A systematic investigation of the Landau-Levich problem of a viscoplastic material (Carbopol 980 is
presented). The validity of the Landau-Levich scaling is assessed via integral scale measurements of
the width of the coating films thickness performed for various Carbopol concentrations in the presence
and in the absence of wall slip. To gain a deeper insight into the physical origins of the experimentally
observed deviations from the classical 2/3 scaling a full characterisation of the time resolved flow field
around the moving solid is performed and the main differences of the viscoplastic flow patterns with
respect to their Newtonian counterpart are highlighted. The experimental investigation is complemented
by a preliminary asymptotic analysis performed within the framework of the Bingham model

Burghelea T

Castelain C

Moyers-Gonzalez M

Sweid M

Wilson PL

UC Research Repository

21ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013On the Landau-Levich problem for a viscoplastic fluidMaria Sweid a, Miguel Moyers-Gonzalez b, Phillip L. Wilson b,CathyCastelain a, Teodor Burghelea aa. LUNAM Université, Université de Nantes, CNRS, Laboratoire de Thermocinétique, UMR 6607, LaChantrerie, Rue Christian Pauc, B.P. 50609, F-44306 Nantes Cedex 3, Franceb. Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800,Christchurch 8041, New ZealandRésumé :Une étude expérimentale systématique du problème de Landau-Levich pour un fluide viscoplastique(Carbopol 980) a été réalisée. La validité de la loi asymptotique de Landau-Levich est observée grâceaux mesures, en présence ou absence de glissement, à léchelle intégrale de la largeur du film déposédépaisseur différente suivant la concentration en Carbopol. Pour avoir une meilleure compréhensionentre les origines physiques des déviations expérimentales observées par rapport au lois classiques en2/3, une étude complète du champ de vitesse et de sa dépendance temporelle a été réalisée autourde lobjet se déplaçant. Les principales différences entre les structures de lécoulement dans le cas dunfluide viscoplastique comparé à celle(s) observée(s) dans le cas dun fluide newtonien ont été mises enévidence. Létude expérimentale a été complétée par une analyse asymptotique préliminaire en utilisantle modèle rhéologique de type Bingham.Abstract :A systematic investigation of the Landau-Levich problem of a viscoplastic material (Carbopol 980 ispresented). The validity of the Landau-Levich scaling is assessed via integral scale measurements ofthe width of the coating films thickness performed for various Carbopol concentrations in the presenceand in the absence of wall slip. To gain a deeper insight into the physical origins of the experimentallyobserved deviations from the classical 2/3 scaling a full characterisation of the time resolved flow fieldaround the moving solid is performed and the main differences of the viscoplastic flow patterns withrespect to their Newtonian counterpart are highlighted. The experimental investigation is complementedby a preliminary asymptotic analysis performed within the framework of the Bingham model.Mots clefs : yield stress ; Landau-Levich problem ; scaling1 IntroductionCoating problems are of a major importance from both a theoretical and a practical perspective. Inparticular, the phenomenon of coating of solid bodies with non Newtonian fluids is relevant to manyprocesses related to food industry, cosmetic industry etc. The first and probably the best known studyof the scaling properties of the coating phenomenon with a Newtonian fluid in the limit of negligibleinertia and small capillary numbers is due to Landau and Levich, [3]. Their central result was thescaling law of the coated fluid thickness with the capillary number : h ∝ Ca2/3. The capillary numberhas defined as Ca = ηVpσ where η, Vp and σ stand for the viscosity of the fluid, the speed of theplate and the surface tension coefficient, respectively. For the case of Newtonian fluids there existsa significant number of investigations, both theoretical and experimental which confirm the classicalresult of Landau and Levich and extend it, [4]. There exist, however, much fewer studies concerningwith the Landau-Levich problem for non-Newtonian fluids. The main theoretical difficulty arrises in thiscase from the dependence of the fluid viscosity on the field of the velocity gradients and, if thixotropiceffects are present, on the time. The present study concerns with an experimental and theoretical121ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013investigation of the Landau-Levich coating of a solid plate withdrawn at a constant speed from a bathfilled with a yield stress fluid. The goal of the study is two-fold. First, by measurements of the widthof the coating fluid layer, the prediction of Landau-Levich will be assessed. To gain a further insight,local measurements of the flow fields around the moving plate are performed. The experimental studyis complemented by a preliminary theoretical analysis.2 Experimental sectionThe experimental setup is schematically illustrated in Fig. 1(a). A solid plate P is withdrawn at aconstant speed Vp from a fluid contained in a acrylic made rectangular tank FT. The flow around themoving plate is illuminated laterally by a laser sheet obtained by passing a laser beam emitted by thesolid-state laser L through a cylindrical optics block CO.(a)1E-3 0.01 0.1 11E-41E-30.010.1110(s-1)(Pa)  (b)Figure 1 – (a) Schematic view of the experimental setup : L - laser, CO - cylindrical optics bloc, P -moving plate, FT - fluid tank, DB - digital balance, CCD - digital camera, PC - personal computer.The insert illustrates the free meniscus formed around the moving plate. (b) Rheological measurementsperformed with a polished geometry (□, ■) and with a rough geometry (⋄, ♦). The full/empty symbolsrefer to the increasing/decreasing branch of the flow curves. The full line is a Herschel-Bulkley fit thatgives τy = 0.64± 0.003(Pa), K = 0.4± 0.002(Pasn), n = 0.54± 0.001. A 0.1% Carbopol solution hasbeen used.The working fluids are seeded with a minute amount of polyamide particles and the flow is visualisedfrontally with a digital camera CCD. A long sequence of images is acquired and individual image pairsare passed to a multi-grid Particle Image Velocimetry (PIV) algorithm.As working fluids we have used aqueous solutions of Carbopol 980 at three distinct (weight) concen-trations : c = 0.1%, c = 0.2% and c = 0.3%. The rheological behaviour of each solution has beeninvestigated by performing controlled stress linear ramps for both increasing and decreasing valuesof the applied stress. To test the role of the wall slip, such measurements have been performed withboth smooth and rough surfaces as illustrated in Fig. 1(b). A more comprehensive of the yieldingbehaviour of the Carbopol gels has been reported in Ref. [5]. In addition to the local measurementsof the flow fields around the moving plate, integral measurements of the width of the fluid film havebeen performed. For this purpose the fluid tank has been placed on a digital balance and the mass ofwithdrawn fluid was measured in real time which. Using the assumption of the homogeneity of coating,the measurements of the withdrawn fluid mass provide an indirect mean of estimating the thicknessof the coating fluid layer. The surface tension coefficient σ of Carbopol solutions decreases with thepolymer concentration, [1].221ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 20133 Results3.1 Experimental resultsTo validate the experimental setup and the measurement techniques, we have studied the coatingof plate with Newtonian fluids (solutions of sucrose at various concentrations). Measurements of thedependence of width of the coating film on the capillary number Ca performed with several sugarsolutions are presented in Fig. 2(a). The capillary number Ca has been varied over nearly two decades by1E-4 1E-3 0.011E-51E-4h(m)Ca  (a)x (m)z(m)0.01 0.02 0.03 0.04 0.050.0050.010.0150.020.0250.030.035(b)Figure 2 – (a) Dependence of the film thickness on the capillary number for sugar solutions withvarious concentrations : (□) - c = 50%, (◦) - c = 55%, (△) - c = 60%, (▽) - c = 65%. The full lineis a guide for the eye, Ca2/3. (b) Flow field measured for a 65% wt sugar solution. The edge of thevertically moving plate corresponds to x = 0 and the free surface of the fluid to z = 0.varying the speed Vp of the plate, the viscosity and surface tension coefficient of the fluid (by modifyingthe concentration of sucrose). The error bars have been calculated by repeating each measurement threetimes and computing the standard deviation. The data acquired for the Newtonian fluids follows fairlywell the classical Landau - Levich scaling ∆h ∝ Ca2/3 which is an indicator for the reliability ofour measurement technique. Slight deviations from this scaling might be attributed to both the finitesize of the withdrawn plate and the limited accuracy of the digital balance. A typical flow patternaround the plate withdrawn from a 65% sugar solution is illustrated in Fig. 2(b). The flow is organisedin two counter-rotating vortices, one near the moving plate and the other in the vicinity of the freefluid surface. This flow organisation was reproduced for various Newtonian solutions and for variouswithdrawing speeds and is consistent with previous experimental observations, [4].Similar measurements of the dependence of the width of the coated fluid layer performed with severalCarbopol solutions on the capillary number Ca within a wide range of capillary numbers are presentedin Fig. 3(a). The raw measurements of the width of the film versus the speed of the withdrawn plateVp are presented in the inset.As the rheological behaviour of the Carbopol solutions is strongly nonlinear (see Fig. 1(b)) the calcula-tion of the capillary number is not straight forward and deserves a separate discussion. Measurementsof the flow patterns around the withdrawn plate such as the one presented in Fig. 3(b) allowed thedetermination of the velocity gradients in its direct proximity. By connecting the measured velocitygradients with the rheological measurements (such as the one illustrated in Fig. 1(b)) one can assess alocal (near the moving plate) viscosity which is ultimately used for the calculation of the capillary num-ber Ca. For each of the solutions we have investigating the data acquired in the presence of slip with asmooth plate (the full symbols) scales differently with Ca than the data acquired in the absence of slipwith a rough plate (the empty symbols). This can be understood in terms of significant differences inboth the stress magnitude and its distribution near the moving plate induced by the presence of wall321ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 20130.01 0.1 10 1001E-51E-41E-3Ca0.05Ca0.331E-3 0.011E-51E-41E-3  h (m)Vp(m/s)h(m)CaCa0.66  (a)x (m)z(m)0.01 0.02 0.03 0.04 0.05 0.06 0.070.010.020.030.04(b)Figure 3 – (a) Dependence of the film thickness on the capillary number for Carbopol solutions withvarious concentrations : (□) - c = 0.1%, (◦) - c = 0.2%, (△) - c = 0.3%. The full lines are guides forthe eye, ∆h ∝ Caα with the scaling exponents α indicated in the inserts. The raw dependencies ofthe film thickness on the plate’s speed are presented in the upper inset. The full/empty symbols referto smooth/rough plates. (b) Flow field measured for a 0.3% wt Carbopol solution. The edge of thevertically moving plate corresponds to x = 0 and the free surface of the fluid to z = 0.slip which originate from different rheological responses of the material in the presence/absence of wallslip and in a range of small rates of shear, see Fig. 1(b). The data acquired with and without slip mergeonly a critical capillary number Cac which increases monotonically with the Carbopol concentration.Beyond this critical value of the capillary number a power law scaling of the width of the coated fluidfilm is observed, ∆h ∝ Caα, with the exponent α monotonically decreasing with the concentrationof Carbopol (the yield stress), Fig. 3(a). This rather unexpected fact is, most probably, related to asignificant re-organisation of the flow pattern due to the plasticity of the material. Indeed, preliminaryobservations of the flow field indicate the presence of a single large scale vortex located in the vicinityof the moving plate, Fig. 3(b), as opposed to the two-vortices flow pattern observed with a Newtonianfluid, Fig. 2(b).3.2 Dimensional analysisFocusing on flow regimes close to the yield point we model the fluid as a Bingham fluid such thatthe fluid viscosity is given by µ = η + τy/γ̇. Then for the dynamic meniscus thickness δh we seek afunctional relationship∆h = f(Vp, σ, g, τy, η, ρ) , (1)where g is the acceleration due to gravity.Taking Vp, σ, g as a set of dimensional parameters with independent units in the length-mass-time class,we can rewrite (1) in nondimensional terms asH = F (Yi,Ca,Bo) , (2)whereH =h0gV 2p, Yi =τyV2pgσ, Ca =ηVpσ, Bo =ρV 4pgσ(3)are the nondimensional thickness, yield number, capillary number, and Bond number, respectively, andthe function F is unknown at this stage.421ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 20133.2.1 Limit as Yi → 0We approach a Newtonian fluid in the limit as Yi → 0 with all other nondimensional parameters fixedand of O(1). A Newtonian fluid certainly has a dynamic meniscus of finite non-zero thickness δh underthe conditions of order unity capillary and Bond number, thus following [2]∆h =V 2pgF1(Ca,Bo) when Yi ≪ 1 (4)for some function F1.In this Newtonian limit we expect the dynamic meniscus thickness to tend to zero as both η and ρtend to 0 (independently), and so we conclude that for some positive and real α, β,∆h ∼V 2pgCaαBoβ when Yi ≪ 1, Ca ≪ 1, Bo ≪ 1. (5)The exponents α, β cannot be derived from a dimensional analysis, even in principle. However, weknow from the classic result of Landau and Levich [3] that for Newtonian fluids in the small capillarynumber regime∆h ∼(σρg) 12(ηVpσ) 23, (6)which is consistent with (5) when α = 2/3, β = −1/2, and thus∆h ∼V 2pgCa23√Bo. (7)which is also confirmed for the case of a viscoplatic fluid with low yield stress (c = 0.1% in our case)as can be seen in Figure 3.3.2.2 Limit as Yi → ∞The experimental data shows that when τy, and hence Yi, take on large values while the remainingnondimensional groups remain fixed and of order unity, the dynamic meniscus thickness approaches afinite non-zero limit, suggesting that∆h ∼V 2pgF2(Ca,Bo) when Yi ≫ 1 (8)for some function F2. However, the experimental data also shows that in this large-Yi regime thethickness h0 is independent of plate speed Vp, hence we must consider a power-law monomial of theform∆h ∼V 2pg√Bo(CaBo14)δ, (9)which can be rewritten as∆h ∼ aCaδ , (10)where a = V 2p /g√Bo =√σ/ρg is the capillary length, and Ca = Ca/Bo14 is a renormalized capillarynumber.Once again, the value of δ in (10) cannot be determined by dimensional analysis. But for any givenexperiment, the following should be a constant :δ =log(∆ha)logCa. (11)521ème Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013c(%,wt) σ(mN/m) η(Pas) Ca δ0.2 27 0.1 0.45 0.690.3 6 0.150 2.1 0.42Table 1 :Physical properties of Carbopolsolutions.c=0.2 %,  = 0.1 Pas, s= 27mN/m0.0 0.5 1.0 1.5 2.00.00.5c=0.3 %,  = 0.15 Pas, s= 6mN/ma(mm)  Fig. 4 :Dependence of the estimated scalingexponent δ on the capillary length a.Using the physical parameters given in Table 1, we can estimate the values of the scaling exponent δ atvarious capillary lengths obtained in our experiments, Fig. 4. Bearing in mind that the origin shouldnaturally be part of this dependence (the added empty symbol in Fig. 4), the linear dependence ofscaling exponents is likely to be valid in the limit of large yields numbers. However, to fully validatethis result, some additional experiments performed for larger concentrations of Carbopol are needed.4 Conclusions, outlookAn experimental investigation of the coating of a solid surface with a viscoplastic fluid (Carbopol 980)by combined measurements of the thickness of the coating film and of the flow patterns around thesolid object is presented. The experimental setup and the measuring techniques have been validatedby studying the coating of a solid object withdrawn at a constant speed from various Newtonianfluids (sucrose solutions with various concentrations). For each of the Newtonian solutions we haveinvestigated the classical Landau-Levich scaling of the width of the coated layer ∆h ∝ Ca2/3 has beenfound, Fig. 2(a). Measurements of the film’s thickness performed in a wide range of capillary numbersCa with several Carbopol solutions reveal a clear departure from the classical Landau-Levich scaling :an increase of the polymer concentration (yield stress) practically suppresses the dependence of thefilm’s width on Ca. Below a critical value of the capillary number, the presence of wall slip stronglyaffects this dependence. An increase of the Carbopol concentration (yield stress) results in a gradualsuppression of the scaling of the film’s width with the capillary number.Through careful dimensional analysis we have shown that the Newtonian scaling with Ca is recoveredif τy ≪ 1. Following our experimental results, when τy ≫ 1 we have shown that the witdh of film willbe proportional to the capillary length with no dependence on the speed of the plate Vp. This lineardependence derived via scaling arguments is fairly well supported by our experimental results.Références[1] Gookhyun Baek, Seokwon Kim, Jeongin Han, and Chongyoup Kim. Atomization characteristics ofimpinging jets of gel material containing nanoparticles. Journal of Non-Newtonian Fluid Mechanics,166(21â22) :1272 – 1285, 2011.[2] G.I. Barenblatt. Scaling. Cambridge University Press, 2003.[3] L. Landau and B. Levich. Dragging of liquid by a plate. Acta Physicochim. URSS, 17 :42 – 54,1942.[4] H. C. Mayer and R. Krechetnikov. Landau-levich flow visualization : Revealing the flow topologyresponsible for the film thickening phenomena. Physics of Fluids, 24(5) :052103, 2012.[5] A. M. V. Putz and T. I. Burghelea. The solid-fluid transition in a yield stress shear thinningphysical gel. Rheol Acta, 48 :673–689, 2009.6

On the Landau-Levich problem for a viscoplastic fluid

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On the Landau-Levich problem for a viscoplastic fluid

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