An important practical feature of simulating droplet migration computationally,

using the lubrication approach coupled to a disjoining pressure term, is

the need to specify the thickness, H, of a thin energetically stable wetting layer,

or precursor lm, over the entire substrate. The necessity that H be small in

order to improve the accuracy of predicted droplet migration speeds, allied to the

need for mesh resolution of the same order as H near wetting lines, increases the

computational demands signicantly. To date no systematic investigation of these

requirements on the quantitative agreement between prediction and experimental

observation has been reported. Accordingly, this paper combines highly ecient

Multigrid methods for solving the associated lubrication equations with a parallel

computing framework, to explore the eect of H and mesh resolution. The solutions

generated are compared with recent experimentally determined migration

speeds for droplet 

ows down an inclined plane

Gaskell, P.H.

Jimack, P.K.

Koh, Y.Y.

Lee, Y.C.

Thompson, H.M.

English

An important practical feature of simulating droplet migration computationally, using the lubrication approach coupled to a disjoining pressure term, is the need to specify the thickness, H-*, of a thin energetically stable wetting layer, or precursor film, over the entire substrate. The necessity that H-* be small in order to improve the accuracy of predicted droplet migration speeds, allied to the need for mesh resolution of the same order as H-* near wetting lines, increases the computational demands significantly. To date no systematic investigation of these requirements on the quantitative agreement between prediction and experimental observation has been reported. Accordingly, this paper combines highly efficient Multigrid methods for solving the associated lubrication equations with a parallel computing framework, to explore the effect of H-* and mesh resolution. The solutions generated are compared with recent experimentally determined migration speeds for droplet flows down an inclined plane.</p

Koh, Y. Y.

Lee, Y. C.

Gaskell, P. H.

Jimack, P. K.

Thompson, H. M.

Heriot Watt Pure

Droplet migration: Quantitative comparisons with experiment

Y. Y. Koh

Y. C. Lee

P. H. Gaskell

P. K. Jimack

H. M. Thompson

Crossref

White Rose Research Online

promoting access to White Rose research papers    White Rose Research Online   Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper accepted for publication in European Physical Journal.   White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/4975/     eprints@whiterose.ac.uk  EPJ manuscript No.(will be inserted by the editor)Droplet migration: Quantitative comparisons withexperimentY.Y. Koh1, Y.C. Lee2, P.H. Gaskell2, P.K. Jimack3, and H.M. Thompson2,a1 INTI International University College, Negeri Sembilan, Malaysia 718002 School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom LS2 9JT3 School of Computing, University of Leeds, Leeds, United Kingdom LS2 9JTAbstract. An important practical feature of simulating droplet migration compu-tationally, using the lubrication approach coupled to a disjoining pressure term, isthe need to specify the thickness, H∗, of a thin energetically stable wetting layer,or precursor film, over the entire substrate. The necessity that H∗ be small inorder to improve the accuracy of predicted droplet migration speeds, allied to theneed for mesh resolution of the same order as H∗ near wetting lines, increases thecomputational demands significantly. To date no systematic investigation of theserequirements on the quantitative agreement between prediction and experimentalobservation has been reported. Accordingly, this paper combines highly efficientMultigrid methods for solving the associated lubrication equations with a parallelcomputing framework, to explore the effect of H∗ and mesh resolution. The so-lutions generated are compared with recent experimentally determined migrationspeeds for droplet flows down an inclined plane.1 IntroductionDroplet migration and wetting phenomena are ubiquitous throughout science and engineeringand are crucial to several natural, engineering and manufacturing processes. A key issue inrelated theoretical studies is the alleviation of the stress singularity at dynamic wetting linesand for which several models have been proposed, see for example [1]. Analytical investigationsapart [2], numerical solutions, based on lubrication theory coupled with a disjoining pressureterm, to alleviate the stress singularity at dynamic wetting lines, have increasingly appeared,which explore droplet motion: (i) on chemically- and topographically-heterogeneous substrates[3,4]; (ii) driven by external body or Marangoni forces [5,6].Despite the above successes, an important feature of using the disjoining pressure term, isthe need to specify the thickness of a thin energetically stable wetting layer, or precursor film,H∗, over the whole of the substrate, which experimental evidence suggests lies in the broadrange 1−100 nm [7]. To improve computational accuracy it is necessary to use small H∗ values inorder to achieve droplet migration speeds commensurate with experimentally observed ones [8].This requirement increases the computational demands significantly since mesh resolution nearwetting lines must be of the same order as H∗ in order to avoid highly inaccurate, oscillatoryor even negative film thicknesses being predicted. No systematic investigation into these effectson the quantitative agreement with experimental data has been performed to date. However,the need for such is highlighted in a recent comparison of numerical solutions [5] with theexperiments of Podgorski et al [8] and Le Grand et al [9] for droplet migration down an inclinedplane, see Fig 1. Although good qualitative agreement can be obtained without highly resolveda e-mail: h.m.thompson@leeds.ac.uk2 Will be inserted by the editorXZYH*HαLn_t_oHoDropletInclined solid surface(a) (b) (c) (d) (e) (f)Fig. 1. (a) Schematic of droplet motion down an inclined plane; computed contour plots showingdifferent modes of droplet migration with varying inclination angle, α: (b) 20o (U = 0.2m/s), (c) 35o(U = 0.27m/s), (d) 40o (U = 0.03m/s), (e) 45o (U = 0.036m/s) and (f) 60o (U = 0.048m/s).grids (O(105) grid points), a recent study by Koh [10] has shown that far greater grid densitiesare required to obtain grid- and precursor film thickness-independent solutions even for simpledroplet motions.This paper combines recent advances in the development of highly efficient Multigrid meth-ods [4] for solving the associated lubrication equations used to model droplet migration withthe implementation of parallel computing architectures to present a quantitative comparison ofdroplet migration speeds with those measured in the recent experiments of Podgorski et al [8]and LeGrand et al [9].2 Problem DescriptionThe problem considered is illustrated schematically in Fig. 1(a) showing the motion of a dropletmigrating down an inclined plane. Assuming that the ratio, ε = H0/L0, relating the charac-teristic droplet thickness to the extent of the substrate, is small, the Navier-Stokes equationsreduce to the simpler, non-dimensional lubrication equations of the form:∂h∂t=∂∂x[h33(∂p∂x−Boεsin α)]+∂∂y[h33(∂p∂y)], (1)p = −∇2(h) − Π(h) + Bo cos α(h) , (2)where h and p are the non-dimensional dependent variables for film thickness and pressure,α is the substrate inclination angle, and, Bo = ρgL20/σ, is the Bond number. The disjoiningpressure term is:Π(h) =(n − 1)(m − 1)(1 − cos θs)h∗(n − m)ε2[(h∗h)n−(h∗h)m], (3)where θs is the equilibrium static contact angle, σ is surface tension; constants n and m arethe exponents of the interaction potential, chosen to be 3 and 2, respectively. Further detailscan be found in [4].A contact angle hysteresis model is built in, based on the droplet motion history; in whichstatic advancing and receding contact angles, θs,a and θs,r, are assigned to equation (3), de-pending on the motion of the droplet [10].3 Method of SolutionFollowing [4], a system of differential-algebraic equations is obtained by application of finite-difference discretisation on a quadrilateral mesh with equal spacings in the x and y directions.Will be inserted by the editor 30 0.5 1 1.5 2Bo sin α00.0050.010.0150.020.0250.03CaPodgorski et al (2001)h* = 0.01h* = 0.005h* = 0.004 (2049 x 513)h* = 0.004 (4097 x 1025) LH(a) (b)Fig. 2. Quantitative comparison between experimental data and numerical predictions of dropletmigration on an inclined plane: (a) Podgorski et al [8]; (b) LeGrand et al [9]. The plot in (a) showsthe accuracy of the numerical solution improves with smaller pre-cursor film thickness, h∗, and (b)delineates the change in droplet shape with varying droplet speed.Time integration is performed using an implicit, second-order scheme with adaptive step-sizeselection based upon a local truncation error estimate. At each time step, the nonlinear dis-cretized equations are solved using a full approximation storage and full Multigrid scheme(FAS-FMG)[11]. Parallel implementation of the solver is achieved via a geometric decomposi-tion of the computational domain, based on a partitioning strategy of the coarsest grid used.Its basic philosophy follows that employed in [12] and is described in greater detail in [10].4 Results and DiscussionThe data from two experimental studies investigating the effects of inclination angle, α, ondroplet shape and wetting speed is compared with corresponding results obtained using thelubrication-based parallel Multigrid solver. The first case considers data obtained by Podgorskiet al [8] where a silicon oil droplet of radius 1mm and volume 8.3mm3 with ρ = 924 kg/m3,µ = 0.00915 Pas and σ = 0.0205 N/m at 25oC, is used (see Fig. 1). Their reported staticadvancing and receding contact angles, θs,a = 50o, and, θs,r = 40o, respectively were employedin the simulations. The latter were performed on a rectangular domain (385 × 129 grid points;H∗ = 26µm) and predictions of the effects of varying inclination angle, α, on droplet shapewith Lo = 4 mm, are seen to be in excellent agreement with the data of Podgorski et al [8].They display smooth rounded contact lines and angled corner shaped tails at low velocities,and cusp formation and pearling at the higher velocities induced at high inclination angles.Of even greater interest is the effect of α on steady droplet speed, data for which is givenin the form of a Capillary number (Ca = µU/σ) versus Bo sin α graph; Fig. 2(a) shows thatthere is a strong dependence of the over-prediction of droplet speed on the value of h∗. Withh∗ = 0.01 (corresponding to H∗ = 26µm), speeds are over-predicted by a factor between 6 and2. A 60% reduction in h∗ to 0.004 (H∗ = 10.4µm) leads to a significant improvement; withpredicted values being reduced to a factor between 2 and 1.34, respectively. The effect of griddensity on solution accuracy is shown more clearly in the left half of Table 1. As expected,significantly larger computational resources are required to obtain grid-independent solutions;for the latter case, over 4 million grid points are needed. Even greater resources are requiredfor h∗ values nearer the physically realistic range [10].The experiments of LeGrand et al [9], consider the motion of a 6 mm3 oil droplet of radius2 mm with ρ = 936 kg/m3, µ = 0.01 Pas and σ = 0.0201 N/m on a surface with θs,a = 51oand θs,r = 46o. Sample simulations were performed with h∗ = 0.005 (H∗ = 2.4µm) and Lo = 8mm, on a rectangular domain with over 1 million grid points. The right half of Table 1 providesa comparison between the experimentally and numerically obtained values of Ca as a function4 Will be inserted by the editorBo sin α Ca(Pod) Cagrid1Cagrid1Ca(P od)Cagrid2Cagrid2Ca(P od)Bo sin α Ca(LeG) Cagrid1Cagrid1Ca(LeG)0.32 0.0012 0.0030 2.45 0.0028 2.21 0.39 0.0014 0.0019 1.360.47 0.0020 0.0035 1.76 0.0032 1.61 0.52 0.0024 0.0029 1.210.62 0.0030 0.0046 1.53 0.0042 1.41 0.64 0.0032 0.0040 1.250.91 0.0043 0.0065 1.50 0.0061 1.42 0.75 0.0038 0.0048 1.261.04 0.0054 0.0084 1.56 0.0080 1.47 0.86 0.0048 0.0057 1.201.31 0.0077 0.0110 1.43 0.0104 1.35 0.97 0.0055 0.0067 1.211.49 0.0095 0.0142 1.49 0.0135 1.42 1.06 0.0060 0.0073 1.211.57 0.0121 0.0166 1.49 0.0162 1.34 1.16 0.0065 0.0080 1.23Table 1. Comparison between experimental and predicted Ca values as a function of Bo sin α. (Left)Data of Podgorski et al [8], Ca(Pod), and (Right) of LeGrand et al [9], Ca(LeG). Grid 1 has 2049 × 513uniformly spaced grid points and grid 2 has 4097 × 1025 points.of Bo sin α. The quantitative agreement between the two is very good, where speeds are beingpredicted to within 20% of many of the experimental values.Le Grand et al [9] also measured the length, L, and maximum height, H , of the droplet asa function of Ca. A comparison between their experimental results and numerical predictionis given in Fig. 2(b). They show that, H , remains effectively constant as L increases at higherα. The predicted lengths, L, are typically 50% larger than those reported in experiments atsmall Ca, whereas the agreement between the predicted heights, H , is consistently good. Theshapes of the droplets that result are also indicated in Fig. 2(b), where data without any shadingindicates that the droplets are rounded, as shown in Fig. 1(b), whereas rounded shading, squareshading and triangular shading represents droplet shapes with corner (Fig. 1(c)), cusp (Fig.1(d) and 1(e)) and pearling (Fig. 1(f)) characteristics. The data shows that corner to cusp topearling transitions as observed experimentally, occurs at smaller droplet speeds than predictednumerically. Finally, note that the cusp angle at which pearling transition occurs has alsobeen considered numerically by Koh [10], who obtained good agreement with the experimentsreported in [9].5 ConclusionAlthough lubrication theory displays good qualitative agreement with experimental observationof droplet migration when relatively coarse grids are used to solve the model equations, bothprecursor film thickness, h∗, and grid density have an important influence on the predictedaccuracy of key features such as droplet speed. In that, much finer grids are required togetherwith small precursor film thicknesses to obtain grid independent solutions commensurate withexperiment.References1. T.D. Blake, K.J. Ruschak, Liquid Film Coating (Chapman and Hall, 1997), 63-97.2. A. Oron, S. Davis, S. Bankoff, Rev. Mod. Phys. 69, (1997) 931.3. L.W. Schwartz, R.R. Eley, J. Colloid Interface Science 202, (1998) 173-188.4. P.H. Gaskell, P.K. Jimack, M. Sellier, H.M. Thompson, Int. J. Num. Meth. Fluids 45, (2004) 1161-1186.5. L.W. Schwartz, D. Roux, J.J. Cooper-White, Physica D 209, (2005) 236-244.6. L.W. Schwartz, R.V. Roy, R.R. Eley, H.M. Princen, J. Eng. Math. 50, (2004) 157-175.7. A.L. Bertozzi. Notices of the AMS. 45(6), (1998), 689-697.8. T. Podgorski, J. Flesselles, L. Limat, Phys. Rev. Lett. 87, (2001) 036102(1).9. N. LeGrand, A. Daerr, L. Limat, J. Fluid Mech. 541, (2005) 293-315.10. Y.Y. Koh, PhD thesis, University of Leeds, 2007.11. Y.C. Lee, H.M. Thompson, P.H. Gaskell, Comp. Fluids 36, (2007) 838-855.12. C.E. Goodyer, M. Berzins, Concurrency and Computation 19, (2007) 369-396.

Droplet migration: quantitative comparisons with

experiment

An important practical feature of 
simulating droplet migration computationally, using the lubrication approach coupled to a 
disjoining pressure term, is the need to 
specify the thickness, H*, of a thin energetically stable wetting layer, or
precursor film, over the entire substrate. 
The necessity that H* be small in order to improve the accuracy of predicted droplet migration speeds, allied to
the need for mesh resolution of the same order as H* near wetting lines, increases the computational
demands significantly.
To date 
no systematic investigation of these requirements on the quantitative agreement between prediction and experimental observation has been reported. 
Accordingly, this paper combines highly efficient
Multigrid methods for solving the associated lubrication equations with a parallel computing framework,
to explore the effect of H* and mesh resolution.
The solutions generated are compared with recent experimentally 
determined migration speeds for droplet flows down an inclined plane

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Droplet migration: quantitative comparisons with experiment

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