A Numerical Sceme for Particle-Laden Tin Film Flow in Two Dimensions Mattew R. Mata a,, Andrea L. Bertozzi a a Department of Matematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, California, 90095-1555 Abstract Te pysics of particle-laden tin film flow is not fully understood, and recent experiments ave raised questions wit current teory. Tere is a need for fully two-dimensional simulations to compare wit experimental data. To tis end, a numerical sceme is presented for a lubrication model derived for particle-laden tin film flow in two dimensions wit surface tension. Te sceme relies on an ADI process to andle te iger-order terms, and an iterative procedure to improve te solution at eac timestep. Tis is te first paper to simulate te two-dimensional particle-laden tin film lubrication model. Several aspects of te sceme are examined for a test problem, suc as te timestep, runtime, and number of iterations. Te results from te simulation are compared to experimental data. Te simulation sows good qualitative agreement. It also suggests furter lines of inquiry for te pysical model. Keywords: adaptive timestepping, alternating direction implicit, coupled system, particle-laden, surface tension, tin film 1 2 4 5 6 7 8 9 10 11 12 1 14 15 1. Introduction In recent years, te problem of numerically solving gravity-driven tin film flow for clear fluids as ad ample work done in bot one and two dimensions. However, te case wen te film contains particles suspended witin it as received less attention, especially in two dimensions. Te evolution of a clear fluid down an inclined plane is modeled using a single partial differential equation and numerical scemes ave been derived using finite differences [9, 16] and finite elements [2]. For similar equations, suc as spreading tin films, tere are metods for finite elements in one dimension [10, 11, 7] and for finite differences in two dimensions [5]. Te incorporation of particles into suc a flow leads to anoter variable in te model, namely te particle concentration, and an accompanying equation related to te evolution of te particles. Te result is a system of equations tat requires a different approac from te clear fluid case to formulate a practical numerical sceme, due to te coupling of te equations. An active area of researc in te last decade as been te development of numerical metods for iger-order tin film equations including complex fluids described by systems of equations. Related problems include metods for coupled systems of nonlinear parabolic equations [22, 26]. Principal corresponding autor Email addresses: mattewmata@mat.ucla.edu Mattew R. Mata, bertozzi@mat.ucla.edu Andrea L. Bertozzi Preprint submitted to Journal of Computational Pysics April 22, 2011
16 17 18 19 20 21 22 2 24 25 26 27 28 29 0 1 2 4 5 6 7 8 9 40 41 42 4 44 45 46 47 48 49 50 51 52 5 54 55 56 57 58 Te sceme presented ere is, in part, inspired by recent models for surfactants [4] and tin films [5]. We coose an Alternating Direction Implicit ADI sceme as a tractable metod for implicit timesteps, because surface tension introduces a severe restriction on te timestep in te case of explicit scemes. Tis ADI approac also allows for an implicit sceme wile avoiding to ave to solve te large sparse linear algebra problems by an iterative metod, suc as GMRES, tat result from linearizing te two-dimensional operators in Newton s metod [5]. ADI is also amenable to parallelization. Wile ADI scemes for numerically solving parabolic equations date back to te 1950 s [27], teir use in iger-order problems is rater new, e.g., [5], and not all tat wellstudied. However, te ease of parallelization makes suc scemes a viable coice for multiprocessor platforms. Since teir inception, ADI scemes ave been extended to andle parabolic problems wit mixed derivative terms [2, 8, 24, 0], variable coefficients [15, 5], and ig-order terms [5]. Te ideas present in tese scemes can be combined to create an efficient way to numerically solve te particle-laden tin film flow equations. Te nonlinearity and iger-order terms are andled in a similar manner to Witelski and Bowen [5], wic dealt wit tin film equations, and te remaining terms are treated as in Warner et al. [4], wic devised a semi-implicit sceme for surfactants. Tis combined approac is fine-tuned to draw out better efficiency, via adaptive timestepping and an iterative procedure witin eac timestep. At te cost of te extra calculations due to te iterative nature of te sceme, te timestep needed for stability can be improved over recent metods. Te result is an efficient metod to simulate te continuum model in two dimensions. Te full pysics of particle-laden tin film flow is not well understood. Recent experiments, and teir comparison to te model, ave raised questions. We present suc a comparison in tis paper, were te results sow qualitative agreement. In particular, by performing two-dimensional simulations, we are able to observe finger formation and compare directly wit experiments. Te development of quantitatively correct models for tese systems is an ongoing active area of researc. Tus, tere is a need for accurate, fully two-dimensional simulations of te model, suc as in te case of mudslides and oil spills. Te paper is organized as follows: Section 2 presents te system of evolution equations for te flow. In Section, te numerical sceme for tis system is derived. Section 4 covers te adaptive timestepping sceme implemented in te code. A complete explanation of te spatial discretization is given in Section 5. Te practicality and implementation of a moving reference frame in te simulations are discussed in Section 6. Numerical simulations are presented in Section 7. We compare te results generated from te numerical sceme to an experiment using silicone oil and glass beads in Section 8. Finally, in Section 9, we provide a discussion of te results and future work. 2. Model Te results from experiments indicate tat particle-laden tin film flows exibit tree distinct regimes, based on te initial particle concentration and angle of inclination [6]. For low concentrations and angles, te particles settle to te substrate wit clear fluid flowing over te top. Te beavior after sedimentation is similar to clear fluid experiments, suc as tose performed by Huppert [14]. Hig concentrations and angles cause a particle-ric ridge to emerge at te front of te flow. Medium concentrations and angles lead to a particle concentration wic appears to stay well-mixed trougout te duration of te experiment. Based on Cook [5], tis beavior likely 2
59 60 belongs to one of te two previously mentioned regimes, but may not ave evolved to te point were tis distinction can be made. Figure 1: Te coordinate system and variables considered in tis problem. x is in te plane, in te direction of te flow; y is in te plane, perpendicular to x; and z is normal to te plane. is te film tickness and ϕ is te particle concentration. 61 62 6 64 65 66 67 Te evolution equations for te flow are based on te regime were te inclination angle and particle concentration are bot ig enoug to induce te formation of a particle-ric ridge. Te equations are formulated in terms of te tickness of te film,, and te particle concentration by volume, ϕ see Figure 1. Te equations for modeling tis regime were first derived in Zou et al. [6]; re-derived in Cook et al. [6], using conservation of volume rater tan mass; and modified in Cook et al. [7], adding in a sear-induced diffusion term to correct for an instability affecting ϕ. Te dimensionless system [7] is t + v av = 0, 1 68 69 70 71 72 7 74 75 ϕ t + [ϕ v av + 1 ϕv rel F diff ] = 0. 2 Te orientation for 1-2 is suc tat x lies in te plane and is parallel to te direction of te flow, y is across te plane and perpendicular to x, and z is normal to te plane. A one-dimensional form of te problem considers only te x-direction, wile two dimensions includes bot x and y. Te two velocity terms, v av and v rel, are te volume-averaged velocity of te fluid and te velocity of te particles relative to te liquid, respectively. We use te term liquid to refer to te substance tat te particles are suspended in and fluid to refer to te mixture as a wole. In Equation 2, v av + 1 ϕv rel is te individual velocity of te particles [6] and F diff is sear-induced diffusion of te particles.
76 77 78 79 80 81 82 8 84 85 86 87 88 89 90 91 92 9 94 95 96 97 98 99 100 101 102 10 104 105 106 107 108 109 Te volume-averaged velocity of te liquid and te particles togeter is [ v av = 2 2 µϕ 2 Dα µϕ ρϕ 5 ] 8 µϕ ρϕ + ρϕ µϕ 2ˆx, were te terms in come from surface tension, te effects of gravity normal to te inclined plane, and te effects of gravity parallel to te inclined plane. Te density of te fluid as a wole is ρϕ = 1 + ρ f ϕ; ρ f = ρp ρ l ρ l is te difference in te densities between te particles and te liquid. Te function µϕ = 1 ϕ/ϕ max 2 [18, 1] is te effective fluid viscosity, were ϕ max is te maximum packing fraction of particles, assuming te particles are speres. For tis problem, te maximum packing fraction as been empirically determined to be 0.58, wile te teoretical value is 0.64 []. Dα = Ca 1/ cot α [] is a modified capillary number, were Ca is te capillary number of te liquid and α is te angle of inclination of te plane on wic te fluid is flowing α = 0 corresponds to te plane being orizontal wile α = π/2 to vertical. Te settling velocity of te particles, relative to te velocity of te liquid, is a combination of tree factors, assumed to be multiplicative, v rel = V s fϕwˆx. 4 Te coefficient V s = 2 a2 ρ f in 4 is te Stokes settling velocity of a single spere settling in a viscous liquid, were a is te dimensionless particle radius. A indered settling function, in tis case te Ricardson-Zaki function fϕ = 1 ϕ 5 [29], accounts for te effect of sedimentation. Te particles settling parallel to te substrate is modeled using a wall effects function, w = A/a 2 / 1 + A/a 2 2 wit A = 1/18. Tis function is an approximation to a metod of images solution to a single spere falling parallel to a vertical wall [1]. Tis as te property tat it is near 0 for small and near 1 for large. Since te system 1-2 is fourt-order and contains iger-order terms but 4 does not, v rel is not regularized. Tis leads to an instability affecting te particle concentration in numerical simulations [7]. To correct for tis, a sear-induced diffusion term 5 was added in, F diff = 2 a2 Ca 1/ ˆDϕ 2 ρϕ ϕ. 5 µϕ Tis beavior can be seen in a one-dimensional example on te domain x : 0 50 wit x = 0.05. Te initial film tickness is a jump, from 1 to 0.05 at x = 25, smooted by yperbolic tangent. Te initial particle concentration is taken to be ϕ = 0.. Tis simulation is similar to tose described in Section 7, and a moving reference frame is used, as discussed in Section 6. By time t = 1000, te solution witout te extra diffusion term as developed an instability near x = 10 Figure 2 wile te one wit it is still stable Figure. Note tat te oscillations trailing te particle-ric ridge, between x = 0 and x = 10 are a result of te discretization of te moving reference frame and are discussed in Section 6. Equation 5 accounts for orizontal diffusion of particles in te fluid caused by orizontal gradients of ϕ and was derived based on results from Leigton [20] and Leigton and Acrivos [21]. Te term ˆDϕ = 1/ϕ 2 1 + 1/2e 8.8ϕ is a dimensionless diffusion coefficient. 4
0.2 0.1 φ 0. 0.29 0.28 0 10 20 0 40 50 x Figure 2: Te numerical solution of ϕ at time t = 1000 witout sear-induced diffusion. By tis time, an instability as developed near x = 10. 0.2 0.1 φ 0. 0.29 0 10 20 0 40 50 x Figure : Te numerical solution ϕ at time t = 1000 wit sear-induced diffusion 5. Te solution is still stable due to te extra term. 110 111 112. Numerical Sceme In te case of a gravity-driven clear fluid flow, te model reduces to a single equation [] for te film tickness,, 5
11 114 115 116 117 118 119 120 121 122 12 124 125 126 127 128 129 10 11 12 1 14 15 16 17 18 19 140 141 142 14 144 145 146 147 148 149 150 151 152 15 t + x + 2 Dα = 0. 6 Solving 6, and similar problems, numerically in one and two dimensions as been performed using several different metods [1, 9, 16, 2, 2, 5]. Including particles in te pysics not only adds a second equation, but couples it to te equation for te film tickness. Te particle-laden case as been solved numerically in one dimension wit metods suc as forward Euler wit upwind differencing [6] and te Lax-Friedrics metod [6] wen te ig-order terms are omitted, and backward Euler wit centered differencing [6] wen te terms are included. Tis system of PDEs in two dimensions poses numerical difficulties beyond tose present in te clear fluid problem. For bot te clear and particle-laden cases, fully explicit scemes typically ave te problem tat an O x 4 timestep, assuming x = y, is needed for stability. One solution is to use an implicit sceme. For te clear fluid and similar problems, te nonlinearity combined wit an implicit sceme amounts to solving te problem at eac timestep using an iterative process, suc as Newton s metod, to converge to te solution [5]. For te particle-laden case, using an implicit sceme typically requires tat bot equations be solved simultaneously, using an iterative process to account for te nonlinearity. Tis results in a linear algebra problem wit twice te number of unknowns and a matrix tat is twice as large in eac dimension, compared to te clear fluid problem. Terefore, solving te particle-laden case leads to larger linear algebra problems to solve at eac timestep and te matrix from Newton s metod will ave a more complex structure tan for clear fluids. Te goal of te sceme presented ere is to circumvent some of te aforementioned difficulties. Te advantages of tis approac, over a purely explicit sceme or implicit wit Newton s metod, is tat te timestep is more lenient tan for a fully explicit sceme and te linear algebra problem tat results from te implicit part of te sceme is reduced to a series of smaller banded matrix solves, wic can be done efficiently and independently for eac equation. Te numerical sceme tat we employ for te particle-laden tin film flow problem is inspired by te scemes presented in Witelski and Bowen [5] for iger-order parabolic PDEs and Warner et al. [4] for surfactants. In Witelski and Bowen, several ADI scemes, based on backward Euler, second-order backward difference formulas, as well as Newton-like scemes, are derived for solving te nonlinear PDE known as te tin film equation, t + f 2 = 0. 7 Te backward Euler-based ADI sceme for 7 uses approximate values of in te nonlinear and mixed-derivative implicit terms. It is suggested to start wit approximations, suc as time-lagged values, for evaluating tese terms and calculating te numerical solution at te timestep. Ten use tis solution for te new approximate values witin te same timestep and recalculate. Tis results in an iterative sceme at eac timestep. However, for solving te tin film equation, it was noted tat te iterations did not provide a noticeable improvement. Warner et al. use tis metod for a coupled system of nonlinear PDEs relating to surfactants. Tey andle te iger-order terms implicitly using Crank-Nicolson, and apply ADI to tis. Te remaining terms, wic are at least second-order in space, are treated explicitly. For te nonlinear and mixed-derivative terms, te values are time-lagged and te problem is solved only once per timestep. In te simulations, x = y = π/100 0.014 required a timestep of O10 5. Our approac is to andle applicable terms implicitly, using ADI, and treat te remaining terms explicitly, as we sow below. Te terms andled implicitly are tose wit spatial derivatives on te 6
154 155 156 157 158 159 160 161 162 16 164 165 166 167 same variable as te time derivative. For example, Equation 1 as te time derivative on, so te terms treated implicitly sould ave spatial derivatives on. Making tis coice allows for te splitting of te two-dimensional operators into to te product of two one-dimensional operators in te derivation of te ADI sceme. Iterations witin eac timestep allow for a larger t to be taken at te cost of some extra calculations. In general, te increase in te size of te timestep outweigs te extra computational work, as sown in Section 7. For Equation 1, te terms µϕ 2 + ρϕ µϕ ˆx 8 can be andled implicitly. Tis is because te spatial derivatives on tese terms are applied to. Of tese terms, some parts of tem will be andled by approximation, as in Witelski and Bowen [5]. Including te first-order terms in te implicit treatment allows tem to be discretized spatially using centered differencing to maintain stability. Solving tis equation numerically assumes tat ϕ is known, or can be approximated, and we are solving for. First discretize te terms in 8 in time wit backward Euler, including te time derivative, Write out te operators in 9 fully, n+1 + t µϕ 2 + ρϕ n+1 µϕ ˆx = n. 9 n+1 + t [ x µϕ xxx + y µϕ yyy ] n+1 ] ρϕ n+1 + x µϕ + t [ x µϕ yyx + y µϕ xxy = n. 10 168 169 170 171 172 Te idea beind te ADI approac is to reduce te implicit part of 10, wit derivatives in bot x and y, to a product of two operators, eac wit only derivatives in eiter x or y. To acieve tis, te terms involving only x-derivatives and only y-derivatives are grouped togeter. Define te operators D x = x µϕ xxx + ρϕ n+1 n+1 µϕ 2 I, D y = y µϕ yyy. 11 Ten replacing te terms in 10 wit te definitions in 11, we ave n+1 + td x + D y n+1 12 ] n+1 + t [ x µϕ yyx + y µϕ xxy = n. 17 174 175 In order to obtain an ADI sceme from 12, note tat I + td x + td y = I + td x I + td y t 2 D x D y and so te left-and side, wit te addition of an O t 2 term, can be written as a product of two one-dimensional operators. 7
176 177 178 179 180 181 I + td x I + td y n+1 t 2 D x D y n+1 1 ] n+1 + t [ x µϕ yyx + y µϕ xxy = n. To andle te nonlinear terms, wic occur in front of derivatives, and mixed-derivative terms in 1, define tem as approximate, denoted by a tilde e.g., n+1. Te approximate terms can be cosen in some reasonable manner, suc as time-lagged or extrapolated. Tis will be discussed in more detail later. Subtract te mixed-derivative terms from and add te O t 2 term to bot sides. Tis leaves a sceme in wic all te terms operating on n+1 are known, as is te entire rigt-and side. I + t D x I + t D y n+1 = n 14 { [ ]} n+1 + t 2 Dx Dy t x µ ϕ yyx + y µ ϕ xxy n+1. 182 For simplicity, define te operators in 14 as L x = I + t D x, Ly = I + t D y. 18 184 185 Subtracting L x Ly n+1 from bot sides of 14, wic cancels te O t 2 term, yields L x Ly n+1 n+1 n+1 = n t µ ϕ 2 ρ ϕ ˆx n+1 + µ ϕ. 15 At tis point, te implicit part of te sceme is complete and te explicit terms can be added back into 15 using forward Euler. 186 Define L x Ly n+1 n+1 n+1 = n t + t { [ Dα µ ϕ 2 ρ ϕ ˆx n+1 + µ ϕ 16 µϕ ρϕ 5 ]} 4 n 8 µϕ ρϕ. 187 188 u = n+1 n+1, wic can be tougt of as a correction term to te approximation of n+1, and 16 can be written as a tree-step process: two one-directional solves 17-18 and an update step 19. L n+1 x v = n t + t { [ Dα µ ϕ 2 ρ ϕ ˆx n+1 + µ ϕ 17 ]} 4 n, µϕ ρϕ 5 8 µϕ ρϕ 8
L y u = v, 18 189 190 191 192 19 194 195 196 197 198 199 200 201 202 20 204 205 206 207 208 209 210 211 212 21 214 215 216 217 218 219 220 n+1 n+1 + u. 19 Since te operators L x and L y involve at most fourt-order terms, te spatial discretization of tem will lead to a five-point stencil in te x- and y-direction, respectively. Tis discretization is discussed fully in Section 5. Along eac row/column of te discretized domain, tis results in a pentadiagonal linear algebra problem. Tis can be solved using a pentadiagonal solver, or a more generic banded matrix solver. To elp wit te inaccuracy in te nonlinear and mixed-derivative terms resulting from approximation, an iterative procedure can be used at eac timestep to improve te solution and size of te timestep. Tis was first suggested for te ADI sceme in te context of tin film equations [5]. Tis procedure amounts to repeating te tree-step process associated wit solving eac equation at eac timestep and updating te approximate solution wit te most recent solution, until te new and approximate solutions sufficiently converge. For example, one would solve 17-19, solve 29-1, and examine ow muc te approximate solution differs from tis computed solution. If tis difference is significant, one can replace te old approximate terms wit te computed solution and solve te same timestep again. Tis process can be continued until te approximate and computed solutions are close. Tis is similar to fixed-point iteration. n+1 For Equation 1, wen entering te timestep, a coice must be made as to te value of and ϕ n+1. Using as an example, two reasonable coices would be a time-lagged approximation, n, wic is a first-order accurate approximation in time, or an extrapolated approximation, 2 n n 1, wic is second-order in time. For adaptive timestepping, tis extrapolation is given by n + t/ t old n n 1, were t is te prospective timestep between t n and t n+1 and t old is te timestep between t n 1 and t n. Wile te second coice of an approximation is second-order, it also requires storing an extra set of data, namely n 1. Oter coices for estimating n+1 and ϕ n+1 based on previous data could be used as well. Wit tis coice made, te tree-step process for eac equation can be implemented, obtaining a solution for n+1 and ϕ n+1. We refer to te case wen te solution obtained ere is accepted as performing One Iteration. However, at tis point, te approximation can be redefined, n+1 = n+1, and te process repeated. Tis can be continued until convergence between te approximate and new solution, or equivalently wen te correction term u is small in a cosen norm. We refer to tis case as Iterations since te problem is solved iteratively for eac timestep. For 2, te ADI metod is applied to ϕ as a wole, since te time derivative is on tis term. Te applicable terms in te equation are [ ] ϕ Dα ρ 2 ρϕ f µϕ ϕ + ϕ µϕ 2 + 1 ϕv s fϕw ˆx. 20 As wit 1, te time discretization of 20 is based on a backward Euler metod ϕ n+1 + t [ ϕ Dα ρ 2 f ρϕ +ϕ µϕ 2 + 1 ϕv s fϕw µϕ ϕ n+1 ˆx] = ϕ n. 21 9
221 222 22 224 225 226 227 228 229 Writing out te operators in 21 explicitly, ϕ n+1 tdαρ f [ x ϕ 2 ϕ 2 µϕ xϕ + y ρϕ + t x [ϕ µϕ 2 + 1 ϕv s fϕw ] n+1 µϕ yϕ 22 ] n+1 = ϕ n. Define te operators in 22 involving only x-derivatives and only y-derivatives as D x and D y, respectively. ϕ 2 n+1 [ ] n+1 ρϕ D x = Dαρ f x µϕ x + x µϕ 2 1 ϕv s fϕw I, 2 n+1 ϕ D y = Dαρ f y µϕ 2 y. Using 2, te equation can be compactly written as ϕ n+1 + t D x + D y ϕ n+1 = ϕ n. 24 Note tat tere are no mixed-derivative terms to andle in 24. Te left-and side can be written as te product of two one-dimensional operators, incurring an O t 2 term in te process. I + td x I + td y ϕ n+1 t 2 D x D y ϕ n+1 = ϕ n. 25 Add te O t 2 term to bot sides of 25, and make all terms tat occur nonlinearly at time t n+1 approximate, as before. I + t D x I + t D y ϕ n+1 = ϕ n + t 2 Dx Dy ϕ n+1. 26 Define 20 21 L x = I + t D x, Ly = I + t D y and subtract L x Ly ϕ n+1 from bot sides of 26 to obtain L x Ly ϕ n+1 ϕ n+1 = ϕ n+1 ϕ n [ t Dα ρ ϕ 2 f ϕ + µ ϕ ϕ ρ ϕ n+1 µ ϕ 2 + 1 ϕv s f ˆx] ϕw. Te remaining terms can be incorporated into 27 via forward Euler. L x Ly ϕ n+1 ϕ n+1 = ϕ n+1 ϕ n [ t Dα ρ ϕ 2 f ϕ + µ ϕ ϕ ρ ϕ n+1 µ ϕ 2 + 1 ϕv s f ˆx] ϕw 28 [ 2 2 t ϕ µϕ 2 Dα µϕ 5 ] n 8 µϕ ρϕ F diff. 27 10
22 2 Define w = ϕ n+1 ϕ n+1. Ten 28 can be written out as te tree-step process 29-1: L x v = ϕ n+1 ϕ n [ t Dα ρ ϕ 2 f ϕ + µ ϕ ϕ ρ ϕ n+1 µ ϕ 2 + 1 ϕv s f ˆx] ϕw 29 [ 2 2 t ϕ µϕ 2 Dα µϕ 5 ] n 8 µϕ ρϕ F diff, L y w = v, 0 24 25 26 27 28 29 240 241 242 24 244 245 246 247 248 249 250 251 252 25 254 255 256 ϕ n+1 ϕ n+1 + w. 1 Te spatial operators in te L x and L y terms are at most second-order, and spatial discretization leads to a tree-point stencil in eac direction. Similar to 17 and 18, a tridiagonal solver or banded matrix solver can be used to solve along eac row/column. Solving te system, as a wole, at eac timestep can be ten acieved by solving 1 using 17-19 for n+1, solving 2 using 29-1 for ϕ n+1, ten recovering te particle concentration as ϕ n+1 = ϕ n+1 / n+1. Note tat eac solve only uses values n, n+1, ϕ n, and ϕ n+1, all of wic are known. Tis sceme can be solved in oter possible ways. One migt coose to use, after solving 1, n+1 in lieu of an approximation for n+1 for solving 2. Alternatively, te equations could be solved in te opposite order. 4. Adaptive Timestepping We use an adaptive timestepping sceme to advance te solution. Te sceme utilizes te solution at consecutive timesteps t n 1, t n, t n+1. Based on a measure of error, it decides weter or not to accept te new solution, and if it is reasonable to increase te size of te timestep. Tis is a modification of te sceme used in Bertozzi et al. [4], in wic it serves as an estimate of a dimensionless local truncation error in time. Consider te solution of te film tickness,, at times t n 1, t n, and t n+1. Calculate e n+1 = n+1 n / n and e n = n n 1 / n. Te modification from te original metod is to divide by te value n at eac point rater tan n max = max i,j { n i,j}, since it produces a better-working adaptive sceme for tis problem. Denote te timestep going from time t n to t n+1 as t and from t n 1 to t n as t old. Ten define Error = en+1 t e n t old. 2 Tis provides a dimensionless estimate of te local truncation error in time, accumulated over te grid. Te solution will be accepted if tis error is less tan some tolerance, denoted Tol 1. If te error is less tan a smaller tolerance, Tol 2 < Tol 1, for a fixed number of steps, te timestep is increased by a scale factor. If te error is larger tan Tol 1, te maximum number of iterations witin a timestep 11
257 258 259 260 261 262 26 264 265 266 267 268 269 270 271 is surpassed, or te solution becomes negative, te timestep is reduced by a factor of 2. An example for Tol 1 and Tol 2 would be 10 7 Area of Domain and 10 9 Area of Domain respectively, were te difference in te tolerances are at least an order of magnitude apart to prevent te error from alternating between too large to accept and small enoug to increase te timestep. Te form of tese tolerances were cosen to make it convenient for various size domains witout aving to cange te tolerances manually for eac domain. Since 2 only takes into account one of te two variables, tis error can be computed for ϕ, or merely ϕ, as well. Tese two errors can be combined into an overall measure of te error by taking te maximum of te two, or by some oter reasonable combination suc as adding te two errors togeter or coosing a separate set of tolerances for eac. 5. Spatial Discretization We use centered finite differences for all spatial discretizations. Using te notation, i+1/2,j i,j + i+1,j /2, te fourt-order term in 1 is µϕ 2 i+1/2,j µϕ i+1/2,j xxx,i+1/2,j i+1/2,j + µϕ i+1/2,j yyx,i+1/2,j i,j+1/2 + µϕ i,j+1/2 xxy,i,j+1/2 i,j+1/2 + µϕ i,j+1/2 yyy,i,j+1/2 i,j i 1/2,j µϕ i 1/2,j xxx,i 1/2,j i 1/2,j µϕ i 1/2,j yyx,i 1/2,j i,j 1/2 µϕ i,j 1/2 xxy,i,j 1/2 i,j 1/2 µϕ i,j 1/2 yyy,i,j 1/2 / x / x Here, te tird derivatives are calculated at alf-grid points by differencing consecutive standard second-order approximations. Two representative examples are / y / y. 272 Te two second-order terms are discretized as xxx,i+1/2,j i+2,j i+1,j + i,j i 1,j / x, 4 xxy,i,j+1/2 i+1,j+1 2 i,j+1 + i 1,j+1 / x 2 5 i+1,j 2 i,j + i 1,j / x 2 / y. µϕ ρϕ i+1/2,j µϕ i+1/2,j ρϕ i+1,j i+1,j ρϕ i,j i,j i,j+1/2 + µϕ i,j+1/2 ρϕ i,j+1 i,j+1 ρϕ i,j i,j i,j i 1/2,j µϕ i 1/2,j ρϕ i,j i,j ρϕ i 1,j i 1,j i,j 1/2 µϕ i,j 1/2 ρϕ i,j i,j ρϕ i,j 1 i,j 1 / x 2 6 / y 2, 12
27 274 275 276 277 278 279 280 281 282 28 284 285 286 287 288 289 290 4 µϕ ρϕ i,j 4 i+1/2,j µϕ i+1/2,j ρϕ i+1,j ρϕ i,j 4 i,j+1/2 + µϕ i,j+1/2 ρϕ i,j+1 ρϕ i,j 4 i 1/2,j µϕ i 1/2,j ρϕ i,j ρϕ i 1,j 4 i,j 1/2 µϕ i,j 1/2 ρϕ i,j ρϕ i,j 1 / x 2 7 / y 2. Te advective term is discretized using a standard centered-differencing sceme. Te terms in 2 are discretized in te same manner since many of tem are similar to tose in 1. Te fourt- and second-order terms tat come from v av are discretized as in -7, wit replaced by ϕ. Bot advective terms are discretized via standard centered differencing. Te sear-induced diffusion term is discretized te same way as 6-7. + ˆDϕ i+1/2,j 2 i+1/2,j ρϕ i+1/2,j µϕ i+1/2,j ˆDϕ i,j+1/2 2 i,j+1/2 ρϕ i,j+1/2 µϕ i,j+1/2 ˆDϕ 2 ρϕ µϕ ϕ i,j ϕ i+1,j ϕ i,j ˆDϕ i 1/2,j 2 i 1/2,j ρϕ i 1/2,j ϕ i,j ϕ i 1,j µϕ i 1/2,j ϕ i,j+1 ϕ i,j ˆDϕ i,j 1/2 2 i,j 1/2 ρϕ i,j 1/2 ϕ i,j ϕ i,j 1 µϕ i,j 1/2 Centered differencing is not used for te moving reference frame, if one is employed. Instead, a second-order upwind differencing sceme is used, wic will be discussed in te next section. 6. Reference Frame Te area of interest in te simulations is near te front of te flow, were effects like te capillary and particle-ric ridges occur. Wit a fixed reference frame, te spatial domain would need to be taken as te entire area over wic te flow would evolve, leading to large portions of te domain were no cange is occurring. Tis issue can be easily addressed by using a moving reference frame. To implement a moving reference frame, we add an extra term to eac equation, s x on te left-and side of 1 and sϕ x on 2. Here, s > 0 is te speed at wic te moving reference frame travels. Zou et al. [6] approximate te front speed by removing all terms from te equations wic are iger tan first order, leaving only te advective terms. Tey observe tat tese terms capture te large scale dynamics, including te speed of te socks, and te ridges tat develop in and ϕ. Tis leaves a 2 2 system of conservation laws of te form / x 2 / y 2. t + [F, ϕ] x = 0, 8 ϕ t + [G, ϕ] x = 0, 9 F, ϕ = ρϕ µϕ, G, ϕ = ρϕ µϕ ϕ2 + ϕ1 ϕv s fϕw. 1
291 Te initial conditions for 8-9 are x, 0 = { l, x 0, r, x > 0, 40 ϕx, 0 = { ϕ0 l, x 0, ϕ 0 r, x > 0. 41 292 29 294 295 296 297 were l and r in 40 and 41 are te initial film tickness and te eigt of te precursor b, respectively, and ϕ 0 in 41 is te initial particle concentration of te fluid. Tese initial conditions specify a Riemann problem [19]. From te initial sock in bot equations, an intermediate state emerges, i, ϕ i. Te weak form of tis system produces two Rankine-Hugoniot jump conditions, wic define te sock speeds, aead and beind te intermediate states. For s 1, te speed of te sock beind te intermediate state, and s 2, te speed aead, tese conditions are given by s 1 = F i, ϕ i F l, ϕ l i l = G i, ϕ i G l, ϕ l ϕ i ϕ l, 42 s 2 = F r, ϕ r F i, ϕ i r i = G r, ϕ r G i, ϕ i ϕ r ϕ i. Figure 4: Te intermediate states tat develop in te film tickness left and particle concentration rigt for te first-order system of equations. 298 299 00 01 02 0 04 05 06 Te intermediate states and socks can be seen in Figure 4. Tis nonlinear system 42 of four equations and four unknowns, i, ϕ i, s 1, and s 2, can be solved via Newton s metod. For te simulations sown in Section 7, our reference frame speed is an average of te two speeds, s = s 1 + s 2 /2. Te discretization of te terms for te moving reference frame is done explicitly using forward Euler combined wit second-order upwind-differencing, s x s i+2,j + 4 i+1,j i,j. 2 x Tis was cosen over explicit first-order upwind and implicit centered differencing. For a test run to time t = 10 wit no variation in te y-direction, implicit centered differencing produced te igest particle-ric ridge, but introduced small oscillations aead of te flow tat were approximately 2% 14
07 08 09 10 11 12 1 14 15 16 17 18 19 20 21 22 2 of te eigt of te ridge. First-order upwind was dissipative and lead to te ridge being 28% smaller tan implicit centered differencing. Te effects of coosing second-order upwind appear to be some minor dissipation, about 17% as compared to implicit centered differencing, and dispersion, wic was not seen in tis test case, beind te particle-ric ridge. Te moving reference frame can be used for bot te one- and two-dimensional cases see Figures 5 and 6. To demonstrate tis, simulations were run under te same conditions as tose in Section 7. Te teory-based solution for te problem witout iger-order terms 8-41 aligns well wit te one-dimensional numerical solution for te full problem. Te two-dimensional solution for te full problem wit a perturbation to te initial film tickness leads to a finger tat moves faster tan te one-dimensional case and te trougs, to te sides of te finger, move slower. Tis can be viewed more succinctly in Figures 7 and 10, were te contours of te perturbed two-dimensional case are sown. Te position of te finger runs aead of te one-dimensional front, wic is approximately at x = 15, wile te trougs lag beind. Te averaging of te front position was first investigated by Huppert [14] for experiments involving clear fluids. Bot simulations start wit te same volume and, after an initial transient, te average front positions for te one- and perturbed two-dimensional case measured at = 0.5 stay close to eac oter Figure 8. Figure 9 sows te position of te finger and te troug in te two-dimensional case over time. 1.75 1.5 1.25 1 0.75 0.5 0.25 0 0 5 10 15 20 25 0 x Figure 5: Comparison of teory and simulations at time t = 100 for te film tickness, : teory witout igerorder terms solid line, one-dimensional solution to te full problem dased line, perturbed two-dimensional finger dotted line, and perturbed two-dimensional troug dot-dased line. Te domain in te y-direction is 15 units long, wit te finger slice taken at y = 7.5 and te troug slice taken at y = 1 see Figure 7. 24 25 26 7. Simulations A rectangular domain is used wit te x-direction oriented down te inclined plane and te y-direction across te inclined plane. In all cases, te particle concentration is initially taken to be 15
0.2 0.1 φ 0. 0.29 0 5 10 15 20 25 0 x Figure 6: Comparison of teory and simulations at time t = 100 for te particle concentration, ϕ. Te labels are te same as in Figure 5. 15 1 15 1.6 0.9 1.4 10 0.8 0.7 10 1.2 0.6 1 y 0.5 y 0.8 5 0.4 0. 5 0.6 0.2 0.4 0 0 5 10 15 20 25 0 x 0.1 0 0 5 10 15 20 25 0 x 0.2 Figure 7: A contour plot of te simulation at times t = 0 left and t = 100 rigt for te film tickness,, in te perturbed two-dimensional case. Te perturbation in two dimensions leads to a fingering instability not seen in te one-dimensional case. 27 28 29 0 1 2 ϕx, y, 0 = ϕ 0, were 0 ϕ 0 ϕ max. Tis corresponds to aving a well-mixed initial fluid. Te film tickness far beind te contact line is set at x, y, 0 = 1 and aead of te flow, a precursor of eigt x, y, 0 = b is assumed. At te contact line, a perturbation to a linear front can be applied to induce beavior suc as a fingering instability. Te parameters in te model are taken to be: a = 0.1, ρ f = 1.7, Ca = 10, α = π/4. Te constant ϕ max is taken to be 0.67, in line wit te simulations in Cook et al. [7]. Te initial timestep is set to t = 10 6 and te mes widt is 16
60 Flat Perturbed Average Front Position 50 40 0 20 0 20 40 60 80 100 Time t Figure 8: Te average front position of te film tickness,, of te one-dimensional and perturbed two-dimensional case up to time t = 100. After an initial transient, te average front positions stay close to eac oter. 80 70 Finger Troug 60 Front Position 50 40 0 20 10 0 20 40 60 80 100 Time t Figure 9: Te front position of te film tickness,, of te perturbed two-dimensional case up to time t = 100 along te finger and troug. 4 x = y = 0.05. For te model, two sources contribute to te eigt of te film tickness and particle concentra- 17
15 0.12 0.1 10 0.08 0.06 y 0.04 5 0.02 0. 0 0 5 10 15 20 25 0 x 0.298 Figure 10: A contour plot of te simulation data at time t = 100 for te particle concentration, ϕ, in te perturbed two-dimensional case. Te perturbation leads to a particle-ric ridge tat outlines and begins to fill in te finger. 5 6 7 8 9 40 41 42 4 44 45 46 47 48 49 50 51 52 tion near te front of te flow. Te first is te iger-order terms, suc as surface tension, wic produce smoot ridges in bot and ϕ. Second, even witout tese terms, an intermediate state at te front emerges for bot variables, iger tan eiter of teir respective left or rigt states. Tese eigts are dependent on te precursor b. Te eigt of te precursor in te following simulations is cosen to be te same as x. In general, te coice of precursor as a small effect on te speed of te flow, but a large effect on bot te film tickness and particle concentration. To illustrate tis, Table 1 sows te eigt of te intermediate states for bot and ϕ as well as te speeds of te trailing and leading socks obtained from te teory-based solution to te system of conservation laws 8-41 see Section 6 for a more in-dept discussion. Te intermediate film tickness i and particle concentration ϕ i increase as te eigt of te precursor b decreases. For te sock speeds, a smaller precursor leads to te trailing sock speed s 1 staying relatively te same, but te leading sock speed s 2 slows down and approaces s 1. Tese results agree wit te previous ones related to solving te system of conservation laws [6, 6]. For tis model, te smallest precursor for wic a solution exists is b 9 10 4 [6]. A precursor close to tis case, b = 0.001, produces socks speeds wic are close togeter and an intermediate particle concentration near te maximum packing fraction. An alternative settling function tat permits solutions wit smaller precursors, f B ϕ = 1 ϕ/ϕ max 5, is examined in Cook et al. [6]. 5 54 55 56 Te boundary conditions for are Diriclet in front and beind, in te x-direction, te flow and Neumann on te sides, in te y-direction. Te same is employed for ϕ. In addition, all tird derivatives in, normal to te boundary, are set to 0. More specifically, for a rectangular domain wit lengt X 0 and widt Y 0, te boundary conditions are 18
b i ϕ i s 1 s 2 0.1 1.0165 0.07566 0.4592 0.510221 0.05 1.0478 0.1558 0.45914 0.48782 0.025 1.07107 0.01 0.45901 0.471418 0.0125 1.1427 0.56006 0.459289 0.465441 0.00625 1.28276 0.96078 0.459294 0.462488..... 0.001 9.14247 0.65545 0.459788 0.459916 Table 1: Te intermediate states and sock speed solutions from Equation 42 based on te precursor tickness b. As te precursor decreases, bot i and ϕ i increase and te sock speeds converge. 0, y = 1, xxx 0, y = 0, X 0, y = b, xxx X 0, y = 0, y x, 0 = 0, yyy x, 0 = 0, y x, Y 0 = 0, yyy x, Y 0 = 0, ϕ0, y = ϕ 0, ϕx 0, y = ϕ 0, ϕ y x, 0 = 0, ϕ y x, Y 0 = 0. 57 58 Te simulations are all run using moving reference frames, wit te speed of te frame determined as in Section 6. 8 7 Simulation Data N 6 Speed Up 5 4 2 1 1 2 4 5 6 7 8 Number of Processors N Figure 11: Te speed-up gained by going from 1 to N processors using OpenMP. Te line y = N is sown as a point of reference. 59 60 61 Te code is written in parallel using te C++ OpenMP package. Tis coice of parallelization was made since te majority of calculations are done via for loops and OpenMP works well wit loop-eavy code. Tis includes te calculation of all finite differences and te solves along rows and 19
62 6 64 65 66 67 68 69 70 71 72 7 74 columns associated wit te ADI part of te sceme. Tis is especially useful since rows/columns can be solved independently of eac oter for eac equation. In addition, writing special solvers for linear systems of equations across multiple processors [25, 28] is avoided by tis approac. Te speed-up attained using N processors is calculated by dividing te runtime for one processor by te runtime for N processors Speed-Up = Time1 Processor/TimeN Processors. Based on Figure 11, te scaling is close to linear up to 4 processors, wit a small drop-off in performance as te number increases. Tis almost-linear beavior is a result of all of te code, outside of a few minor calculations and te recording of te data, being amenable to parallelization. To test some preferences tat need be cosen a priori in te simulation, we conducted sorttime tests to gauge te effectiveness of eac approac. Te ones considered ere are a weter to time-lag or extrapolate te approximate terms and b weter or not to perform iterations past a single solve to improve te approximate terms, and terefore te solution at eac timestep see Table 2. a Approximate Terms Time-Lagged Extrapolation b Number of Iterations One Iteration Iterations Table 2: Te two coices to be made wen implementing te numerical sceme. One must coose weter to a time-lag or extrapolate te approximate terms and b weter or not to perform additional iterations past te initial solve. 75 76 77 78 79 80 81 82 8 84 85 86 87 88 Consider an initial condition of ϕ 0 = 0. and a front perturbed from Riemann initial data, x, y, 0 = 1 far beind te front, x, y, 0 = 0.05 far aead of te front. At te jump from fluid to precursor, te sape of te front is given as x front = X 0 /2 cos2πy/y 0. Tis initial data is ten smooted using yperbolic tangent and matced to te boundary condition see Figure 14. Tis as te effect tat te initial timestep can be taken more leniently. We ran tis initial simulation for eac of te four combinations in Table 2 to time t = 1 and te maximum timestep allowed, average number of iterations per timestep, and te total runtime, in seconds, are listed in te table below Table. Tis and Table 4 provide some global measures to compare te different scemes rater tan illustrating convergence studies for any particular metod. Te coice of t = 1 was made as te timestep canges dramatically over tis time interval and can provide insigt as to wat metods seem practical for long-time runs. Since adaptive timestepping is utilized ere, te tolerances are tuned so as to ensure tat te simulation stays stable, not only to time t = 1 but for some time afterwards as well it is taken up to t = 100 in tis case, wic is te lengt of te long-run simulations. t max Avg. Iter. Runtime Time-Lagged and One Iteration 0.00056841 1.0 518.2 Time-Lagged and Iterations 0.0018296 2.20997 601.468 Extrapolation and One Iteration 4.0774 10 5 1.0 19596.1 Extrapolation and Iterations 0.004868 1.29668 76.60 Table : Results for time t = 1 based on various coices for implementation. 20
89 90 91 92 9 94 95 96 97 98 99 Using Iterations performs well for bot coices of approximate terms in tat te total runtimes are low, te maximum timesteps are large, and te number of iterations stays close to 1. Between tese two, Extrapolation and Iterations does best, wit nearly one fewer iteration required per timestep, on average, and a runtime tat is 7% sorter. Performing One Iteration, te runtime for Time-Lagged is in between te two cases wit Iterations, but for Extrapolation, it performs poorly, producing a runtime tat is to 52 times worse tan te oter tree options. Tis is due to te small maximum timestep tat is associated wit tis approac, wic is 14 to 119 times smaller tan te oter tree. At tis point, it makes sense to discard te Extrapolation and One Iteration approac due to its excessive runtime and explore te remaining ones. Under te same conditions, we ran a longer simulation, tis time to t = 100. Using te best remaining options, we can glean some idea as to wic ones will work best for a longer simulation. t max Avg. Iter. Runtime Time-Lagged and One Iteration 0.00107169 1.0 17811. Time-Lagged and Iterations 0.002917 2.95498 115.8 Extrapolation and Iterations 0.0106161 2.01204 64.9 Table 4: Results for time t = 100. 400 401 402 40 404 405 406 407 408 409 410 411 412 41 414 415 416 417 418 419 420 421 422 42 424 425 Comparing Tables and 4, te maximum timestep for eac approac as increased. Using Iterations, te average number as gone up in for bot Time-Lagged and Extrapolation. However, te average number of iterations per timestep for Extrapolation is approximately one iteration fewer tan for Time-Lagged. Also te runtime takes about 2.9 times longer for Time-Lagged compared to Extrapolation. One can see te benefit of performing iterations instead of using a smaller timestep in comparing te results for Time-Lagged and One Iteration and Time-Lagged and Iterations. Time- Lagged and One Iteration advances te solution approximately te same time forward wit tree timesteps as Time-Lagged and Iterations does wit one timestep and tree iterations. However, doing two extra timesteps costs more tan two extra iterations, as seen in teir respective runtimes. Tis is because te explicit terms do not need to be re-calculated for eac iteration wile tey do for eac timestep. Terefore, te only two options wic make sense to use are te ones involving Iterations. Of tese, Extrapolation is te clear favorite. In Figure 12, we see tat by time t = 8, all tree approaces ave settled into a respective timestep. Te timestep for Extrapolation and Iterations does best, followed by Time-Lagged and Iterations and Time-Lagged and One Iteration. Te timestep for Extrapolation and Iterations is.2 times better tan Time-Lagged and Iterations and 9.9 times better tan Time-Lagged and One Iteration. Te benefit of te larger timestep for bot approaces wit Iterations is partially offset by te need for extra calculations related to te iterations. Figure 1 sows te number of iterations required trougout te simulation. For Extrapolation and Iterations, te increase in iterations approximately between times t = 20 and t = 0 corresponds to te finger forming and stretcing out aead of te flow in te film tickness and te particle-ric ridge growing iger and outlining te finger. Wile te number of iterations jumps once to and ten back down to 2 for Extrapolation and Iterations, it remains constant at for Time-Lagged and Iterations. Te cost of storing extra data and performing a small computation to find te extrapolated approximations seems a small price to pay to save one iteration per timestep, wic includes recalculating values involving te approximate terms and performing te ADI solves. 21
0.012 0.01 Timestep t 0.008 0.006 0.004 Time Lagged and One Iteration Time Lagged and Iterations Extrapolation and Iterations 0.002 0 0 5 10 15 20 Time t Figure 12: Te adaptive timestep up to time t = 20. Te timestep, t, is recorded in intervals of 0.25 for te tree cases. Extrapolation and Iterations as a significantly larger timestep tan eiter Time-Lagged and One Iteration or Time-Lagged and Iterations. Number of Iterations 4 2 Time Lagged and Iterations Extrapolation and Iterations 1 0 20 40 60 80 100 Time t Figure 1: Te number of iterations up to time t = 100. Te iterations are recorded in intervals of 0.25 for te two cases. Using Extrapolation and Iterations does better tan Time-Lagged and Iterations in terms of fewest number of iterations. 426 427 428 429 40 41 Using te simulation data up to t = 100, we can examine te effects of te initial perturbation grapically. For te film tickness, a small capillary ridge forms in te center of te perturbation Figure 15 and begins to stretc out aead of te bulk flow Figures 16 and 17. Tis is te well-known fingering instability present in tin film flows. For te particle concentration, a particleric ridge initially forms at te contact line Figure 15 and, as te fingering instability evolves, outlines te sape of te finger Figures 16 and 17. Directly beind te ridge, a pocket of lower 22
Figure 14: Te initial film tickness. It is perturbed by a cosine wave along y and smooted along x by yperbolic tangent. Figure 15: Film tickness left and particle concentration rigt at time t = 25. A small ridge forms in bot, wit te igest point in te perturbation. 42 4 44 45 46 concentration forms. Te interior of te finger is slowly encroaced upon by te particles tat ave accumulated near te back and sides of te finger. Tis can be seen in Figure 17 as an interior layer along te inside of te particle-ric ridge. It is possible tat tis penomenon is not pysical, meaning tat it occurs only in te simulations and not in te experiments, and may be a result of te current model not containing all of te necessary pysics. 2
Figure 16: Film tickness left and particle concentration rigt at time t = 50. particle-ric ridge form. A fingering instability and Figure 17: Film tickness left and particle concentration rigt at time t = 100. Te fluid finger stretces out aead of te bulk flow. Te particle-ric ridge increases in concentration and as a iger concentration in and around te fingering instability. 47 48 49 440 441 442 44 444 445 446 447 8. Comparison to Experiments Experiments for particle-laden tin film flows ave been compared in one dimension to te solution, bot analytically and numerically, for constant-volume clear fluid flows. Te average front position for clear fluids is given by a power law, were te location of te front scales like Ct 1/, were C is a scaling constant [14]. Ward et al. [] compare te average front position of te flow to tis scaling and find agreement for particle concentrations below ϕ = 0.45 and deviations at later times for iger concentrations. Grunewald et al. [12] compare te average front position to a re-derived one-dimensional model, based on results from Huppert [14] wit a precursor, and to experiments and numerical solutions of te one-dimensional problem. Te Ct 1/ scaling appears valid for concentrations of 0.25 to 0.45, and te scaling constant for experiments and numerics are witin 20% of te teoretical constant. We seek to compare te numerical solution in two 24
448 449 450 451 452 45 454 455 456 457 458 459 460 461 462 46 464 465 466 467 468 dimensions to images of experiments, taking into account tat variations occur across te front of te flow. We use 1000 cst polydimetylsiloxane PDMS, a silicone oil, for te liquid component of te fluid. For te particles, glass beads wit diameters in te range of 250 425 µm are used. Te two components are ten well-mixed and released down an inclined plane from a reservoir. Tis corresponds to a constant-volume experiment, wereas our simulations are constant-flux. Te approximation of a constant-volume problem by a constant-flux one may be invalid at early times, as te eigt of te fluid will be canging rapidly. However, te eigt of te flow canges slower at later times, at wic point a constant-flux approximation may be valid. Te experiment, wic we will compare to simulation, is a fluid of approximately 90 cm containing a volume wic is 5.9% particles. Te plane is inclined at a 2-degree angle. Te fluid is allowed to flow down te plane, wic is 14 cm across and 90 cm down. In te experiments, te flow starts out close to uniform across te front, away from te edges, and over time develops instabilities, in te form of fingers stretcing out aead of te bulk flow. Since, for simulations, starting wit a uniform front along te y-direction leads to a uniform solution, we start te simulation some time after te start-time to add a perturbation to te initial data, wic induces te type of beavior seen in te latter stages of te experiments. In order to avoid simulating te problem over te entire domain, we truncate te solution near te front and treat te problem locally as being constant-flux. We are interested in te dynamics of finger formation during wic time te film tickness only canges by at most 20%, so a local approximation by constant-flux is reasonable. Figure 18: Te initial condition of te experiment, used for comparing wit te simulation. At tis point, te front of te flow as begun to develop perturbations, wic will lead to fingering instabilities. 469 470 We use two images, taken tree minutes apart, to compare wit te simulation Figures 18 and 20. Te first image is taken wen te front of te flow as reaced approximately 5 cm down 25
471 472 47 474 475 476 477 478 479 480 481 482 48 484 485 486 487 488 489 490 491 492 49 494 te plane. Te sape of te front is parabolic-like wit two large perturbations at eiter end of te front. In between, smaller perturbations exist wic lead to fingering instabilities. Te two outer perturbations lead to longer and ticker fingers tan te smaller inner perturbations. We take a front similar to tis in our simulation. Te scales for a constant-flux problem can be taken from Cook et al. [6], wic are te same as for te clear fluid case. Te eigt scale is taken to be 0 = 1 mm. Te lengt scale is x 0 = l 2 0 1/, were te capillary lengt, l, is l = γ/ρ l g. Te constants are γ, te coefficient of surface tension; ρ l, te liquid density; and g, te component of gravity parallel to te inclined plane. Te time scale is t 0 = µ l /γx 0 l 2 / 2 0, were µ l is te dynamic liquid viscosity. Te capillary number is given by Ca = µ l x 0 /γt 0 = 2 0/l 2. Te scales, given tese parameters, are 0 = 0.001 m, x 0 = 0.0016196 m, l = 0.00205041 m, t 0 = 0.925 s, and Ca = 0.079286. Using tis, we can construct an initial condition wic resembles te experiment and will produce similar results. Tis is done by measuring te features of te initial image and creating a similar condition. Wile te flow in te experiment is asymmetric, we take a symmetric initial condition in te simulation wic as features tat are approximately, in bot location and size, te same as in te experiment. Te track is taken to be 86.75 units wide rounded up to te nearest 0.05 increment, wic is te value of x, y, wic corresponds to te 14 cm wide track. Te precursor in te simulation is set to b = 0.05, as in te previous simulations. A moving reference frame is used since tis is taken to be a constant-flux problem locally. Te speed of te moving reference frame is approximately s = 0.4198, calculated as in Section 6. Running a simulation over te course of tree minutes leads to a distance traveled for te frame of approximately 10.69 cm, were te actual displacement, based on experiments, is around 12 cm, so using te constant-flux assumption seems to produce a decent approximation of te distance te fluid will flow. Figure 19: Te initial condition for te film tickness,, used in te simulation. Tis is an artificially-created starting condition to be representative of te state sown for te experiment. Te eigt is in mm. 26
495 496 497 498 Te initial data is generated using a sine wave to form te two large perturbations and te space away from te edges. Te tree fingers tat develop between tese two perturbations are simulated wit a cosine wave of small amplitude, 0.25 in dimensionless units Figure 19. Te simulation is run to t = 19.06, te equivalent of tree minutes of real-time. Figure 20: Te evolution of te experiment after tree minutes. Te fingering instability starts to form at te front. 499 500 501 502 50 504 505 506 507 508 509 510 511 512 51 514 515 516 517 518 Over te course of te tree minutes, te exterior of te outer fingers in te experiments go from 4 cm and 6.5 cm on te left and rigt, respectively, to 7.5 cm and 12 cm. Te interior of tese fingers go from less tan 1 cm on eac side to about cm. Te interior fingers are not discernable in te initial image. Te flow as a wole, measured from were te fluid touces te walls, as moved about 11 cm down te plane. Te interior fingers in te experiment, extend about 0.5 cm aead of te flow. In te simulation Figure 21, te moving reference frame accounts for 10.69 cm of movement, so te position were te fluid touces te edges as moved approximately 11.4 cm. Te evolution of te fingers in te simulation is sligtly less pronounced tan in te experiments. Tis is likely due to te simulation initially undergoing a transient state were te fluid travels slower tan at later times, wile te transient in te experiment as occurred prior to tis tree-minute interval. Te exterior of te outer fingers is approximately 4.2 cm and interior 1.2 cm. Te interior fingers extend aead of te flow about 0.8 cm. Te tip of te longest finger in te experiments as moved 15 cm wile in te simulations, it as advanced approximately 11.4 cm. Te tips of te fingers, in te z-direction, reac up to 1.7 mm. Te particle concentration cannot be determined accurately at te particle-ric ridge in te experiment, but te increased opacity at te leading edge of te flow indicates an increase in te concentration, relative to te ambient concentration. Tis cange in sade is approximately 1 cm long in te direction of te flow. In te simulation Figure 22, te tickness of te ridge ranges from 0.6 to 1.1 cm, wic is consistent wit te experiments. 27
Figure 21: Te evolution of te film tickness,, in te simulation after tree minutes. Bot te experiment and simulation exibit a fingering instability, but te instability in te simulation is less pronounced. Te eigt is in mm. Figure 22: Particle concentration, ϕ, for te film tickness in Figure 21. 519 520 521 9. Discussion Scemes originally derived for numerically solving ig-order parabolic problems ave recently been extended to ig-order systems, suc as te case of surfactants and particle-laden tin films. 28