Eur. Phys. J. Special Topics 192, 163–173 (2011)c© EDP Sciences, Springer-Verlag 2011DOI: 10.1140/epjst/e2011-01371-9

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Threshold onset of Marangoni convectionin narrow channelsA. Mizev1,a, M. Denisova1, K. Kostarev1, R. Birikh1, and A. Viviani2

1 Institute of Continuous Media Mechanics, UB Russian Academy of SciencesAcad. Koroleva 1, 614013 Perm, Russia

2 Seconda Universita di Napoli, Dipartimento di Ingegneria Aerospaziale Meccanica,via Roma 29, 81031 Aversa, Italy

Received 21 January 2011 / Received in final form 25 January 2011Published online 9 March 2011

Abstract. There are several experimental studies where the Marangoniconvection begins only at a certain difference in the surface tension,i.e. in a threshold way. This effect contradicts a traditional point ofview according to which the surface flow in Newtonian fluids shouldbegin at an arbitrary small difference in surface tension. To explore thisphenomenon in detail we investigated the initiation of the Marangoniconvection at a free liquid surface caused by injection of a droplet ofsurfactant. It was found that the surface motion occurs in a thresholdmanner, i.e. when a surfactant concentration in the droplet approachesa certain critical value. The described phenomenon is more importantin narrow channels and essentially depends both on the purity of thebasic liquid and on the surfactant used. Based on the experimentalresults, a hypothesis about an important role of residual impuritiescontained in basic liquids which can thoroughly change a surface rhe-ology was suggested. The theoretical model taking into account specialrheological properties in the free surface is considered. The results of thenumerical simulation are in a good agreement with the experimentalobservations.

1 Introduction

The development of experimental facilities and techniques, related to the introductionof digital and computer technologies, has led to a disclosure of some phenomenain interfacial hydrodynamics, which contradict fixed notions. An example of suchdiscrepancy is the threshold development of the Marangoni convection taking place insurfactant solutions with small inclusions of gas or liquid [1] or in narrow channels [2],that as a rule was associated with the formation of a homogeneous diffusion layer of asurfactant along the entire interface. The growth of the initial surfactant concentrationin one of the contacting liquids results both in an intensification of mass transfer andin an increase of the surfactant concentration gradient normal to the surface thatprovokes Pearson instability. The development of the latter leads to a surfactantconcentration difference along the interface initiating a capillary motion [3–5]. On

a e-mail: alex mizev@icmm.ru

164 The European Physical Journal Special Topics

the other hand the threshold onset of Marangoni instability, i.e. when an initial non-uniform distribution of the surfactant takes place along the interface, has been foundin recent studies [6]. Since all the liquids used in the above mentioned experimentsare Newtonian, their abnormal behavior might be attributed to a surface rheologywhich can be different from the volume one. The appearance of special rheologicalproperties at the interface can be related to adsorbing of residual impurities from thecontacting liquids to the interface. Such adsorbed film is able to change the boundaryconditions at the interface. As a result it can give rise either to the impossibilityof a surface flow [7,8] or to a damping of intensity (so-called Plateau-Marangoni-Gibbs effect) [9,10] or a change in structure [11] of the already developed Marangoniflows.The paper presents the results of an experimental study of initiation of solutal

Marangoni convection caused by local injection of a surfactant on a free liquid surface.A theoretical model is proposed, based on special surface rheological properties, todescribe the observed phenomenon. The experimental and numerical data are com-pared, and potential applications of the obtained results to the thermocapillary caseare discussed.

2 Experiment

2.1 Experimental details

The simplest object used to study the influence of impurities on the onset ofMarangoni convection is a free horizontal surface of water. Owing to its relativelyhigh surface tension (∼ 72 dyne/cm) water surface creates favorable conditions foradsorption of the uncontrollable surface-active impurities contained in it. A certaindose of controllable surfactant locally injected on such a surface should induce asolutal Marangoni convection if the arising shear stress exceeds a certain criticalvalue. A local injection of a surfactant eliminates the problem of Pearson instabil-ity and the use of the horizontal surface allows to neglect the possible confrontationbetween the buoyancy and the capillary mechanisms of motion observed at curvedsurfaces. Moreover, the addition of a small amount of controllable surfactant to thebasic liquid (water) prior to injection of a droplet should make it possible to controlthe properties of the interface.In the present experiments we studied the process of initiation of a solutal

Marangoni convection caused by a local injection of a droplet of aqueous solutionof controllable surfactant with concentration Cd onto the free surface of a purewater or of an aqueous solution of this surfactant with lower concentration Cs.Monoatomic alcohols forming homologous series CnH2n+1OH act as the controllablesurfactant. These alcohols differ in density ρ and surface tension σ, which changein the range 0.78–0.80 g/cm3 and 22–25 dyne/cm respectively, but their surfaceactivity α = −∂σ/∂C increases by 3.2 times with every new hydrocarbon groupCH2 (Traube’s rule [12]).The cuvette filled with a basic liquid has the form of a vertical rectangular cavity

with dimensions of 9.0 × 4.0 cm2 and thickness b ranging from 0.12 to 2.0 cm. Theparallel glass walls of the cuvette form a working cell of the Fizeau interferometer tovisualize the surfactant distribution.The amount of uncontrollable impurities in basic liquid is specified both by the

degree of purification of the water and by the quality of cuvette preparation. Beforeeach test the cuvette was prepared according to the following method. The cuvettewas washed several times with concentrated detergent, after which it was rinsed foran hour with running water and for ten minutes with distillate. Deionized water (with

Interfacial Fluid Dynamics and Processes 165

ba

dc

Fig. 1. Evolution of the concentration field after injection of a droplet of isopropanol aqueoussolution with concentration Cd onto the free surface of water from air (Cd = 6%; the timeinterval after injection t, s: 0.2(a); 0.9 (b)) and from liquid phase (Cd = 12%, the timeinterval after injection t, s: 2.1 (c); 2.6 (d)). The cuvette thickness b = 0.12 cm.

electrical conductivity of 0.1μS) was used in the experiments as a basic liquid andfor the final rinsing of the cuvette (repeated several times). During tests the ambienttemperature and the liquid temperature were (23± 1) ◦C.The experiments were conducted in the following way. The preliminary prepared

(as described above) cuvette was filled with a basic liquid (distilled or deionized wateror alcohol aqueous solution). A mechanical pipette was used to form the dropletof a given alcohol concentration with fixed volume of 5μl with accuracy of 0.1μl.Then the droplet smoothly came into contact with a free surface of a basic liquid(Fig. 1a) or was injected into the liquid volume near its free surface (Fig. 1c). Ifthe injection of the droplet did not lead to the appearance of Marangoni motion thetest was repeated (in a newly cleaned cuvette) with a droplet of a higher alcoholconcentration. The value of the next concentration depended on the alcohol used andvaried in the range between 0.1% and 1.0%. The minimal alcohol concentration underwhich the development of the Marangoni convection was observed was taken as thecritical (threshold) concentration. More determinations of this threshold value wereobtained in order to define the experimental error (see the error bars in Figs. 2–7).All tests have been run just after the cuvette was filled with a basic liquid to avoidany additional contamination by impurities possibly coming from the air.

2.2 Influence of basic liquid properties and channel thickness

Figs. 1a-b present the situation in which a droplet of the isopropanol solution wasplaced on the free surface with the air. In this case, a drop hanging at the end of theneedle was placed slowly at the surface of the liquid. As soon as the drop touchedthe surface, it was sucked into the bulk of water under the action of capillary forcesminimizing the resulting total surface (Fig. 1a). As a result, the surfactant lying onthe droplet surface immediately landed on the surface of the liquid. The higher thesurfactant concentration in the droplet, the smaller the depth to which the drop sinks,and it was the larger the fraction of the drop that contributed to the formation ofthe capillary flow.When a droplet of the isopropanol solution was injected from the liquid phase

(Figs. 1c–d) it did not deform the free surface of the liquid, although the time ittook for the droplet to reach the interface as a buoyant jet (Fig. 1c) was rather long.Despite the fact that the whole bulk of the surfactant solution rose to the surface as

166 The European Physical Journal Special Topics

Fig. 2. Threshold isopropanol concentrationdifference ΔC = Cd − CS versus the initialcontent of isopropanol in the basic liquid fora droplet injection from the liquid phase andfor different degree of water purification: 1 –distillate, 2 – deionized water. The cuvettethickness b = 0.38 cm.

Fig. 3. Threshold isopropanol concentra-tion versus cuvette thickness for a dropletinjection from the gas phase for differentdegree of water purification: 1 – distillate,2 – deionized water.

a single portion it did not have its own surface. The dissolution in the surroundingliquid caused a decrease in concentration of the surfactant solution coming into con-tact with the liquid surface.The experiments showed that the initiation of a capillary motion on a horizontal

free surface of the basic liquid occurred in a threshold manner as in the case of curvedinterfaces [2,6]. Figure 1 presents a series of interferograms of the concentration fieldwhen the isopropanol concentration in the droplet is higher than the threshold valuefor a given thickness of the cuvette. Therefore, as soon as the drop reaches the freesurface, there appears an intensive but rapidly ceased capillary motion in the formof two cells (Figure 1b) or two convective cell chains (Figure 1d) whose occurrence isrelated to the simultaneous appearance of the alcohol onto the free surface in severalplaces (within the boundaries of the jet spreading under the surface).As might be expected, the value of the threshold concentration decreases as the

degree of preliminary water purification increases. Figure 2 presents the dependenceof the threshold concentration difference ΔC = Cd − Cs on the initial content ofisopropanol in the basic liquid for different degrees of water purification. One can seethat the use of deionized water instead of the distilled one reduces the value of thethreshold concentration by 50%.As is evident from Fig. 2, the value of the threshold essentially depends on the

content of the isopropanol in the surrounding water reaching a maximal value in purewater. The obtained results confirm the supposition that the initial adsorbed film atthe free surface is formed by the group of uncontrollable surface-active impurities.Most of them have the boiling point above 100 ◦C and therefore survive during waterdistillation. Half of residual impurities (group A) can be removed by means of an ion-exchange filter and the cleaning of the cuvette walls. The second half (group B) do nothave electrical charges and therefore stay in the solution. The co-linearity of curves 1and 2 in Fig. 2 means that the impurities of group A are not sensitive to changes inthe concentration of the alcohol in water and are held on the fluid surface as separatemolecules. By contrast, the impurities of group B form the so-called viscous-elasticfilm. The initial presence of alcohol molecules in a base fluid can gradually changethe rheological properties of the adsorbed film of impurities.

Interfacial Fluid Dynamics and Processes 167

h

Fig. 4. Threshold isopropanol concentration as a function of the meniscus form. Negativevalues of h correspond to the concave form of the menisci, positive values of h correspondto the convex form of the menisci. The cuvette thickness b = 0.38 cm.

Figure 2 presents the dependence of a threshold concentration of isopropanol inthe droplet on the cuvette thickness for two different degrees of water purification(Cs = 0). One can see that the critical content of the alcohol in the droplet essen-tially increases as the lateral dimension of the cuvette decreases. One of the possiblereasons of this behavior could be an interaction of the molecules of the impuritysubstance with the cuvette walls. Under small thicknesses the influence of the wallscan lead to a structurization of the molecules over the whole area of the adsorbedfilm even in the case of a small concentration of impurities. Such structurization canresult in the formation of a pseudo rigid layer precluding the Marangoni convectiondevelopment. The increase of the cuvette thickness reduces the wall influence. An-other possible explanation of the observed dependence of the critical concentrationon the cuvette thickness could be connected with menisci. Indeed, in a wide cuvette,where the threshold is lower, the relative area of meniscus zone is smaller then in anarrow one. To verify this assumption we performed the additional series of tests inwhich the contact line of the free water surface was fixed. This allowed us to changean overfilling of the cuvette varying the curvature of the water surface from concave(negative values of h in Fig. 4) to convex (positive values of h in Fig. 4) form, whereasthe distance between the walls was kept unchanged. One can see from Fig. 4 that amaximal critical concentration difference is obtained when the surface is flat (h = 0),i.e. when the lateral dimension of the surface is minimal.The results presented in Figs. 5 and 6 indicate that the threshold concentration

thoroughly depends on the direction of the droplet injection. Figure 5 shows that fordeionized water the value of the threshold decreases by 1.5 times in the case whenthe droplet is injected from the gas phase. By increasing the alcohol concentrationin the basic liquid the threshold concentration difference rapidly decreases, and thecurves for the different directions of surfactant injection converge until they coincideat Cs ∼ 4–6% and then tend to an asymptotic value, which in the examined intervalCs remains non-zero (ΔC ∼ 0.5%). It should be noted that an increase in the sur-factant concentration reduces the surface tension of the basic liquid. This explainsthe correlation between the observed tendency towards the reduction of the concen-tration threshold with increase of Cs and the results of investigations concerning thedevelopment of capillary convection on the free surface of gas bubbles in verticallystratified solutions of some surfactants [6].The direction of surfactant injection has a much greater effect on the development

of capillary motion (Fig. 6) when the cuvette thickness is changed and, therefore, of

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Fig. 5. Threshold isopropanol concentrationversus isopropanol content in the basic fluidin the case of drop injection from liquid (1 )and gas (2 ) phase (b = 0.38 cm).

Fig. 6. Threshold isopropanol concentrationversus cuvette thickness in the case of dropinjection from liquid (1 ) and gas (2 ) phase.The basic liquid is deionized water.

lateral dimension of the free surface. In particular, the injection of a droplet from thegas phase reduces the concentration threshold for Marangoni convection in deionizedwater approximately by 3 times for all cuvettes used in the tests.

2.3 Influence of surfactant properties

The above presented results imply that the value of the concentration threshold shouldbe defined by the value of the surface tension difference between the drop and thebasic liquid and also by the ability of the injected surfactant to affect the adsorbedfilm of impurities. To verify the influence of surfactant properties, a series of tests withan aqueous solution of methanol, ethanol, butanol, pentanol, hexanol as surfactants,were carried out. The surface activity of these alcohols increases as the alcohol numberin the homologous series increases, whereas a limiting solubility decreases with theincrease of the number of hydrocarbon groups in alcohol molecule.The value of the concentration threshold of amyl alcohol is plotted in Fig. 7

as a function of the cuvette thickness when the drop is placed on the surface of thedeionized water from the liquid and from the gas phase. A comparison of Figs. 5 and 7shows that an increase in the molecular mass of the alcohol reduces its thresholdconcentration value as well as the ability to affect the impurities from the groupB, because the threshold does not reach the asymptotic value. Figure 8 shows thedependence of the concentration threshold on the number of the hydrocarbon groupsin the molecule of alcohol. One can see that an increase in the surface activity togetherwith a decrease in its solubility in water lead to a reduction of the concentrationthreshold by a factor of ten. But on the other hand, the critical Marangoni numberfor different alcohols has weaker variations and it remains in the order of 107 for allalcohols (Fig. 9). The Marangoni number was calculated asMa = σ′ΔCd/ηD, whereσ′ = dσ/dC, ΔC is the difference in surfactant concentration in the injected dropand in the surrounding liquid, and d is the droplet size. This confirms once more thatthe phenomenon of threshold development of the Marangoni convection is generallydefined by the properties of the impurities, contained in the water, that form thesurface adsorbed film but does not depend on the method used to create a surfacetension gradient along the free surface.

Interfacial Fluid Dynamics and Processes 169

Fig. 7. Threshold pentanol concentrationdifference ΔC = Cd − CS versus cuvettethickness for the droplet injection from theliquid (1 ) and gas (2 ) phase.

Fig. 8. Threshold alcohol concentration ver-sus the number of hydrocarbon groups in analcohol molecule.

Fig. 9. Critical Marangoni number versus the number of hydrocarbon groups in an alcoholmolecule.

A chromatography analysis of the deionized water used in the experiments hasshown the presence of monophthalate, dibutylphthalate and tetradecane in very smallquantities of the order of nanograms per ml. To verify the importance of the impu-rities we repeated the experiment with the water containing monophthalate with aconcentration of approximately 1μg/ml. The threshold became two times higher.

3 Theory

Let us consider a two-dimensional problem on a solutal convection in a plane hori-zontal liquid layer with a free upper boundary. Initially a “cloudlet” of light solublesurfactant is situated under the liquid surface at a certain depth. Let us assumethat adsorption and desorption are the main mechanisms transferring the surfactantmolecules between the solution and the free surface which is considered as a separatesubphase with surface surfactant concentration Γ . The density of the surroundingliquid ρ = ρ0 (1− βC) and surface tension σ = σ0 − σ1Γ depend on the volumeconcentration C and the surface concentration Γ of the surfactant, respectively. The

170 The European Physical Journal Special Topics

experimental results described in the previous section show that the initiation of thesolutal Marangoni convection strongly depends on the shear stress caused by the sur-factant concentration gradient applied along the interface. The latter behaves as asolid boundary (infinite surface viscosity) remaining immovable until the shear stressexceeds a certain critical value. Such special surface rheological properties can bedescribed for example by the following equation:

ηS =η∞

exp {(P − P0)/a + 1} + η0. (1)

Here P = |∂u∂z+Ma · Sc−1 · dΓ

dx| is the absolute value of the shear stress at the free

surface, P0 is its critical value, a – the width of the region corresponding to thetransition of the surface viscosity from an extremely large value η∞ to a minimalone η0.The dimensionless equations of fluid motion (in terms of the stream function ψ,

vortex ϕ and surfactant concentration C) in Boussinesq approximation:

∂ϕ

∂t+∂ψ

∂z

∂ϕ

∂x− ∂ψ

∂x

∂ϕ

∂z= Δϕ−Gr∂C

∂x, Δψ = −ϕ

∂C

∂t+∂ψ

∂z

∂C

∂x− ∂ψ

∂x

∂C

∂z= Sc−1ΔC

Gr =gβC0h

3

ν2, Sc =

ν

D.

(2)

Here we define scales for length, time, stream function, volume and surface concen-tration as h, h2/ν, ν, C0 and C0h respectively, where h – liquid layer thickness,ν – kinematic viscosity, C0 – initial surfactant concentration in the “cloudlet”.The free surface is assumed to be undeformable (vz = 0) and all the boundaries

are impermeable for the surfactant:

z = 1 : ψ = 0

z = 0 : ψ = 0,∂ψ

∂z= 0,

∂C

∂z= 0

x = ±L : ψ = 0, ∂ψ

∂x= 0,

∂C

∂x= 0.

(3)

According to the above made assumptions, the equations of mass and momentumtransport between volume and surface subphase are:

z = 1 : −∂C∂z= KaC −KdΓ

∂Γ

∂t+

∂

∂x

(∂ψ

∂zΓ

)= Sc−1

(DS

∂2Γ

∂x2+KaC −KdΓ

)

BΓ

(∂2ψ

∂t∂z+∂ψ

∂z

∂2ψ

∂x∂z

)= ηS

∂3ψ

∂x2∂z− ∂2ψ

∂z2− Sc−1MaS

∂Γ

∂x

Ka =k1h

D, Kd =

k−1h2

D, B =

C0

ρ, MaS =

σ1C0h2

ηD.

(4)

Interfacial Fluid Dynamics and Processes 171

Fig. 10. Variation of a maximal stream function value with time for different boundaryconditions at the free surface: free surface with Newtonian rheology properties 1 – Ma = 0,2 –Ma = 105, 3 –Ma = 107, 4 - solid surface; free surface with yield strength 5 –Ma = 105,6 – Ma = 107.

Here k1 and Ka – dimensional and nondimensional adsorption coefficient, k−1 andKd – dimensional and nondimensional desorption coefficient, Mas – “surface”Marangoni number defined with σ1 = −∂σ/∂Γ .The first of the above equations (4) states a balance between the diffusion flux

of the surfactant to the surface and the flux carried to the surface phase due toadsorption-desorption process. The second equation describes the time variation ofthe surface concentration at the point of the surface phase caused by its motion,surface diffusion and adsorption-desorption process. The third expression is a dynamicequation, which describes the motion of the surface phase element taking into accountthe finiteness of its mass. The first term in the right-hand side is related to the surfaceviscosity, the second term gives the value of viscous stresses acting from the near-surface layers of the fluid and finally the last term specifies the magnitude of thesolutocapillary force.All calculations were made for the following parameters: Sc = 1500, Gr =

100, Ka = 103, Kd = 10

4. The non-stationary boundary value problem is solvedby the finite difference method on the 80× 160÷ 160× 320 square meshes using theimplicit Crank-Nicolson scheme. The solution of the Poisson equation is found usingthe successive over-relaxation method.Initially the fluid is assumed to be in a quiescent state (ψ = 0, ϕ = 0) and the

surfactant in the fluid is absent (C = 0) except in the circle area with coordinates ofthe centre (0, 0.5), where the surfactant is distributed uniformly. The “cloudlet” comesto the surface forming there a surfactant concentration gradient. Further evolutiondepends on the boundary conditions at the free surface. Three situations have beenexamined:

- free surface with Newtonian rheology properties;- solid surface;- free surface with rheology properties described by (1) for η∞ = 10, η0 = 1,a = 4, P0 = 100.

Figure 10 presents the variation with time of the maximal value of the stream func-tion for different boundary conditions at the free surface for different Marangoninumbers. The curves 1, 2 and 3 illustrate the variation of the convective motionintensity for different Marangoni numbers in the case of Newtonian surface rheology.

172 The European Physical Journal Special Topics

For the surface (curves 4, 5 and 6) described by (1) a convective motion developsinitially as for a solid upper boundary (curve 4). Further evolution depends on theMarangoni number. For relatively small values of the latter (i.e. for a low initialconcentration of the surfactant in the “cloudlet”) the Marangoni convection does notdevelop at all (curves 4 and 5 coincide). The increase of the Marangoni number (curve6) leads to the initiation of solutocapillary convection, but it is less intensive and itstarts later than in the case of free surface with Newtonian rheology (compare curves3 and 6).

4 Conclusions

The results of the experiments presented in this paper support the view that thereason for the observed threshold development of the Marangoni motion at the surfaceof water and water solutions is the adsorption layer of uncontrolled residual surface-active impurities which lead to a surface rheology different from volume one. Someof such impurities can be removed by the preliminary purification of water withdistillation and the use of ion-exchange filters. However, even the remaining partof the uncontrolled surfactants of extremely small concentration will suffice for thegeneration of a rather high concentration threshold. This phenomenon becomes mostimportant in narrow channels or in the case of small inclusions of gas or liquid. Incuvettes wider than 10mm the influence of the residual impurities rapidly decreasesand can be already hardly detected for a cuvette thickness of more than 20mm. Inour opinion this is the main reason why this phenomenon has not been described anddiscussed till now. The studies of Marangoni convection are carried out, as rule, withwide surfaces where the discussed effects can be neglected.Our recent preliminary experiments show that the observed phenomenon takes

place not only in water or water solution systems but can also be met in asurface of liquids with sufficiently lower surface tension, where the formation ofadsorbed films is usually thought to be less probable. For instance we have founda threshold development of Marangoni convection induced by a droplet injection of(i) isopropanol-cyclohexane solution in cyclohexane-water and cyclohexane-air inter-faces; (ii) isopropanol-chlorbenzene solution in chlorbenzene-air interface and (iii)acetic acid- chlorbenzene solution in chlorbenzene-air interface. The obtained valuesof critical Marangoni number (from 103 to 105 for different situations) is found tobe not so high as for water systems described above, but not so small to neglect theeffect.Everything presented above is also connected to thermocapillary convection,

because how the gradient of surface tension is created is irrelevant. For example,we could not initiate thermal Marangoni convection in the surface of deionized waterfilled in a narrow (up to 1 cm) channel. Indeed, the critical Marangoni number fordeionized water is found to be of order of 107 (see Sec. 3) that gives us a temperaturedifference of more than 100K to generate thermocapillary convection in a cuvette of8mm thickness.Although these experiments with nonaqueous systems and with thermocapillary

convection are preliminary and they require additional studies, one can concludethat the phenomenon of threshold development of Marangoni convection should betaken into account in some scientific and technological problems. For instance aninvestigation of the problem discussed in this paper should attract more attentionin connection with the development of microfluidics. An accurate definition of theboundary conditions in any particular situation should be of primary importanceduring the design of various chip-cooling systems which often deal with free coolantsurface in rather narrow channels. Even small critical Marangoni numbers can lead to

Interfacial Fluid Dynamics and Processes 173

the presence of a finite threshold temperature difference for the development of ther-mal Marangoni convection that can dramatically affect heat and mass transfer in acooling system. The effects of the threshold onset of capillary motion described abovecan also be significant in systems with small inclusions of gas or liquid, e.g. duringthe processes of extraction or in chemical reactors. The presence of such a thresholdwas obtained in the early publication of Pratt and Lewis [1] on the extraction of sur-factant from droplets, but the results were misinterpreted. We think that the resultspresented in this paper could help to design and interpret this type of experiments infuture. On the other hand, the experiments demonstrate that adding low molecularsurfactants to liquid systems, in which they play the role of a surfactant, is aneffective way to control the value of the threshold for the onset of the capillary motion.

The work was supported by the RFBR (projects 09-01-00484, 10-01-96009) and by jointproject of SB, UB and F-EB of RAS (116/09-C-1-1005) and by grant for young scientists ofUB RAS.

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