At this point it needs to be mentioned that thermally drivenflows represent only one aspect of the fluid dynamic phenom-ena contributing to the propulsion of Leidenfrost solids on theratchet. In a separate article [16] it is shown that the pressure-driven flow due to the production of vapor from the solid givesrise to a net force that compares favorably with experimentalresults. Typical Reynolds numbers of the pressure-driven flowlie between 1 and 10. Reynolds numbers within that rangeare small enough to use the superposition principle, strictlyvalid only for Stokes flow [17], in an approximate manner.In other words, to a reasonable approximation the two flowphenomena, pressure-driven and thermally driven flow, may beconsidered independently, provided that the Reynolds numberof the thermally driven flow does not exceed that of the Stokesflow. To judge the significance of thermally driven transport,the corresponding shear force onto the solid may be comparedto that derived from pressure-driven flow. For a given ratchetgeometry, the relative importance of thermally driven transportdepends on the value of the parameter B. A rather thickplatelet of dry ice (diameter of 10 mm, thickness ≈ 7 mm), asconsidered in geometry 1, corresponds to B = 40 μm. If thethickness is reduced to about 1.7 mm, B increases to 100 μm.For the former case, the thermally induced shear stress amountsto about 1.4% of the stress due to the pressure-driven flow. Forthe latter case the relative importance of the thermally inducedstress is even smaller, amounting to about 0.3% of the valuedue to pressure gradients.

In contrast to the scenario considered in this work, insituations where no evaporation occurs, thermally driven flowscan be the main source of normal or tangential stresses actingon solid structures. In general, if the boundaries of twosolids at different temperatures are aligned with each otherand separated by a thin gas layer, the impacting moleculesgive rise to normal and tangential stresses between them[8,13,18–20]. These become relevant, for example, when aheated microcantilever approaches a solid surface.

In the present analysis of thermally driven flow, isothermalboundary conditions at the solid walls were assumed. Isother-mal walls correspond to an infinite thermal conductivity ofthe substrate, so the question of what will happen if realisticmaterial properties are considered arises. Corresponding ex-periments are usually performed on metal surfaces. Owing tothe finite thermal conductivity λ of the substrate, a temperaturegradient will form along the surface of the ratchet, with the tipsbeing colder than the troughs. This will give rise to thermalcreep flow in the classical sense, as described by Eq. (2). Inorder to estimate the magnitude of the temperature gradientalong the incline in the solid right at the boundary to the fluid,the conjugate heat transfer problem was solved using the finite-element software COMSOL MULTIPHYSICS. The heat conduction

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equation was solved inside the solid and inside the gas, whichwas carbon dioxide. The upper surface, the boundary of theLeidenfrost solid, was assumed to be isothermal, while asecond parallel isothermal surface at a distance of 2H + B

below the upper surface served as the lower boundary ofthe computational domain. The same geometry parameters(geometry 1) as before were considered. Even in the presenceof evaporation-driven flow, typical Peclet numbers are ofthe order of 1. Hence, since the goal is to provide anestimate of the corresponding creep flow, it is justified toneglect convective heat transfer. The problem was solved forboth a high thermal conductivity [copper, λ ≈ 370 W/m/K]and a low thermal conductivity metal [stainless steel, λ ≈30 W/m/K]. From the resulting temperature gradients alongthe inclines of the ratchet, the thermal creep velocity isobtained by means of Eq. (2). The latter can be translated into adimensionless shear stress acting on the Leidenfrost solid usingH as a length scale. For the Knudsen number corresponding to1 bar (Kn ≈ 4.7 × 10−4) the resulting shear stress values areabout 1.2 × 10−11 (copper) and 1.5 × 10−10 (steel). These aremuch smaller than the average dimensionless stress resultingfrom the thermally driven flows shown in Fig. 2, amounting toabout 2.72 × 10−8. Therefore, it follows that when taking intoaccount the finite thermal conductivity of the ratchet substrate,the resulting thermal creep flow can be neglected comparedto the thermally induced flow already present for isothermalboundaries and even more so compared to the pressure-driven flow originating from sublimation of the Leidenfrostsolid.

Up to now, in this work only situations with comparativelylarge ratchet teeth (H/B > 1) have been considered. Recently,the case of shallow teeth was studied [6,11], with the result thatin this situation a very pronounced propulsion of Leidenfrostobjects is observed. Specifically, in the work of Ok et al.[11] propulsion speeds of Leidenfrost droplets of up to about40 cm/s are reported. In that case the distance between thedroplet surface and the tips of the ratchet teeth (correspondingto the parameter B) was not measured. The ratchet teeth areonly shallow in the sense that the absolute values of H wereof the order of several micrometers in some cases; the ratioH/L, however, was comparable to that of geometry 1. Thereare indications that in many of these experiments H/B was ofthe order of 1 or even smaller than that. The indications comefrom the fact that local contacts between the droplet and thetips of the ratchet occur, as suggested in [11].

To study the thermally driven propulsion in the caseH/B < 1 the gas flow was simulated in a domain defined bygeometry 2, using the same boundary conditions as before.Knudsen numbers between 0.02 and 0.06 were considered.Each of the Monte Carlo simulations took about 1 weekon a 3.3 GHz CPU. As a result, the absolute values ofthe dimensionless shear stress are found to be about oneorder of magnitude smaller than in the case of geometry1. However, the net flow obtained with geometry 2 is sosmall that the shear stress data scatter around zero. Therefore,the conclusion is that the collective velocities of the thermalstress slip flow are too small to be extracted from the MonteCarlo simulations with statistical significance. This resultindicates that the thermally driven flows and correspondingshear stresses diminish when H/B � 1. This is in agreementwith the expectation that the effect should vanish in the limitH/B → 0. In view of the results obtained by Ok et al. [11] thisrules out thermally driven flows becoming large for H/B �1 as a potential cause for the observed large propulsionvelocities.

IV. CONCLUSIONS

It can be stated that thermally driven flows are insignificantfor the propulsion of Leidenfrost solids on ratchets; theyonly play a minor role when compared to the pressure-drivenflow due to the sublimation of the solid [16]. It should benoted that situations are conceivable (beyond the scope ofLeidenfrost phenomena) in which the upper wall in Fig. 1is simply a solid wall with no mass flux emerging from it.Such situations have been considered in [8]. Especially atlarge Knudsen numbers, specific choices of the wall boundaryconditions for the reflection of molecules result in substantialmomentum and mass fluxes between the two surfaces. Thosemay be utilized in energy-conversion applications or for theconstruction of novel micropumps. In that sense, while theproblem studied in this article is only of limited relevancefor Leidenfrost objects, it may be important in a differentcontext.

ACKNOWLEDGMENT

This work was partially supported by the German ResearchFoundation (DFG), Grant No. KL1105/17-1.

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