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J. Non-Newtonian Fluid Mech. 165 (2010) 596–606

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

nfluence of yield stress on the fluid droplet impact control

lireza Saïdia, Céline Martina,b,∗, Albert Magninb

Laboratory of Pulp and Paper Science and Graphic Arts (LGP2), UMR 5518 CNRS - Grenoble Institute of Technology (Grenoble INP), 461 rue de la Papeterie,8402 St Martin d’Hères, FranceLaboratoire de Rhéologie, CNRS UMR 5520, Grenoble Institut National Polytechnique, Université Joseph Fourier Grenoble I, CNRS, BP 53, 38041 Grenoble Cedex 9, France

r t i c l e i n f o

rticle history:eceived 1 August 2009eceived in revised form9 November 2009ccepted 25 February 2010

eywords:ield stress fluidsrop

a b s t r a c t

The impact and the spreading of a drop of the yield stress fluid on a solid surface have been experimentallyinvestigated. A yield stress fluid chosen as a model fluid can shed some light on the manner in whichit is possible to control the impacted drop’s profile. Several gels based on polymer concentration wereprepared to obtain different levels of yield stress. Their shear rheological behaviours were characterizedand their flow behaviours were modeled using Herschell–Bulkley equation. Droplets were impacted ina wide range of velocities upon a dry and smooth polymethylmetacrylate substrate. Their dynamics onthe impacted surface were captured using a high-speed camera. The spreading and recoil of drops arestudied and their behaviour was compared to that of a Newtonian fluid at each impact velocity.

Influence of the yield stress level and intensity of inertia on the transient and final stages of drops

mpactpreading

impact were studied. It was shown how the increasing yield stress dictates the drop formation and alsoled to an emphasis of the inhibition of spreading and the weakening of retraction in the case of highinertial impacts. It was also noticed that the magnitude of the gravitational subsidence observed for thelow impact velocities, is governed by the initial non-spherical shape of droplets. Dimensionless numberswere defined in the case of yield stress fluids, allowing us to compare the effects of forces present in the

standce on

process and better undercharacterized. Its influen

. Introduction

As the impact and the spreading of fluids is of great importancen many industrial applications and everyday life activities, thishenomena has been investigated widely from both theoretical andxperimental points of view [1]. Although the materials involvedn industrial practices like spray-cooling, coating processes, spray-orming and printing, contain highly complex fluids from the pointf rheology, most studies concern the impact of Newtonian fluidsroplets on solid surfaces [2–8].

Indeed, very little is known about the effect of non-Newtonianroperties on the solid–liquid contact line dynamics [9]. How-ver, there is a growing interest in non-Newtonian liquids in thisesearch area. Earlier studies have focused on the control of droplet

eposition with the objective of limiting splashing and bouncing.hese two phenomena strongly limit the efficiency of drop depo-ition on non-wetting solid surfaces [9]. Polymer additives haveeen proposed to prevent droplet rebound by influencing the fluid

∗ Corresponding author at: Laboratory of Pulp and Paper Science and Graphic ArtsLGP2), UMR 5518 CNRS - Grenoble Institute of Technology (Grenoble INP), 461 ruee la Papeterie, 38402 St Martin d’Hères, France. Tel.: +33 476826918;ax: +33 476826933.

E-mail address: Celine.Martin@pagora.grenoble-inp.fr (C. Martin).

377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2010.02.020

the phenomena observed. Wall slip of gels on the PMMA substrate wasthe drop spreading has been discussed.

© 2010 Elsevier B.V. All rights reserved.

rheology. The presence of small amounts of flexible polymers suchas polyethylene oxide (PEO) and polyacrylamide (PAM) may clearlyinhibit the rebound and slow down the retraction of aqueous dropson non-wetting surfaces. Crooks et al. [10,11], Bergeron et al. [12]and Roux et al. [13] cited the elasticity and the non-Newtonianelongational viscosity �e of dilute polymer solutions to explain thedeceleration of the droplet retraction. Bartolo et al. [9] attributethe same phenomena to non-Newtonian normal stresses gener-ated near the moving contact line of the droplet. They showedthat the contact line dynamics is governed by the competitionbetween the surface tension that drives the retraction and elas-tic normal stresses generated by polymers, which counter it. Theuse of a viscoelastic surfactant system, is also effective in orderto prevent the rebound of fluid droplets on hydrophobic surfaces[14]. Increasing the surfactant concentration in solutions generatesenhanced reduction of the maximum height reached during retrac-tion, despite an equivalent maximum diameter achieved during thespreading stage. The inhibition of rebound seems to be related to arapid restructuration of the fluid after impact even though a loss offluid structure occurs during the spreading phase.

Among the non-Newtonian effects, the yield stress has beenrarely studied in this area. Nevertheless, many fluids of industrialsignificance have been shown to exhibit flow properties intermedi-ate between those of a solid and a liquid. When the applied stress isless than a certain critical value, the yield stress, such fluids do not

an Flu

flltifyotyyvptpseoeto

racytdihdwshaae

rfla

set

uoc

srsaeestes

2

2

e

noting that every rheometric characterization was carried on usingroughened tools to avoid slip.

Fig. 1 shows the flow curves of used gels. These steady flowcurves can be plotted based on the Herschell–Bulkley model (Eq.(1)). Thus shear yield stress, consistency, and power-law exponent

A. Saïdi et al. / J. Non-Newtoni

ow but deform. When the yield stress is exceeded, the fluid flowsike a truly viscous material with finite viscosity [15]. This impor-ant class of fluids encompasses quite a wide range of materialsncluding concentrated suspensions, pastes, foodstuffs, emulsions,oams, and composites. Nigen [16] has shown that the presence ofield stress at high impact velocities may lead to the suppressionf the drop retraction and the cover of a maximal surface. Never-heless the results of this study are limited to a single value of theield stress. Luu and Forterre have also investigated the impact of aield stress fluid in a recent work [17]. They observed an irreversibleiscoplastic coating for clay suspensions irrespective to the wettingroperties of the substrate. By contrast, for Carbopol gels, deforma-ions during the impact remain elastic even above the yield stress,rovoking an elastic spreading and recoil on the super-hydrophobicurfaces. A minimal model of inertial spreading, taking into accountlastoviscoplastic behaviour has also been proposed. However, allf these results have been obtained by neglecting the capillaryffects, suggesting that bulk rheological properties are controllinghe spreading issue. On the other hand, this study is mainly focusedn the mechanisms that control the maximal spread.

German and Bertola have conducted one of the latest explo-ation on the impact process of yield stress fluids [18]. This workttempts to compare the impact of glycerol drops of variable vis-osity, Xanthan shear-thinning drops and a commercial hair gelield stress drops. While the impact of shear-thinning drops seemso be similar to that of Newtonian drops, yield stress drops mayisplay central drop peaks at the end of inertial spreading. The

nfluence of yield stress magnitude on the inertial spreading stageas thus been established by measuring the size of these centralrop peaks. The peaks correspond to an undeformed inner regionhere shear stress effects are not strong enough to overcome yield

tress effects. However, this deduction cannot be established atigh impact velocities where drops completely deform and peaksre no longer observed. Once again, this work is largely devoted tonalysis of the spreading inertial phase and does not mention thelastic aspect of used polymer solutions.

In the present study, we have focused on yield stress fluids toeach, through the exploitation of the intrinsic properties of theuid, the control of final drop shape. For that, several Carbopol gelss a well-characterized model fluid are used.

The use of these elastoviscoplastic gels (without a large yieldtress magnitude) allows us to take into account rheological prop-rties as well as surface tension effects on the drop formation andhe impact process.

A surface tension relatively close to that of aqueous matrices lets establish a better comparison between the retracting behaviourf these gels and Newtonian fluids (as water or glycerol) widelyited in literature.

Through this study, recent results about the influence of yieldtress on the spreading inertial phase have been confirmed. Theole of the yield stress in drop formation and the influence of yieldtress magnitude and that of inertia on the final drop shape havelso been investigated. For a better comprehension of the phenom-na, both transient and final stages have been discussed through thevolution of dimensionless numbers defined in the case of a yieldtress fluid. A precise rheometric characterization was carried outo measure the volume properties and identify the interface prop-rties such as slip of gels on the solid substrate. The influence oflip effects on the drop impact process is discussed.

. Experimental set-up and test fluid

.1. Fluid model and characterization

The yield stress fluid used is a Carbopol gel. Piau [19] gives anxtensive study on this material. Carbopol neutralized physical gels

id Mech. 165 (2010) 596–606 597

which have been used in numerous applications over the past 40years, are inexpensive and easy to prepare. They are also quite pop-ular for researchers involved in rheology and non-Newtonian fluidmechanics. The reticulated polyacrylic acid resins are delivered inthe form of more or less agglomerated and poly-dispersed parti-cles. An acid pH is obtained when the resin is mixed with waterand the turbid dispersion hydrates slowly. When neutralized witha suitable base, the carboxylate groups ionize and, as a result ofmutual ionic repulsion, the molecules adopt a greatly expandedconfiguration [19].The characterization of yield stress fluids’ prop-erties should be undertaken with caution [20]. In particular, it iswell known that the yield stress fluid can slip on solid interfaces[19,21]. This slip occurs when the shear stress is close to the yieldvalue. This wall slip must be removed for the characterization ofvolume properties of the fluid. Specific rheometric procedures mustbe implemented [20]. On the other hand, the wall slip must becharacterized. Wall slip is more likely to exist during the spreadingand the retraction of the drop on the solid substrate. It depends onparticular conditions at the interface (roughness, hydrophobicity,etc.). In this section, the volume properties will be studied. The slipeffects will be characterized in Section 2.2 devoted to the substrate.

In simple shear, the viscoplastic behaviour of these gels is oftenrepresented by the Herschel–Bulkley viscoplastic model [22]:⎧⎪⎨⎪⎩

� = 2

(k(√

−4DII)n−1 + �o√

−4DII

)D if −�II > �2

0

D = 0 if −�II ≤ �20

(1)

with �0 being the yield stress, k the consistency index, n the power-law index, D the rate of strain tensor, � the extra stress tensor, II thesecond invariant of corresponding tensor.

For the preparation of fluids at different levels of yield stress,the correlation and scaling laws relating the variations of the yieldstress and the concentration of polymer established by Piau [19]were taken into account. Rheological properties of the preparedgels were obtained using an ARG rheometer (TA instruments) at25 ◦C. Cone-plate geometry with a 49 mm diameter and 4◦ 30 angleconfigurations was used. The method that we have adopted todetermine the yield stress is to extrapolate the flow curve in asteady state into low shear rates. In this case the yield stress valuedepends on the level of the lowest shear rates for which measure-ments have been done [20,21]. For a better accuracy, very low shearrates (10−4 s−1) were explored.

Dynamic oscillatory tests were conducted using plate-plategeometry with a 25 mm diameter and 1% strain rate. It is worth

Fig. 1. Flow curves of Carbopol gels at 25 ◦C modeled by Herschell–Bulkley equation(dotted curves).

598 A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606

Table 1Rheological parameters of prepared Carbopol gels.

Gel 1 2 3 4 5 6 7

�0 (Pa) 0.14 1.3 5.8 12 21.1 28.3 34.3

opst[

TseftmtCcb

cltwdaattsm

pawtaoaw

2

pstsavccPrOwlbog

k (Pa s−n) 0.5 1.2 3 5.5 9.1 10.8 12.8n 0.47 0.43 0.42 0.38 0.34 0.36 0.35G (Pa) 5.5 19 40 99.4 234 270 337

f each gel were identified (Table 1). Carbopol gels have more com-lex behaviours [20]. They exhibit an elasticity below the yieldtress [19,20]. Table 1 gives the values of shear elastic modulus inhe linear strain domain. These gels also show normal shear stresses21].

Yarin et al. [23] studied the elongational properties of these gels.he simple shear and uniaxial elongation agree fairly closely for thehear-thinning behaviour of the Herschell–Bulkley model. How-ver, the yield stress in uniaxial elongation has not been studiedor these gels. Recently, Tiwari et al. [24] found similar results tohose of Yarin et al. on suspensions of carbon nanotubes. For this

aterial, they show that the yield stress in elongation is higherhan in simple shear flow. The same study should be conducted onarbopol gels. Note that the Herschel–Bulkley model is limited toonditions of steady flow and does not represent the viscoelasticehaviour.

Due to the presence of yield stress, the static surface tensionould not be determined using classical methods. In the inner capil-ary method, the yield stress fluid does not rise into the tube [16]. Inhe case of the pendant drop method, break-up of the droplet occurshen the gravitational forces exceed surface tension. During therop formation of a yield stress fluid, the yield stress effects mustlso be taken into account. Indeed, for the high yield stress materi-ls, where a cylindrical aspect is obtained at the exit of the capillary,he surface tension effects are negligible [25]. Being located in aransitional phase of drop formation, where gravity, surface ten-ion and yield stress effects still play a role, classical calculationethods of surface tension do not appear to be appropriate.On the one hand, the Carbopol micro-gels used here do not

ossess surfactant properties and on the other hand, a very smallmount of polymer was used in the preparation of gels (≤0.14%eight concentration). The theoretical evaluation of the surface

ension of Carbopol gels (based on the surface tension of polyacryliccid and that of water) shows that the surface tension of gels is therder of 71 ± 2 mN/m. This value is relatively close to that of thequeous matrix. The surface tension of the aqueous matrix of gelsas therefore taken into consideration in the calculations.

.2. Substrate

Films of polymethylmethacrylate (PMMA) of 0.05 mm thicknessrovided by Goodfellow (as reference ME301200) were used as aolid substrate. PMMA is an amorphous, transparent and colourlesshermoplastic polymer. The average roughness of the used sub-trate is Ra ≈ 0.03 �m. This smooth substrate is rather hydrophobicnd allows a moderate contact angle [2] with drops of water. Thealues that we measured are the order of 81 ± 1.5◦ at 25 ◦C. Toharacterize the slip effects on the PMMA substrate, rheometricharacterizations were performed using a plane covered with theMMA film and a rough cone. Fig. 2 displays that gels 3 and 5 exhibit,espectively, yield stress values of 5.8 and 21.1 Pa on rough surfaces.nce rheometry carried out on a PMMA plane, filmgels begin to slip

−1

hen the shear rate falls below 10 s . The shear stress at the wall isower than the yield stress. These typical behaviours are discussedy Magnin and Piau [20] and Piau [19]. This suggests that slip effectsbserved in yield stress fluid flows [19,26] may affect the Carbopolel drop impact process at low impact velocities or during the final

Fig. 2. Flow curves of gels 3 (5.82 Pa of yield stress) and 5 (21.1 Pa of yield stress)obtained on a rough (sandpaper) and a PMMA substrate.

moments of the relaxation stage. As will be discussed later, someof our results verify this hypothesis.

2.3. Visualization system

We used a drop measurement device (DMD) type ejection sys-tem [27] to study the impact of fluid drops on a solid surface. Thisapparatus possessing a computer-controlled, motorized syringe,generates millimeter drops which are formed under their ownweight. A high-speed camera with 7 s recording frames, recordsup to 2000 frames/s with a resolution of 240 × 140 pixels. Imageanalysis software is used to calculate automatically contact angle,diameter and drop height during the impact process. Thus, thedynamic behaviour of droplets on a solid substrate has been char-acterized.

To ensure the symmetry of the droplets generated, non-bevelledneedles of 0.1 mm internal diameter were used. By modifying thefall height of drops, gel drops were impacted at five different veloc-ities: 0.3, 0.6, 1.4, 2 and 3 m/s.

The volume and the diameter of the drop are determined fromimages obtained before impact. The drop is assumed to be axisym-metric and composed of several superimposed disc layers havingeach a thickness of 1 pixel. The diameter of each disc correspondsto the distance between two points (X2i, X1i) forming the contourof the drop on the same line. By reasoning in pixels, the volume ofeach disc is:

˝i = �

4(X2,i − X1,i)

2 × 1 pixel (2)

And the volume of the drop is equal to:

˝ = �

4

n∑i=1

(X2,i − X1,i)2 × 1 pixel (3)

By using the calibration module of the system that calculates thenumber of pixels constituting 1 mm, it is also possible to determinethe volume in microliters. Once the volume of drops before impacthas been calculated, d0 the initial diameter associated with a sphereof same volume was determined and chosen to make the obtaineddata dimensionless. The equivalent diameter calculated for dropsof different gels were about 2 ± 0.2 mm. For a given yield stress, theerror in the determination of the drop diameter was about 3%.

To ensure a better accuracy, instead of using an average diameterfor each gel, the diameter associated with a single experiment wastaken into account to make dimensionless parameters and calculatethe dimensionless numbers.

A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606 599

Table 2Dimensionless numbers defined and written in the case of a yield stress fluid, where �: density, V: drop impact velocity, d: diameter of the drop before impact, �: liquidsurface tension, and g: gravity’s acceleration.

Dimensionless number Formula Corresponding ratio Dimensionless number Formula Corresponding ratio

Weber We = �d0V2

�Inertial effects

Capillary effects Oldroyd Od = �0K(V/d0)n

Yield stress effectsViscous effects

Reynolds Re = �dn0

V (2−n)

kInertial effectsViscous effects Ohnesorge Oh = KV (n−1)d

(1/2−n)0√

��Viscous effects√

Inertial effects√

Capillary effects

SCa SCa = d0�0 Yield stress effects SGr SGr = �0 Yield stress effects

C

3

weiOflSuw0eyi0bsficdco

4

wa

4

tagsaarisagtgdtl

dsa

Despite the time required for the formation and detachmentof gel drops, we have not recorded any specific oscillations onthe drops’ surfaces during the detachment, nor during the firstmoments of flight.

� Capillary effects

InSV InSV = �V (2−n/2)√�0k(1/d0)n

Inertial effects√Viscous effects

√Yield stress

. Dimensionless numbers

In studies on the fluid drop impact process, it is useful to reasonith dimensionless numbers that quantify the influence of differ-

nt forces present in the phenomena. The most commonly usedn the case of Newtonian fluids are the Weber, Reynolds, capillary,hnesorge, Bond and Froude numbers. In the case of yield stressuids, these numbers have been written and others were defined.ome of them used in this article are presented in Table 2. Val-es of different dimensionless numbers show that experimentsere conducted in a range of 3 ≤ We ≤ 250, 1.25 ≤ Re ≤ 530 and

.03 ≤ Oh ≤ 1.26. SCa numbers (comparing yield stress to capillaryffects) vary from 0.003 to 0.94, while SGr numbers (comparingield stress to gravity effects) alter from 0.007 to 1.7. These exper-ments cover effects occurring in a range of Oldroyd numbers from.008 to 0.44 at different impact velocities. Values of InSV num-ers (which compare inertial effects to those of viscosity and yieldtress) vary from 1.89 to 5806 at different impact velocities. Andnally CaSV numbers (comparing capillary effects to those of vis-osity and yield stress) vary from 0.47 to 41.4. Note that for therops of water and very low yield stress gels, the experiments werearried out below the value of the capillary length (�c =

√�/�g)

f aqueous solutions (≈2 mm).

. Experimental results

In this section, the influence of yield stress on the drop formationill be investigated. Then in the second part the influence of inertia

nd yield stress on the impact process will be characterized.

.1. Drop formation

In the yield stress range studied, the drop formation is con-rolled by a complex equilibrium between surface tension, gravitynd yield stress. The drop swells and falls under its own weight. Ineneral, when a pendant drop is formed at the end of a capillary, ittretches the filament of fluid connecting the drop to the capillarynd ultimately falls. This phenomena can be used to determine thepparent elongational viscosity of a fluid [28]. But in the yield stressange studied, the filament becomes thinner very quickly breakingmmediately thereafter (Fig. 3). According to Goldin et al. [29] diluteolutions of Carbopol, despite their yield stress, seem to have thebility to promote break-up. The relatively easy break-up of theseels could be explained by assuming that these solutions regainheir viscosity only slowly compared with the time scale of therowth of disturbances (in capillary jets). Furthermore, detachedrops present specifically sharp edges at their upper part just afterheir pinch-off, which indicates that the surface tension effects are

ocally greater compared to those of yield stress [16].

Although the drops break-up relatively easily, the time of gelrop formation is relatively long as compared to a Newtonian fluiduch as water for the same flow rate. Firstly, the thin threads, whichppeared just before the detachment, have a longer life than those

�gd0 Gravitational effects

aSV CaSV = �d(n−1)0√

kVn�0d0n

Capillary effects√Viscous effects

√Yield stress

of Newtonian liquids. The stretching motion during the formationof thread which causes an increase in viscosity due to structuralreorientation, may be the source of thread longevity [29]. Thisincreased viscosity takes a finite time to relax.

Secondly, the time required for the formation of a drop (dropswelling) varies between 30 s and 1 min depending on its yieldstress magnitude. As a matter of fact, the formation of drops andtheir pinch-off from the needle takes longer when the yield stressincreases. This is particularly noticeable when the yield stressreaches a value beyond 28 Pa. This could be attributed to therespective role of surface tension and yield stress during the dropformation process [25].

With characteristic values of d0 = 2 mm, � = 1000 kg/m3,� = 71 mN/m, numbers SCa and SGr can be written, respectively,as follows: (�0d0/�) ≈ (�0/35), (�0/�gd0) ≈ (�0/20). It shows thatbeyond 20 Pa, yield stress effects become predominant comparedto the effects of gravity and surface tension.

Fig. 3 depicts the detachment moment of a drop of the largestyield stress magnitude examined here. As can be seen, the gel dropof 34 Pa yield stress pinch-off from the needle without forming anoticeable filament. Accordingly, the formation of studied Carbopolgel drops shows no elongational characteristics. Beyond the yieldstress scale studied in this work, the almost cylindrical drops formquite a long filament before they pinch-off from the capillary. Alimit case can be calculated for the effects of very high yield stress.The length of the cylindrical drop is about:

l ≈ �0

�gd0(4)

Fig. 3. The formation of a gel drop with a yield stress of 34.3 Pa and its pinch-offfrom the needle.

600 A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606

F capill

mhbs

BiBiodstpy

dtvoBbIfivolm

tortoa

TV

ig. 4. Configuration of drops of different gels, 0.5 ms after their pinch-off from the

Fig. 4a shows the shape of drops 0.5 ms after their detach-ent from the needle. While drops of lower yield stress gels

ave an almost spherical shape, with rising yield stress, dropsecome increasingly elongated and take an increasingly prolatehape.

Furthermore, we recorded quite a surprising volume variation.y increasing yield stress and therefore a less spherical shape, it

s expected that the volume increases following the same trend.ut as Table 3 shows, the volume of the drop of 34 Pa gel reduces

n quite a spectacular manner. This fact is also easily noticeablen the pictures of drops in flight (Fig. 4). The SCa number value forrops of 34 Pa gel is approximately 0.94. This value shows that yieldtress effects are as important as those of surface tension duringhe drop formation process. One could suggest, this unexpectedhenomenon occurs when approaching a limit beyond which theield stress begins to govern the events.

Image analysis shows a change in the configuration of the dropsuring the flight as seen in the images of Fig. 4(b–f) which illus-rate the shape of drops 0.5 ms before impact. Indeed, at low impactelocities (0.3 and 0.6 m/s), drops of different gels preserve theirriginal form (acquired after their detachment from the capillary).ut with the increase of impact velocity (distance travelled by dropsefore impact), drops adopt a more spherical form during the flight.

t is noteworthy that drops of lower yield stress start to change con-guration at lower velocities. The situation intensifies to an impactelocity of 3 m/s. Drops all become almost spherical. Only the dropf the gel with a yield stress of 34 Pa escapes the rule. Despite itsess elongated shape, it preserves a non-spherical shape in the final

oments of its flight.Although air resistance may explain this change in configura-

ion, this phenomenon must be related to structural relaxation time

f Carbopol gels [29]. Actually, these fluids possess a structuralelaxation time during which the gel structure is reformed. It seemshat the time of flight gives enough time for gel drops to relax. Obvi-usly, the relaxation amount varies depending on the time of flightnd the yield stress magnitude. We will see later, how well these

able 3olume average of drops of different Carbopol gels.

�0 (Pa) 0.14 1.3 5.8V (�l) 4.426 ± 0.05 4.64 ± 0.07 4.44 ± 0.1

ary (a), and 0.5 ms before their impact on the substrate at different velocities (b–f).

initial drop shapes influence the outcome of events, particularly atlow impact velocities.

We wish to emphasize that the drop formation issue is not themain purpose of this study, and we will use these brief explanationsas an indicator to better understand the events that follow.

4.2. Impact process: influence of the intensity of inertia

To observe the events, the diameter (D) and height (H) ofdrops during the impact process were measured over time withan interval of 0.5 ms. The obtained data were subsequently madedimensionless by the following ratios:

D∗ = D

d0and H∗ = H

d0(5)

To ensure the reproducibility of the phenomena, tests on theformation and the impact and spreading process of gel drops wererepeated 9 times at two different speeds of acquisition (1000 and2000 fps). As a first step and with the objective of studying the effectof inertia, the dynamic evolution of gel drops at different impactvelocities was analyzed.

Fig. 5a shows a drop of Newtonian fluid such as water, impact-ing on a dry smooth solid surface at different velocities. It alwaysreaches the same final profile despite a distinct inertial spreadingstage. This is completely in agreement with the results cited forthe water or water/glycerin mixture in the literature [4,7]. Maxi-mum spreading diameter increases with inertial dominant effects(higher Weber number) and more oscillations are observed duringthe relaxation stage. Despite this behaviour observed at differentimpact velocities, the final shape is almost similar. This situationis completely different for a yield stress fluid. Although very low

yield stress gels behave almost like Newtonian fluids, beyond ayield stress of 5.8 Pa, final shapes of drops become dependent onthe impact velocity (Fig. 5b).

Here again, different maximum diameters are achieved at var-ious impact velocities and they increase with the Weber number.

12 21.1 28.3 34.34.57 ± 0.09 4.84 ± 0.11 5.27 ± 0.2 3.59 ± 0.08

A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606 601

Fdd

Brvadstistitm

4

s

immediately recovers to some extent its height. At this level, thecolumn of fluid formed on the substrate begins to subside under theeffect of stress generated by hydrostatic pressure [16]. The startingpoint of subsidence for each gel is highlighted by an arrow in Fig. 7b.

ig. 5. Dynamic behaviour of the dimensionless diameter of water drops (a), gelrops with a yield stress of 5.8 Pa (b) and 34.3 Pa (c) impacting a PMMA substrate atifferent Weber number values.

ut due to the presence of the yield stress and the weakness ofetraction, the final diameters become distinct at different impactelocities. So we can talk about the dependence of final diameter,s well as maximum diameter, on the impact velocity. The depen-ence of the final shape on inertial effects increases with the yieldtress magnitude. As it will be shown later, at a higher yield stress,he role of gel drops surface tension is increasingly weakened dur-ng the retraction stage. For yield stresses above 28 Pa, the yieldtress is so high that retraction is paralyzed almost instantly andhe spreading diameter is equivalent to the final diameter of thempacting drop (see Fig. 5c). Final diameters are completely dis-inct to each other, and their dependence on the impact velocity is

ore obvious.

.3. Impact process: influence of yield stress magnitude

The study of the dynamic evolution of drops of different gelshows that they behave distinctively at various impact velocities.

Fig. 6. Visualization of a gel drop with a yield stress of 28.3 Pa impacting a PMMAsubstrate at 0.3 m/s (We = 2.8).

Fig. 6 presents the visualization of a gel drop with a yield stressof 28.3 Pa impacting a PMMA film at 0.3 m/s (We ≈ 2.7). At this rel-atively low velocity, the oscillations observed in the spreading ofwater drops are suppressed upon the presence of any yield stressin the gels. The dynamic behaviour of gel drops of different valuesof yield stress is also presented in Fig. 7a and b.

As Fig. 7a shows, another behaviour is observed from ayield stress of 5.8 Pa. Referring to Fig. 4, at this relatively lowimpact (deposition) velocity, drops completely preserve their non-spherical and elongated shape that they had during their shortflight.

When the drop impacts the substrate, it deforms and a verylow lateral movement is observed. The drop deforms slightly fromthe bottom and while its lateral movement is obviously stopped, it

Fig. 7. Dynamic behaviour of dimensionless diameter (a) and dimensionless height(b) of gel drops with different values of yield stress impacting a PMMA substrate at0.3 m/s (We = 2.8).

602 A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606

Fig. 8. Dynamic behaviour of dimensionless diameter of gel drops with differentviw

Wwtc

db

wsnvragfiiath

tsIomDafiy

d

inertial dimensions of drops depend also on the level of the yieldstress.

By increasing yield stress, the inertial diameter decreases andthe inertial height increases. But we also noticed that despite a

alues of yield stress impacting a PMMA substrate at 0.6 m/s (a). Dimensionlessnertial and final configuration (D∗

max , H∗max, final D*, final H*) of Carbopol gel drops

ith different values of yield stress according to InSV number, We = 9.81 (b).

hile the deposition process appears to be controlled at this veryeak inertial regime, high columns of fluid (because of the ini-

ial prolate shape of drops) subside under their own weight and areeping movement induced by gravity is observed thereafter.

We will subsequently explain how this creeping flow variesepending on the initial shape of drops and could be influencedy slip effects at low shear rates.

By increasing the impact velocity to 0.6 m/s, while remainingithin a range of relatively low velocity (We ≈ 9.8) and the non-

pherical configuration of flight drops, some changes are worthoting. In fact, a fairly rapid inertial spreading is followed by aery slight retraction and thereafter very weak oscillations. Theetraction is almost non-existent for the gels of higher yield stressnd oscillations are suppressed (Fig. 8a). As the yield stress ofels exceeds 5.8 Pa, the values of the initial spreading and thenal diameters become equivalent, i.e. drops reach their final state

mmediately after the initial spreading. As the results show, leavingdeposit scheme and entering in a register of very low spreading,

he drops spread so that despite a slight rebound in the inertialeight, a subsidence under the effect of gravity no longer occurs.

At this impact velocity, except the inertial height and diameter,he final shape of deposited drops also depends on the level of yieldtress. Fig. 8b clearly illustrates this observation. By the decrease ofnSV number (weakness of inertial effects as compared to thosef yield stress and viscosity), maximum diameter decreases andinimum height increases leading to the reduction of spreading.espite a very low rebound in the height, the final diameter remainslmost equivalent to the maximum spreading diameter and the

nal shape of the drops prove to be completely dependant on theield stress level of each gel.

When increasing the impact velocity even more (1.4–3 m/s),rops spread to a maximum diameter then retract to their final

Fig. 9. Visualization of a Carbopol gel drop with a yield stress of 34.3 Pa impactinga PMMA substrate at 3 m/s (We = 240).

equilibrium state. Since a lamella is formed during the spreadingstage, we can speak with certainty about an inertial spreading. Itappears that the inertial forces have become large enough to over-come the yield stress and the consistency, and thus spread the drop(Fig. 9). As discussed in Section 4.1, by increasing impact velocity,drops of different gels (except that of Gel 7) appropriate a less pro-late configuration during the last moment of their flight. Despite anon-spherical shape before impact (Fig. 4), a peak [18] is no longerobserved during the inertial spreading stage of a gel drop with 34 Paof yield stress.

Indeed, at high impact velocities, as a drop of yield stress fluidspreads, its velocity reduces and the stresses decrease until fallingbelow their yield value. When the limit is reached, the spreadingstops suddenly [16]. As the curves of Fig. 10a show, for gels withhigher yield stress values, spreading occurs faster and the retractionis also inhibited in the same manner as the spreading. As has beenobserved in the case of lower velocities, at higher impact velocities,

Fig. 10. Dynamic behaviour of dimensionless diameter of gel drops with differentvalues of yield stress impacting a PMMA substrate at 3 m/s (a). Dimensionless inertialand final configuration (D∗

max , H∗max, final D*, final H*) of Carbopol gel drops with

different values of yield stress according to InSV number, We = 241 (b).

A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606 603

ing a

daiigfihnicert

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5

5

fiwialthb

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studied here is about Ks ∼ 4763. K number values belonging toexamined gel drop impact are well below the general critical valueof Ks. In other words, with such consistencies, impact velocities andsurface roughness, we are far from creating instabilities or evenany splashing phenomenon. Ensuring that we observe an impact

Fig. 11. Final shapes of drops of various gels impact

istinct inertial stage, the final shapes of drops of different gelspproach each other. The observed phenomenon intensifies withncreased velocity. This is explained by a difference in the retract-ng velocity and accordingly the retraction magnitude of differentel drops. Fig. 10b presents dimensionless maximum (inertial) andnal diameter and dimensionless minimum (inertial) and finaleight of different Carbopol gels impacting drops, regarding to InSVumber. Spreading diameters of drops decrease with weakened

nertial effects as compared to those of yield stress and viscosity. Itan be clearly noted that drops of various gels have different lev-ls of retraction depending on their yield stress. The decrease ofetraction with the increase of yield stress and the equivalency ofhe final shape of different gel drops can also be seen clearly.

Images of Fig. 11 display the final equilibrium state of differentel drops impacting the PMMA surface at various Weber values.lthough the results of the study are dimensionless, these imagesighlight the results and analysis on the final form of drops. Weotice that drops of each gel reach distinct final form at differenteber. This effect increases with the rising yield stress. By com-

aring the final shape of different gel drops at a given Weber, aistinction between their final forms is observed for the lowest

nertial regimes (among those studied here). At the higher Weberbeyond We = 57), the final form of drops of various gels becomeloser and closer with the increase of inertial effects.

. Discussion

.1. Impact scheme

At high impact velocities (We ∼ 250), some instabilities asnger-shape perturbations at the outer rim of the liquid lamellaere detected during the inertial spreading stage of a water drop

mpact (0.5–2.5 ms after impact). Inertial spreading is followed byviolent recoil before the drop relaxes by undergoing some oscil-

ations. This fingering phenomena [30] is also sometimes referredo as splashing [1] or as its beginning. The dimensionless K groupas been widely used in literature to characterize the transitionetween spreading and splashing:

= We Oh−2/5 (6)

here We and Oh are, respectively, Weber and Ohnesorge num-ers. Indeed there is a critical value of Kc, beyond which splashingccurs [31]. Although the presence of any yield stress leads to dis-ppearance of disturbances observed during the inertial spreading

PMMA substrate at different Weber number values.

of water drops, we wanted to verify the influence of the yield stresslevel on the limit between spreading and splashing.

The K number values were thus calculated for gel drops impact-ing the substrate at different inertial regimes. The value of the Knumber belonging to different gel drops alters between 2.3 and1009 at different impact velocities. The K number values belongingto drops of each gel regarding to InSV number evolution are pre-sented in Fig. 12. Indeed, at a given inertial regime, by the increaseof yield stress, K number and therefore the splashing probabilitydecreases.

These values are far below those evoked in literature. Accordingto Stow and Hadfield [32], at a roughness smaller than the thicknessof drop spreading lamella, the critical value of the K number for apolished surface (Ra ∼ 0.05 �m) is 3.4 × 109. Mundo et al. [31] alsopropose Kc = 1.1 × 107 for 2.5 < Ra < 78. Based on the expression ofthe critical value established by Cossali et al. [33]:

Ks = 649 + 3.76(

d0

Ra

)0.63

(7)

and taking into account the roughness of our substrate(Ra ∼ 0.03 �m) and an equivalent diameter (d0 ∼ 2 mm) for dropsof different gels, the critical value obtained for the entire system

Fig. 12. Evolution of K group values at different Weber (We ∼ 2.84, 9.85, 56.23,117.37, 252.6) according to InSV number.

6 an Flu

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04 A. Saïdi et al. / J. Non-Newtoni

attern, the influence of yield stress in inertial-gravitational andapillary-inertial regimes will be discussed thereafter.

.2. Inertial-gravitational regime

As mentioned in Section 4.1, prolate drops which are gentlyeposited on the substrate, subside under the hydrostatic pres-ure effect. To assess the magnitude of the creeping flow at lowmpact velocity, we have determined the diameter gained duringhe gravitational movement by the relation (8):

Df − Din

d0(8)

here Df is the final diameter and Din is the inertial spreading diam-ter. In the same way the height decreased during the subsidencehich can be assessed by the relation (9):

(Hs − Hf )d0

(9)

here Hs is the height of the column of the fluid just before subsi-ence and Hf is the final height.

The dimensionless diameter gained and the dimensionlesseight decreased according to the dimensionless number SGr (com-aring yield stress to gravitational effects) are presented in Fig. 13a.o calculate this number, we took into account the height ofeposited drops just before the start of the subsidence.

As it can be seen in Fig. 13a, the magnitude of the creeping move-ent totally depends on the initial shape of drops. Indeed, at this

ery low impact velocity, gel drops still have their original shapeefore impact. They are more elongated with the increase of yieldtress (Fig. 4) and constitute, therefore, higher fluid column, when

ig. 13. Dimensionless diameter gained and height lost by drops of different gelsuring the gravitational movement (a) and hydrostatic pressure generated by theuid column of drops of different gels according to SGr number (b), with Df the finaliameter, Din the inertial spreading diameter, Hs the height of the fluid column justefore subsidence and Hf its final height.

id Mech. 165 (2010) 596–606

deposited on the substrate. Once the subsidence takes place, themagnitude of the creeping movement increases with the columnheight of fluid. An exceptional case, however, is noticed. A drop ofthe Gel 7 due to the decrease in its volume (Table 3), once depositedon the substrate, forms a shorter fluid column. Hence, a decrease inthe magnitude of movement in diameter gained, as well as in heightlost, is observed for the drop of the gel with a yield stress of 34.3 Pa.

This observation may also be highlighted by the value of thestress due to the hydrostatic pressure (�gHs) generated by the fluidcolumn of each gel deposited on the substrate (Fig. 13b). Onlygel drops generating a subsidence are presented in this figure.The hydrostatic pressure of the fluid column with a yield stressof 34.3 Pa is equivalent to that of the fluid column with a yieldstress of 21.1 Pa, which results in an almost equivalent extent ofgravitational movement of drops of these two gels (Fig. 13a).

The extent of gravitational motion can also be accentuated bysome slip effects (Fig. 2) observed after subsidence occurs. An upperlimit of shear rate can be approximated by ((Dt − Dt−1)/ht) everysecond throughout the creeping motion.

Calculations show that shear rates are always below 20 S−1 fordrops of various gels. These values at the onset of subsidence caneven decrease to lower than 10 S−1 and this depends on the dropheight. Indeed, the higher the height of the deposited drops, thelower the value of the induced shear rate. Drops of greater heightsmay therefore slip more. Slip effects of Carbopol gels on PMMAseem to promote the creeping motion. It is no surprise that thedrop of Gel 6 with the greatest height has the largest gravitationalmovement.

Not only the drop subsides under hydrostatic pressure effects,but the gravitational movement proportional to the column heightof a deposited drop is also intensified by slip effects in low shearrates.

5.3. Capillary-inertial regime

As a matter of fact, when a single drop impacts a solid surface,the drop spreads until it reaches a maximum diameter, and thenit rapidly retracts to the region of initial impact to finally reachingan equilibrium state. The spreading and recoil of the drop repre-sents a continual trade-off between inertial effects (associated withthe mass or size of drops and its impact velocity), capillary effects(which depend on the surface tension and solid surface characteris-tics), viscous dissipation and in some cases gravitational forces [14].For the studied gels, the presence of yield stress should certainly betaken into account as an opposing effect in the expansion as wellas the retraction stage. As a matter of fact, the fluid is subjected toshear both in spreading and retraction [9].

The dimensionless inertial diameter and height of gel dropsimpacting the PMMA substrate at different Weber numbers,according to the InSV number, are presented in Fig. 14a and b. Sinceat given velocity, the initial diameter of the drop was the real vari-able parameter, an equivalent Weber (based on the average of num-bers calculated for each gel) was considered to facilitate the pre-sentations. According to Fig. 14, the maximum diameter decreasesand the minimum height reached at the end of the inertial stageincreases, with the reducing value of InSV number. According to thedefinition of InSV number (Table 2), these results mean the weak-ening of inertial driving effects compared to yield stress and consis-tency opposing effects, when the yield stress of the fluids increases.

On the same curves (Fig. 14), it is also possible to note the influ-ence of inertial effects. At given yield stress, by the extent of inertial

effects (growth of Weber values), the maximum diameter increasesand the minimum height decreases, meaning a greater spreading.A change in the slope of these curves can also be noticed. Declina-tion points of these curves at a given Weber are identical for thoseof inertial diameter and height evolution regarding to InSV num-

A. Saïdi et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 596–606 605

FCI

bctotrWftino

tytw

Ft

ig. 14. Dimensionless inertial diameter (a) and dimensionless inertial height (b) ofarbopol gel drops impacting the PMMA substrate at different Weber, according to

nSV number.

er. Taking into account these points and assuming that there is aritical InSV at which a regime change is observable, we were ableo correlate this critical value of InSV number with the evolutionf Weber number (Fig. 15). Indeed, with growth of inertial effects,he critical value of InSV increases. Comparing the yield stress cor-esponding to these critical InSV values, we noticed that for a given

eber there is a yield stress (among those studied in this work)rom which a significant resistance to the inertia is observable andhe spreading is quickly paralyzed. Obviously with the increase innertial effects (the Weber number), a higher value of yield stress isecessary to slow them down. The curves of the dynamic evolutionf impacting drops confirm this analysis.

While being aware of the elastoviscoplastic nature of used gels inhis study, it is extremely difficult to separate the respective role ofield stress and elasticity effects in this framework. Especially sincehe value of elastic modulus detected for different gels increasesith their yield stress magnitude (Table 1).

ig. 15. Established correlation between the assuming critical InSV and the evolu-ion of Weber number.

Fig. 16. Dimensionless retraction according to CaSV number of Carbopol gel dropsimpacting the PMMA substrate at different Weber number values.

In spite of elastic properties of Carbopol gels, the presence of sur-face tension effects and the use of a rather hydrophobic substrate,neither giant spreading nor recoil [17] have been observed. Even thepresence of elasticity, surface tension and substrate hydrophobiceffects should promote the retraction, but in practice the retrac-tion is increasingly inhibited. One might think that the coupling ofseveral fairly complex phenomena is the cause of such observation,especially since normal stress differences at the contact line [9] mayenhance this kind of dissipation during the retraction stage.

Although one cannot achieve decoupling of all these phenomenain the present investigation, but regarding to experimental resultsby the increase of yield stress, the retraction is increasingly inhib-ited (Fig. 10a and b). It suggests that when the decreasing shearstresses decrease lower than their yield value, the retraction slowsdown.

As regards the mechanism of retraction, in the literature, onetalks rather about the retraction velocity [9,12]. The decrease ofthe slope of dynamic evolution curves during the retraction stage(Fig. 10a); shows that the rise of yield stress leads to the reductionof retraction drop velocity. It may not be convenient to talk aboutthe extent of retraction in this type of study, but given the complex-ity of the behaviour of different gels during the retraction stage, wedecided to present the intensity of the retraction of drops of dif-ferent gels, in terms of CaSV number (Fig. 16). Retraction intensitymay be approximately evaluated by Eq. (10):

Dmax − Dfinal

d0(10)

where Dmax is the maximum diameter at the end of spreadingstage and Dfinal is the diameter gained after the relaxation stage.Of course only velocities generating a retraction are presented inthese curves. As it can be seen, decrease in the retraction only occurswith the reduction in the value of CaSV number. This illustratesthe weakening of driving capillary effects as compared to oppos-ing yield stress and consistency effects that counter the retraction.We will now be able to better explain why at high impact veloci-ties, drops of different gels resulted in a more or less similar finalshape. As it has been mentioned previously, the drop of a higheryield stress fluid reaches a lower maximum spread. And further-more, as it has been shown, the same drop loses much less of itsdiameter during the retraction stage. It seems to remain frozen atthe position gained during its spreading. It is therefore possible toreach a final diameter similar or even larger compared to a dropwith a lower yield stress (Fig. 10a and b).

As a matter of fact, viscous dissipation is proportional to theliquid viscosity because a droplet with higher viscosity mustovercome more dissipation energy during its spreading process.Accordingly, the final spread (diameter) for a given impact velocitydecreases inversely with an increase in the liquid viscosity [34]. At

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06 A. Saïdi et al. / J. Non-Newtoni

igh impact velocities, fluids having various levels of yield stresshow a completely distinct behaviour compared to Newtonian flu-ds with different viscosities. This is explained by the retractionxtent of different fluids drops. Thus significant inertial regimesllow us to highlight the presence of yield stress and its influencen the impact process.

. Conclusion

Through these series of experiments at low and high impactelocities, we observed the impact and spreading of fluids with dif-erent levels of yield stress. At low impact velocities (low inertia),he amplitude of the creeping movement is governed by the initialorm of deposited drops. The intensity of the hydrostatic pressureenerated by the column of fluid formed on the substrate governshis movement. Wall slip of the yield stress fluid on the solid sub-trate influences the flow of the drop when the shear rate is small.

In the case of high impact velocity (high inertia), we demon-trated that the presence of yield stress, causes the spreadingnhibition and the weakening of retraction. This phenomenon isccentuated with increasing yield stress.

The use of yield stress fluids can help to control and optimisehe profile of deposited fluid drops allowing us to cover a greaterurface while controlling the extent of spreading.

cknowledgments

The authors wish to thank the Grenoble Institute of TechnologyGrenoble INP) for supporting this research project and CTP (Centreechnique du Papier) for their collaboration and the use of the DMDpparatus. A great thank you also to Dr. F. Girard (of CTP) for herooperation, her contribution and inspiration during the tests.

ppendix A. Supplementary data

Supplementary data associated with this article can be found, inhe online version, at doi:10.1016/j.jnnfm.2010.02.020.

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