Modeling moving droplets: Benjamin P. Bryant Andrew Bernoff and Anette Hosoi, Advisors

Transcription

1 Modeling moving droplets: A precursor film approac by Benjamin P. Bryant Andrew Bernoff and Anette Hosoi, Advisors Advisor: Advisor: May 23 Department of Matematics

2 Abstract Modeling moving droplets: A precursor film approac by Benjamin P. Bryant May 23 We investigate te beavior of moving droplets and rivulets, driven by a combination of gravity and surface sear (wind). Te problem is motivated by a desire to model te beavior of raindrops on aircraft wings. We begin wit te Stokes equations and use te approximations of lubrication teory to derive te specific tin film equation relevant to our situation. Tis fourt-order partial differential equation describing te eigt of te fluid is ten solved numerically from varying initial conditions, using a fully implicit discretization for time stepping, and a precursor film to avoid singularities at te drop contact line. Results describing general features of droplet deformation, limited parameter studies, and te applicability of our implementation to te long-term goal of modeling wings in rain are discussed.

3 Table of Contents List of Figures iii Capter 1: Introduction Previous work Our metods Capter 2: Derivation of Relevant Equations Fundamental equations Tin film equation derivation Nondimensionalization Capter 3: Numerical Metods Discretization and iteration tecniques Precursor film Numerical verification Capter 4: Results Droplet pases Parameter studies Two drop coalescence Inevitable sear dominance Comparison of 1-d and 2-d simulations Potential surface rougness modeling

4 Capter 5: Conclusions and Future Work 32 Appendix A: Appendix 34 A.1 Discussion of Code Bibliograpy 37 ii

5 List of Figures 2.1 Volume Element Initial condition Blob pase Tetraedron pase Initial dip pase Ridge extension Tip singularity Two-drop coalescence Sear dominance d vs 2-d Unlikely sapes Precursor film and trajectory control iii

6 Acknowledgments Tanks are due to Professors Bernoff and Hosoi for teir guidance, to Joe Malone and Dmitriy Kogan for teir elping me get over te initial urdles of learning C, and of course, to my parents. iv

7 Capter 1 Introduction Te beavior of tin films, droplets and rivulets is of interest in many applications. Tese include suc diverse areas as coating flows used in te creation of microcips and oter electronic devices, sintering in metallurgy, spray coating processes, eat excange, and oil recovery [15, 4]. We are motivated by te problem of water flowing on an aircraft wing or oter possibly inclined surfaces seared by wind. In tis problem, droplets accumulate on te surface as a result of rain impacting te wing. Tey are ten seared by te wind and coalesce into a spanwise film, wic is convected back along te wing, and eventually breaks up into rivulets, wic are driven farter back off te wing or break up into larger drops, wic in turn run off te wing s trailing edge. Te presence of tis penomenon on te wing surface as generally undesirable effects on wing performance, in tat it tends to decrease lift and increase drag under most fligt conditions [2, 13, 12]. Tompson found in wind tunnel tests tat performance degradation correlates wit increasing contact angle (and terefore decreasing surface wettability), presumably due to te flatter profile presented by wetting films [14]. He also found tat wetting surfaces gave rise to longer convecting regions, but tat for a given contact angle, a longer film convection region more negatively affected performance tan a sort region. Tis implies tat an optimal metod for alleviating degradation in fligt performance is to use a wettable surface, but take steps to minimize te lengt of te film convection region, peraps by controlling te ac-

8 2 tion of te droplets prior to teir coalescing into a film. Tis leads to te need for modeling te motion of a droplet subjected to surface sear from te surrounding gas wile experiencing a body force due to gravity, wic is te primary aim of tis tesis. In te remainder of capter 1, we will discuss previous efforts related to tis problem and briefly describe our approac. In capter 2 we derive te relevant equations to describe te situation we are modeling. Capter 3 outlines te numerical metods used to implement our model. In capter 4 we discuss results, before concluding in capter 5 wit a summary and suggestions for future work. 1.1 Previous work Drops and rivulets ave previously been modeled in many different situations, using many different tecniques. Most relevant to our aim is te work of Durbin [5], wo developed a model describing te sape of a drop being seared by a ig-reynolds number stream. However, e solves tis using only a pressure distribution, ignoring sear by assuming te droplet is of equal or greater tickness tan te boundary layer. In te less aeronautical regime, Wilson et al [15] analytically examined a non-newtonian fluid driven by gravity or sear stress using a contact line model and certain lubrication approximations. Tey obtained similarity solutions for two-dimensional steady profiles in te transverse direction, in te limits of strong and weak surface tension. Moriarty and Scwartz [9] developed a precursor film model of a droplet running down an inclined surface tat tey modified to account for sear effects as well. Teir model uses te metod of matced asymptotic expansions, accounting for surface tension only at te critical area near te front of te advancing drop, were curvatures are igest. Tey ten matc tis wit a simpler model for te flatter, receding face of te drop. A larger issue wit teir model is tat

9 3 teir equations describe eigt as a function of time and one spatial dimension only, and terefore are in reality modeling ridges of fluids and not drops. (For future reference, tis sort of modeling is referred to as one-dimensional. It sould be noted tat, because te independent variables in te model are used to track a eigt, one-dimensional modeling produces two-dimensional drops, and twodimensional modeling results in tree-dimensional drops.) Li and Pozrikidis [8] modeled a droplet adering to a surface surrounded by Stokes flow. Wile an interesting problem, tis is quite distinct from our aims, because tey modeled two immiscible fluids of equivalent viscosity, as opposed to a liquid-gas interface. Also, teir model uses a pinned contact line, and terefore it is impossible for te drop to move. Scleizer and Bonnecaze [11] used a boundaryintegral metod to study a drop between two plates subject to pressure or sear stress. Teir model is also one-dimensional, and explicitly tracks a moving contact line. It is not, terefore, wat we are seeking to do, but will prove useful in making comparisons after our model is developed. Lastly, Dimitrakopoulos and Higdon [4] studied te conditions required for te displacement of a two-dimensional droplet from a surface, using a contact line model. Our aim is to track te eigt of a searing droplet as a function of time and two spatial dimensions, and it appears tis as so far not been done. 1.2 Our metods Our approac to te problem is to restrict our attention to very wetting drops, and utilize te simplifications of lubrication teory. Tus we assume our drops are governed by te well-studied tin film equation. For our situation it takes te form [ 3 t = 3 ( G S 2 ) ] + ( Γ 3 + C 2), (1.1) x were is te eigt of te fluid and eac term represents a contribution from different forces, wic are detailed in te derivation found in capter 2. Te above

10 4 equation is a specific example of te more general class of tin film equations, wic ave been modeled fairly extensively in te context of lubrication teory, were it is usually presented in it s one dimensional form [1, 6]. We plan to, among oter tings, test it s applicability to droplet modeling. Te primary goal of our researc is to successfully simulate droplet deformation under bot gravity and surface sear stresses, describing eigt as a function of time and two spatial dimensions. Tis will be accomplised by numerically solving te applicable tin film PDE, using a precursor film to account for contact line singularities. We will also examine te potential of tis metod of implementation for larger-scale modeling of droplets on a wing. Numerical details, results and furter goals are discussed in later capters. First owever, we must derive te equation we plan to model.

11 Capter 2 Derivation of Relevant Equations 2.1 Fundamental equations We begin wit te Stokes equations wic describe flow at very low Reynolds numbers, and derive te tin film equation relevant to our situation. Te Stokes equations are te Navier-Stokes equations wit te inertial and time-derivative terms removed, since tey are assumed to make a neglible contribution relative to te terms we consider, wic are tose describing pressure gradients, sear motion, and body forces: p = µ 2 u + F, (2.1) were te body force F is simply te force due to gravity. For our purposes, we assume gravity can act in te z and x direction only. Tis allows us to model flow on a plane inclined at any angle, wit z normal to te surface of te plane. Taking tis into account and expanding te above equation into separate parts, we ave p x = µ(u xx + u yy + u zz ) + ρg sin θ (2.2) p y = µ(v xx + v yy + v zz ) (2.3) p z = µ(w xx + w yy + w zz ) ρg cos θ, (2.4) were u,v and w are te x,y and z components of u respectively, θ is te angle of inclination below orizontal (tat is, positive θ indicates sloping down to te rigt), and subscripts denote partial differentiation wit respect to te subscripted

12 6 variable. Also, for convenience, te x and z gravity terms ρg sin θ and ρg cos θ will from ereon be referred to as just M and N, respectively. In wetting film and droplet applications suc as ours, we recognize tat te caracteristic eigt (H) is different (specifically, muc muc less) tan tan te lengt (L) for variation in x and y. To furter simplify te equations, we take advantage of tis relation by examining te relative scales of elements of te 2 u term. We partially nondimensionalize by assigning were ɛ = H L x = Lx, y = Ly, z = ɛlz, t = T t (2.5) 1 (tis is known as te tin film approximation) and variables appearing on te rigt side of te equation are nondimensional. Substituting tese gives: p x = µ LT (u xx + u yy + 1 ɛ u zz) M 2 (2.6) p y = µ LT (v xx + v yy + 1 ɛ v zz) 2 (2.7) p z = µ LT (ɛw xx + ɛw yy + 1 ɛ w zz) N, (2.8) From tis, we see tat among te terms contained in 2 u, u zz and v zz are dominant, since all oters are at least O(ɛ) smaller, and most are smaller still. Retaining tese terms of te original equations leaves us wit governing equations of: and also te continuity equation, p x = µu zz + M (2.9) p y = µv zz (2.1) p z = N, (2.11) u x + v y + w z =. (2.12) Te boundary conditions are determined by te specific situation we are trying to model, wic will be discussed below.

13 7 2.2 Tin film equation derivation We model te effect of wind flowing past by simply imposing a constant sear stress in te x direction at te free surface, and a zero sear stress in te y direction at te surface. Tis is expressed as µu z = C and µv z =, at, te eigt of te film. We also must account for pressure differences due to surface curvature. Since we are utilizing te lubrication approximation, it is valid to assume tat te curvature is small, and tus use 2 as te first order approximation to curvature. Tus at te surface we also ave p = σ 2, were σ is te surface tension coefficient, and te Laplacian is two-dimensional. At te fluid-solid interface, we ave te standard no-slip condition, u =. Expressed togeter, we ave: z = : µu z = C, µv z =, p = σ 2 (2.13) z = : u =, (2.14) Integrating (2.11) wit respect to z and satisfying te surface tension boundary condition, we find p = N(z ) σ 2. (2.15) We ten differentiate wit respect to x and equate it wit (2.1) giving µu zz = (N σ 2 ) x M. (2.16) Integration wit respect to z and satisfaction of boundary conditions gives: µu z = [ (M + ( N + σ( 2 )) x )](z ) + C (2.17) Here C is te sear coeffecient, not a constant of integration. Integrating one more time, we arrive at an expression for u: u = 1 µ [ [ (M + ( N + σ( 2 )) x ) ] ( 1 ] 2 z2 z) + Cz. (2.18)

14 8 A similar process undertaken wit respect to y yields a similar equation, less te surface sear and x component of gravity terms: v = 1 µ [ (N σ( 2 )) y ] ( 1 2 z2 z) (2.19) In order to arrive at an equation describing only te eigt of te fluid, we apply conservation of mass to an element of te fluid, as sown below. As x and Figure 2.1: A volume element y approac zero, te volume of te element can be approximated as (x, y) x y. We now satisfy conservation of mass for an incompressible fluid (2.12), and write tat te time rate of cange of volume of te fluid element ( V ), is simply te t net flux of te fluid crossing te faces of te element. Matematically, tis can be expressed as V (x x t = y 2,y) + x (x,y y 2 ) (x+ x 2,y) u dz y v dz x (x,y+ y 2 ) u dz (2.2) v dz (2.21) Dividing troug by x y and taking limits, we see tat tis is equivalent to ( ) ( ) t = u dz v dz x y (2.22)

15 9 Substituting in our previously derived expressions for u and v and evaluating te definite integral, we arrive at [ [[ t = 1 ( )] M + ( N + σ 2 3 ) x ( µ 3 ) + C ] 2 2 wic can be rearranged as t = 1 [ [ ] ( 3 M 3 µ 3 (N σ 2 ) + 3 x + [ ] ] [(N ] σ 2 3 ) y ( 3 ) y (2.23) ) ] + C2. (2.24) 2 x Tis PDE is our equation of interest wic describes te eigt of te fluid as it evolves in time and space. Here we can see different forces accounted for in te different terms of te equation. Te term containing N represents te vertical component of gravity, wic is maximum wen te te surface is flat, and zero wen it is vertical. Te term containing σ is te surface tension term, wile te M term accounts for te orizontal component of gravity, and te C term accounts for surface sear. In te next section we aim to reduce te parameter dependence of te equation in order to analyze it more easily. 2.3 Nondimensionalization We seek to facilitate te exploration of te droplet beavior by minimizing te number of dynamic variables involved, wic we acieve troug te process of nondimensionalization. Tus recognizing differing lengt scales for lengt in te x direction and for eigt in z, we nondimensionalize as follows: = H, l = L l, t = T t, (2.25) were te capital letters are te dimension carrying terms. For notational convenience we drop te tilde, and remember tat every variable below is now dimensionless. t = [ ( T NH 3 3 3µL T )] ( σh3 T MH 2 2 3µL µL 3 + T CH ) 2µL 2 x (2.26)

16 1 If, in non-dimensionalizing, we coose to make te time scaling of equal order as te sear stress term (and tus scale all oter terms wit respect to tat), we ave T 2µL CH, and t = [ 3 ( G S 2 )] + ( Γ 3 + 2) x (2.27) were G te term originates from te z component of gravity, te S term accounts for surface tension effects, te Γ term accounts for te x-component of gravity, and te unmodified term is te surface sear term, to wic all oter terms are normalized. Te form of te remaining scaling terms is G = 2H2 N 2LC To furter nondimensionalize, we set H = Finally, setting L = σ, we ave N Our final equation is ten of te form (2.28) S = 2σH2 3CL 3 (2.29) Γ = 2MH 3C (2.3) 3LC. Tis gives 2N G = 1 (2.31) σ S = (2.32) L 2 N 2LM 2 Γ = (2.33) 3CN G = 1 (2.34) S = 1 (2.35) Γ = ( 2 3C ) σ 4 N 4 M (2.36) t = [ 3 ( 2 )] + ( Γ 3 + 2) x (2.37) We seek to examine te beavior of a droplet subject to tis equation under varying size, sape, and values of Γ. In addition, we will consider te one-dimensional

17 11 form of te equation as well, and compare simulations from one and two dimensions. Te one dimensional form is simply t = [ 3 ( xx ) x + Γ 3 + 2] x (2.38) Neiter of tese equations are analytically tractable, and tis requires tat we use a numerical solver. We discuss ours in te next capter.

18 Capter 3 Numerical Metods Bot te one-dimensional and two-dimensional forms of te relevant tin film equation are clearly difficult to solve wic requires numerical tecniques. We discuss our metods below. 3.1 Discretization and iteration tecniques We discretize te 2-d equation using a centered finite difference sceme and a fully implicit time step. For convenience, we break it into a system of two equations, making te substitution f = 2. Tis gives us a system were G i,j = S i,j = Γ i,j = C i,j = H i,j = i,j n i,j t + G i,j + S i,j + Γ i,j + C i,j = (3.1) F i,j = f i,j 1 ( x) 2 ( i+1,j 2 i,j + i 1,j ) (3.2) 1 ( y) 2 ( i,j+1 2 i,j + i,j 1 ) =, G [ (i+1,j + 8( x) 2 i,j ) 3 ( i+1,j i,j ) ( i,j + i 1,j ) 3 ( i,j i 1,j ) ] (3.3) + G [ (i,j+1 + 8( x) 2 i,j ) 3 ( i,j+1 i,j ) ( i,j + i,j 1 ) 3 ( i,j i,j 1 ) ] S [ (i+1,j + 8( x) 2 i,j ) 3 (f i+1,j f i,j ) ( i,j + i 1,j ) 3 (f i,j f i 1,j ) ] (3.4) + S [ (i,j+1 + 8( x) 2 i,j ) 3 (f i,j+1 f i,j ) ( i,j + i,j 1 ) 3 (f i,j f i,j 1 ) ] Γ [ (i+1,j + i,j ) 3 ( i,j + i 1,j ) 3] (3.5) 8 x 1 [ (i+1,j + i,j ) 2 ( i,j + i 1,j ) 2] (3.6) 4 x

19 13 and subscript denotes te spatial gridpoint. Since tis is a fully implicit sceme, all entries are taken to be at time n + 1 unless explicitly noted in te superscript (as in n i,j). In a fully implicit discretization of a nonlinear system suc as tis, we guess at a solution and Newton iterate until we converge to a solution tat satisfies te system of equations. In order to guarantee convergence (witin an appropriate region), we construct te following linear system: H 1 F 1 H 2 F 2. H N F N dh 1 dh 1 dh 1 DH 1 d 1 df 1 d 2 df df 1 df 1 df 1 DF 1 d 1 df 1 d 2 df f 1 dh 2 dh 2 dh 2 DH 2 d 1 df 1 d 2 df df 2 df 2 df 2 DF 2 d 1 df 1 d 2 df f 2 = dh N dh N d N df N N f N were N = mn is te number of gridpoints (assuming n rows and m columns), and eac gridpoint is assigned te number jn + i. Solving tis system gives us a vector containing approximate corrections i and f i wic we add to eac i and f i, and use te result as our guess for te next time step. We ten reconstruct te system and solve it again. We continue to modify te and f vectors in tis fasion until te norm of te correction vector is below some tolerance. Satisfaction of tis tolerance signifies tat and f are no longer canging significantly, so te current values are ten accepted as one time step, and te process is repeated wit te new solution becoming te guess for te next time step. For furter information on finite difference metods and Newton iteration, see Burden and Faires [1]. Numerically, we take advantage of te fact te Jacobian matrix above is banded, wit alf-bandwidt = 2n + 1. Tus, for a 1 by 1 grid, we would ave 1 gridpoints wit a Jacobian of 4 million points, of wic 86 are stored. It df N d N df N df N

20 14 sould be noted tat even in tis banded form, te majority of entries are zeros because tere is a solid diagonal of alf-bandwidt 3 down te center, and ten one superdiagonal and one subdiagonal eac 2n away (tese result from te mixed derivative). In between tese diagonals are only zeros. If te terms involving te outer diagonals could be approximated in some manner, it is possible significant improvement in speed could be ad, assuming tat te reduced rate of convergence was offset by te increased speed in solving te matrix system. Te code for solving te one-dimensional version of te equation (actually written first) works by te same metod, wit only te discretization being different: Any term tat involves derivatives in te y direction simply becomes zero. Since tere is only one row, points are labeled straigtforwardly and N just equals n. Tis program runs muc muc faster, since te system being solved is of size 2n wit alf-bandwidt of just 3. For more discussion of te code, see te appendix. 3.2 Precursor film It sould be mentioned tat wen numerically modeling a free surface of a fluid contacting a solid surface (as is te case wit a droplet or te front of a tin film), te typical no-slip condition on wic te equations were derived would not allow te contact line to advance. Tis is because all terms in te PDE are a function of eigt, and wen tis is zero, t is always zero. Tere exist two standard metods of circumventing tis issue. One is a Navierslip metod wic relaxes te no-slip condition very near te contact line, advancing it based on te beavior of te rest of te free surface. Te oter metod (wic we adopt) is to use wat is called a precursor film. Numerically, tis exists as a flat seet of fluid wit eigt at least an order of magnitude less tan te eigt of te drop. Te relatively small eigt prevents excessive diffusion of te drop or film, yet allows te object of interest to cange it s boundaries. A precursor film

21 15 cannot give a completely accurate representation of te drop since diffusion will always be playing a role at te boundaries even wen it sould not be, owever it is computationally muc simpler and faster, and in many cases gives (for tin films at least) approximately te same results [3]. 3.3 Numerical verification Tests were run to confirm tat te code preserved volume, tat te dynamic time step did not yield significantly different results tan te fixed time step, and tat varying grid density did not ave significant effects. It was found tat volume was preserved witin 1 6 of te original volume, tat for te 1-d code te dynamic time-step advanced 3 gridpoints (out of 5) farter of a long-time run, and tat wen sifted to account for tat movement, differences in eigt at gridpoints were on te order of 1 3. None of tese indicate a need for concern wit te numerical functioning of te system.

22 Capter 4 Results In an effort to examine wat governs te beavior of moving droplets, we implemented te numerical solvers to solve te one dimensional form, t = [ 3 ( xx ) x + Γ 3 + 2] x, (4.1) and te two-dimensional form, t = [ 3 ( 2 )] + ( Γ 3 + 2) x, (4.2) of our equation wic describes droplet eigt as a function of time and space. Specifically, we looked at wat droplet sapes evolve, and ow tey differ as a function of drop size, sape, and variations in te parameter Γ. We also ran various cecks to test te validity and applicability of our model to our long-term goal of modeling coalescing droplets on te surface of a wing. 4.1 Droplet pases We begin our discussion of simulation results wit a description of te qualitative pases troug wic a searing drop progresses. Our simulations all began wit a Gaussian-like initial condition, te general form of wic is sown in figure 4.1. Tis was generated by allowing a sperical cap to diffuse free of searing forces until te radial velocity of te edge of te drop ad fallen off considerably (since te drop is on a precursor film it will never completely cease spreading and some cutoff must be subjectively decided upon).

23 x y Figure 4.1: Te form of a typical initial condition drop.

24 18 Te first qualitative stage te droplet goes troug is simply a generally sifting of tis initial condition. Wit te exception of were it meets te precursor film, te surface remains everywere convex, wit te downwind side just steepened and te center of mass sifted in te direction of positive x as can be seen in figure 4.2. After tis, te drop develops somewat of a flattened back and takes on wat migt be called a rounded tetraedron sape, were te edges are composed of te contact lines, and tose bounding te back face of te drop, sown in figure 4.3. We saw in te tetraedron figure a tip developing at te front of te drop. As we progress in time, te tip becomes more and more pronounced. Here, in 4.4 we see tis pronounced tip, as well as a dimple beginning to form beind it. Tis dimple is te beginning of wat can be labeled a crescent-saped pase of te droplet, were te curved front of te drop rises above te inner portions, creating a crescent-saped concavity in te back of te drop. Tis general penomenon can be observed wen blowing water droplets along glass surfaces. Figure 4.5 displays a more fully developed crescent sape. Also in tis figure, te tip can be seen to extend beyond te front of te drop, forming a longitudinal ridge as well. Wat occurs beyond tis point is unclear, as more runs need to be performed out in tis pase of te droplet beavior. Wat as appened in a few cases is tat te tip continues tinning longitudinally, to te point tat in te x-direction it is described by only a few gridpoints, as sown in figure 4.6. Tis may be a numerical artifact wic would disappear wit finer grid resolution, or may be te numerical manifestation of some oter penomenon suc as droplet overturning.

25 x y (a) x y (b) (c) Figure 4.2: a) sows a 2-d view of te deformed drop after time t = 3.7. b) and c) sow crosssections in x and y.

26 x y (a) x y (b) (c) Figure 4.3: a) tree dimensional representation at time t = b) and c) sow cross section in x and y.

27 y y (a) x y (b) (c) Figure 4.4: a) 3-d representation at time t = b) and c) are cross-sections in x and y. Note concavity in y cross-section.

28 x 5 2 y (a) x y (b) (c) Figure 4.5: a) tree dimensional representation of drop at time t = 179. Note longitudinal ridge beind tip. b) and c) are cross section in x and y.

29 x Figure 4.6: Note te extremely narrow widt of te tip. Sortly after tis time, te simulation ceases to function properly.

30 Parameter studies Parameter studies were done, varying Γ, dx, and initial eigt, all independently, and also varying eigt and dx simulataneously by te same factor. Aside from a qualitative examination of te results, we measured te time it took to form a depression in te back of te drop, wic was taken as a measure of ow susceptible te drop is to sear-induced deformation. Te standard run to wic comparisons were made ad a dx of.1, and an initial eigt of.23. Our results are summarized in te cart below. Eac pair of columns gives te time to depression formation for te given paramater value, or in te case of and size variation, te scaling of te reference initial condition. Γ t dx t t size t Runs were performed on a wider range of parameters, owever due to various numerical issues not all of tem gave useful data. We can at least reac tentative conclusions from te data we do ave. We find tat, not surprisingly, a stronger driving force decreases te time to becoming crescent saped, and tat widening te drop does te same. Wen canging bot te eigt alone as well as te size of te drop owever, we find tat te smaller te drop, te more stable it is - tat is, te longer it takes for a depression to form. Tis probably is a result of surface tension being able to keep te surface more uniform as its relative contribution increases te smaller te drop becomes, since curvature necessarily becomes greater.

31 Two drop coalescence Since te motivation beind te project involves modeling te beavior of searing droplets wic coalesce, a ceck was made to ensure te droplet coalescence does not present any special simulation problem. We placed two droplets witin close proximity, so tat after a small amount of time tey would diffuse into eac oter, at wic point we would expect surface tension effects to more quickly bring te drops togeter. Tis is indeed wat occurred, as can be seen in figure 4.7. However, it is not visually obvious from tese results tat surface tension is in fact bringing te drops togeter more quickly tan would occur were te drops just diffusing witout interaction. Tus we ran a single drop diffusion to an equal time and superimposed te resulting profiles. Tis clearly sows te interface eigt is being raised muc more quickly wen te drops are interacting. Tis, combined wit te fact tat a precursor film eliminates a need to track te qualitative difference of weter two drops ave touced, indicates tat our model is bot a convenient and capable means for simulating droplet coalescence on a larger scale. 4.4 Inevitable sear dominance Examining te governing equation of our system, one can see tat since te sear and x-gravity terms are ( 2 ) x and and ( 3 ) x respectively, te sear term will dominate for eigts significantly less tan one. Wile tis does model a pysical reality, wen it is combined wit a precursor film model in wic droplets leave tails and continually expand outwards, it implies tat tere will be inevitable sear dominance of te drop. Tis is because all drops will spread out and subsequently (by conservation of mass) decrease in eigt, and at some point te drop will move in te direction of sear, regardless of wic direction gravity is driving it. Indeed, we see tis penomenon in figure 4.8. Tis is a one-dimensional run, were te profile is displayed canging troug time. Gravity is set working to te rigt,

32 x x y y (a) eigt difference single drop x y (b) 4 6 y (c) Figure 4.7: a) double drop initial condition. b) partial merger. c) more complete merger. d) distinguising merger surface tension effects.

33 27 wile sear is directing te drop to te left. We see tat wile te drop is tall, te peak moves toward te rigt, but eventually te entire drop is driven in te opposite direction. It sould be noted tat tis is a limitation in applicability of te model, rater tan an inerent flaw, as te penomenon of flow reversal can be seen wen a large drop of fluid flows down a windsield, spreading into a rivulet wic undergoes flow reversal once sufficiently drawn out. To properly apply our model, one must eiter be modeling a drop wic does in fact leave a trail, or only consider results obtained before te drop as spread significantly. wind gravity time 8 x Figure 4.8: Tis plot sows a one-dimensional droplet evolving over time. It initially moves to te rigt, but as it spreads out sear dominates and it gets sent in te opposite direction.

34 Comparison of 1-d and 2-d simulations As mentioned in te introduction, work as previously been done by sautors modeling droplets eigt as a function of a single spatial dimension only. Anoter aim of our project was to compare te results of simulation in one and two spatial dimensions to see weter one-dimensional modeling (wic is computationally muc simpler) can give useful insigt into real droplet beavior, weter tere are qualitative features it cannot capture, and wat factors need to be borne in mind wen doing 1-d droplet modeling. To compare results, we set te center longitudinal profile of te 2-d initial condition as te initial condition for te 1-d simulation, and set bot Γ and dx to te same values for te one and two dimensional simulation. Eac droplet was ten run to various fixed times and te resulting profiles compared. Examining figure 4.9, we can see tat te droplet profile remains rougly te same, and moves at very nearly te same velocity. Wile in te first plot te 2-d drop profile as lost a sligt amount of cross-sectional area due to outward diffusion in te y-direction, at a later time it appears to ave gained in mass (indeed, integration over te vector determines tat it as). Te 1-d droplet meanwile, conserves mass in bot cases. Wile differences in te center profile of te drop may not be significant, tere is anoter issue to be examined. If we take longitudinal cross-sections of a 2-d drop we find profiles tat would be very unlikely (if not impossible) to develop in a simple 1-dimensional simulation. As sown in figure 4.1, we find odd profiles wit unusual concavities tat mostly likely can only originate from te combined effects of forces in te x and y directions. Tis implies tat it is not feasible to capture all te relevant features of droplet beavior simply using a one-dimensional model. It appears owever, tat 1-d modeling will give a surprisingly accurate estimate of te bounds for te center profile of a corresponding two dimensional drop.

35 d drop.1 2 d d profile d x x (a) (b) Figure 4.9: Here we compare te central profile of a 2-dimensional drop wit a simple 1-d run at two fixed times (a) = 3.7 and b) = 39.6) Figure 4.1: Tese plots are longitudinal cross-sections of a 2-d drop taken off te centerline. Tey sow profiles wic are likely impossible to get using te 1-d model.

36 3 4.6 Potential surface rougness modeling In a tin film application, Kondic and Diez [7] modeled patterned surface rougness by patterned variations in precursor film eigt. Suc variations in rougness can ave te effect of altering contact angle, and if a drop lies across a boundary were one side as a lower contact angle, te drop will tend to move toward tat side. Tis is a potential metod for controlling droplet beavior on te surface of a wing. If our model is to be used to investigate te possibility of controlling droplet trajectories by altering surface rougness, we need to confirm tat canges in te eigt of te precursor film do in fact result in altered drop trajectories. To ceck weter tis was plausible, we ran a zero sear 1-d simulation on a symmetric drop, were te precursor film was of eigt.1 to te rigt of te drop, and.2 to te left of te drop. Te results can be seen in figure A mild skew to te left can be seen as compared wit te initial condition, toug it is very sligt compared to te diffusion in bot directions. We can owever, ceck tat te location of maximum eigt did in fact sift to te left by two gridpoints. Te effect ten, is real, toug a systematic study caracterizing te relations between precursor film eigt, droplet steepness and velocity of diffusion would need to be performed to determine weter canges in surface rougness (or numerically, precursor film eigt) could be used to effectively guide drops.

37 x Figure 4.11: Tis plot illustrates te (sligt) effect differing precursor film eigts can ave on an oterwise symmetric drop. Te drop as spread out and moved sligtly to left, in te direction of te iger precursor film.

38 Capter 5 Conclusions and Future Work We successfully simulated droplet deformation as a result of gravity and surface sear, in bot one and two dimensions. We were able to caracterize general droplet beavior, as well as describe te effect of canging te driving force and size and sape of te droplet. We were able to investigate te applicability and limitations of our precursor film model, and found tat wile te inerent diffusion involved places limits on te applicability of certain results, our model sould be suited to furter modeling of more complex droplet interactions, suc as may be found on te surface of a wing in rain. We also successfully compared runs of one and two-dimensional drops and found tat wile te 1-d model matces quite closely te front and back edges of a two-dimensional drop, tere are certain qualitative features wic 1-dimensional modeling cannot capture. If te project were to be continued, it is recommended tat te 2-dimensional code be converted to use an Alternating Direction Implicit metod (ADI), wic, as described in Witelski and Bowen [16], greatly decreases computational time, by canging te problem to one wic involves solving many muc smaller matrix equations as opposed to one large one. Once tis is done, more systematic study of bot large scale coalescence as well as te ability to control droplet trajectory troug variations in precursor film eigt could be undertaken. It migt also be wise to investigate metods for realistically andling te inevitable diffusion caused by te use of a precursor film. Lastly, a lofty goal to be kept in mind in is tat of simulation of te film formation and break up process. Ideally by starting wit a randomized array of droplets

39 33 we could observe mass coalescence into a film, followed by film breakup and rivulet formation due to instability. Combined wit explorations into canging droplet trajectory, suc a model could give bot teoretically interesting insigts, as well as information useful from an engineering standpoint.

40 Appendix A Appendix A.1 Discussion of Code Code for solving te one dimensional version of our equation was written in Matlab in te summer of 22. In te fall, it was deemed necessary to convert te code to C bot because it would drastically improve te speed of 1-d simulation, and also because it would be necessary to make 2-d simulation practical at all. Te plan for some time was to use an ADI metod for te 2-d code, owever timeline issues and te intricacies involved led to te development of a clumsier but more easily implementable 2-d code wic operates on te same principles as te 1-d code. As discussed in capter 3, it involves a banded matrix wo s alf bandwidt is twice te number of gridpoints in te x-direction, and wo s lengt is two times te total number of gridpoints. Te matrix solving is by far te limiting step in te speed of our program, and scales strongly wit alfbandwidt, and tus wit te number of gridpoints in te x-direction. Currently, te step takes on te order of 1 to 2 seconds, depending upon bot te gridsize as well as te smootness of te surface being modeled. Te 1-d C code can run toug tens of time steps per second. Tus switcing to ADI would very likely decrease te computational time. Since tey use similar metods, te 1-d code and te 2-d code eac ave a similar structure wit similar supporting files. Bot use a vector class, a matrix class and an smatrix class (te two are equivalent in te 1-d code), all of wic were written specifically for te program. Te vector class performs useful functions

41 35 suc as finding te norm of te vector as well as reading in and printing out vectors from ascii files. Te standard matrix class provides a way of conviently referencing a gridpoints in te 2-d simulation, and can store matrices in bot banded and full forms. Te smatrix class is very similar to te matrix class, except it automatically produces an array wose entries are taylored so tat it can be passed to te dgbsv Fortran banded matrix solver (documentation can be found by going to any searc engine and typing dgbsv). Te main files for eac are big2.c and big2d.c - tese are written in a rater ugly script-like fasion, and te b vector, jacobian, solving, updating, time stepping and outputting all occur ere. Eac also as a file benparam., wic contains constant information about te initial timestep, gridspacing, gridsize, and oter parameter values. Wen performing runs in eiter program, te following items must be noted: Te initial condition vector to be read in is named near te top of big2.c and big2d.c. Te name of te output vectors are canged at te bottom of tose two files. Te 2-d simulation outputs a vector of all te gridpoints, as well as a vector of te time and time step at eac output. Attention must also be paid to adjusting te boundary conditions, bot in constructing b (of Ax + b = ) and in constructing te Jacobian. Bot codes ave similar files plotfixer.m and antiplotfixer.m, designed for relating te output and input to matlab. In eac case, plotfixer takes in te name of a run (wic is output as a giant vector) as well as te gridsize (and in te case of 2-d, te time step to be visualized), and outputs a matrix. For te 1-d code, te rows are entries at given time, and time progresses troug te columns. For 2-d code, it is at a specific time, and a mes of te matrix will sow te sape of te 2-d droplet. Antiplotfixer.m takes a matrix and creates a vector wit te columns appended in an appropriate fasion so tat it can be read as an initial condition.

42 36 Lastly, ccavececker.m takes a 2-d run and it s associated time vector and determines at wat time concavity first appeared in te back of te drop. It does tis by taking cross-sections in te y direction and seeing at wat time te maximum is no longer in te middle of te profile. In general, wen viewed by a CS major, te code will likely appear gastly and inelegant, and a knowledgeable person could likely make many improvements to increase it s ease of use. It does owever, function.

43 Bibliograpy [1] R.L. Burden and J.D. Faires. Numerical Analysis. PWS-Kent Pub. Co., [2] B.A. Campbell and M. Bezos. Steady-state and transitional aerodynamic caracteristics of a wing in simulated eavy rain. NASA, TP-2932, Describes te results of wind-tunnel experiments and te effect rain as on te aerofoil properties. [3] J.A. Diez, L. Kondic, and A. Bertozzi. Global models for moving contact lines. Pys. Rev. E, 63, 2. A paper justifying te use of precursor film models. It analyzes bot slip models and precursor models, determining tat for wetting films, precursor models perform just as adequately as slip models, and tat precursor film models are computationally muc more efficient. [4] P. Dimitrakopoulos and J.J.L. Higdon. Displacement of fluid droplets from solid surfaces in low-reynolds-number sear flows. J. Fluid Mec, 336: , dimensional study drop displacement in Stokes flow. Claims to find tat te usefulness of lubrication models is limited, and terefore will be useful to ceck wit once our simulation is running. [5] P.A. Durbin. On te wind force needed to dislodge a drop adered to a surface. J. Fluid Mec., 196:25 222, Solves for te sape of a droplet just before it is dislodged by te force of wind. Interesting, but not terribly useful. [6] L. Kondic. Instabilities in gravity driven flow of tin fluid films. SIAM Review, 45(1):95 115, 23.

44 38 [7] L. Kondic and J. Diez. Flow of tin films on patterned surfaces: Controlling te instability. Pysical Review E, 65, 22. Tis paper describes teir tecnique of varying te precursor film eigt to simulate canges in surface topology, tereby controlling te wavelengt of te tin film breakup. Will most likely use tis metod wen controlling droplet beavior. [8] X. Li and C. Pozrikidis. Sear flow over a liquid drop adering to a solid surface. J. Fluid Mec, [9] J.A. Moriarty and L.W. Scwartz. Unsteady spreading of tin liquid films wit small surface tension. Pys. Fluids A, 3(5): , May Uses an asymptotic matcing metod to model droplet flow down a wall or seared by wind. Except te equations are te tin film equations for 1-d flow. [1] T.G. Myers. Tin films wit ig surface tension. Siam Rev., 4(3): , September [11] A.D. Scliezer and R.T. Bonnecaze. Displacement of a two-dimensional immiscible droplet adering to a wall in sear and pressure-driven flows. J. Fluid Mec, 383:29 54, Studies te flow of a 2-d droplet between two plates in pressure driven or sear flow for adered and slipping droplets. [12] B.E. Tompson and J. Jang. Aerodynamic efficiency of wings in rain. J. Aircraft, 33(6): , Describes experiments performed to determine te effects tat altering wettability as on te aerodynamic performance of te wing. [13] B.E. Tompson, J. Jang, and J.L. Dion. Wing performance in moderate rain. J. Aircraft, 32(5): , Tis article documents oter experiments

45 39 investigating te beaviro of wings in rain, including te effects of boundary layer tripping. [14] B.E. Tompson and M.R. Marrocello. Rivulet formation in surface-water flow on an airfoil in rain. AIAA Journal, 37(1):45 49, Attempts to describe and predict te onset of rivulet formation in te convecting film on a wing by modeling it as occuring wen te free-surface sear stress equals te stress at te liquid solid interface. Does not actually model fluid flow, toug gives suprising agreement wit experiment regarding te location of rivulet formation. [15] S.K. Wilson, B.R. Duffy, and R. Hunt. A slender rivulet of a power-law fluid driven by eiter gravity or a constant sear stress at te free surface. Q. Jl Mec. Appl. Mat., 55(3):385 48, 22. Introduction provides good background on te state of droplet modeling. Actual model is not 3-d, and mainly deals wit profiles at different times. [16] T.P. Witelski and M. Bowen. Adi scemes for iger-order nonlinear diffusion equations. Pre-print, June 22. Describes te numerical metod we plan to implement, and also gives examples of ow to taylor it to specific equations.