3D adaptive finite element method for a phase field model for moving ...

doi:10.3934/ipi.2013.7.xx

Inverse Problems and Imaging Volume 7, No. 3, 2013, X–XX

3D ADAPTIVE FINITE ELEMENT METHOD FOR A PHASE FIELD MODEL FOR THE MOVING CONTACT LINE PROBLEMS

Yi Shi Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, China

Kai Bao Division of Computer, Electrical and Mathematical Sciences and Engineering King Abdullah University of Science and Technology Thuwal 23955-6900, Kingdom of Saudi Arabia

Xiao-Ping Wang Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, China

Abstract. In this paper, we propose an adaptive finite element method for simulating the moving contact line problems in three dimensions. The model that we used is the coupled Cahn-Hilliard Navier-Stokes equations with the generalized Navier boundary condition(GNBC) proposed in [1]. In our algorithm, to improve the efficiency of the simulation, we use the residual type adaptive finite element algorithm. It is well known that the phase variable decays much faster away from the interface than the velocity variables. Therefore we use an adaptive strategy that will take into account of such difference. Numerical experiments show that our algorithm is both efficient and reliable.

1. Introduction. The moving contact line problem, where the interface of the two immiscible fluids intersects the solid wall, has become one of the most interesting research topics in recent years. It has been shown that classical hydrodynamic equations with no-slip boundary conditions can lead to nonphysical singularity in the vicinity of the contact line. Recently, the generalized Navier boundary condition(GNBC) is introduced to solve this problem [1]. A continuum phase field model which couples the Cahn-Hilliard and Navier-Stokes equations with GNBC is proposed. It is shown that the numerical results of the system agree quantitatively with the results from the molecular dynamics(MD) simulation. The model has been used successfully to study several problems involving the moving contact line[2, 8]. Several numerical methods have been developed for the Cahn-Hilliard and NavierStokes system with the generalized Navied boundary condition [5, 6, 7]. In [7], a gradient stable semi-implicit finite difference scheme is proposed. In [6], a finite element method with a time discretization by operator-splitting combined with least-square/conjugate gradient method is developed, and in [5], a multi-mesh adaptive finite element method is introduced to study the phenomena of precursor film. 2010 Mathematics Subject Classification. Primary: 74S05; Secondary: 76D45. Key words and phrases. Phase field, moving contact line, generalized Navier boundary condition, adaptive finite element. 1

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However, most of the simulations based on GNBC are in two dimensions. To carry out full three dimensional simulations and to achieve high numerical accuracy and capture the physical resolution at an acceptable computational cost, an adaptive method seems necessary. This is particularly important for our phase field model because the interface must be thin enough to attain the sharp interface limit. For our phase-field model, it is well known that the phase field variable φ decays exponentially within a thin layer away from the interface. However, the variation of the velocity variable u away from the interface is much moderate. To simulate this different behavior of the solutions efficiently, a multi-mesh adaptive finite element method was developed in [5], which approximates different components of the solution (velocity, pressure and phase variable) on different h-adaptive meshes. To avoid the complexity of using multi-mesh technique, our adaptive strategy in this paper is simpler. We first refine or coarse the mesh using the phase field variable. We then only refine the mesh using the velocity variable until some tolerance has been achieved. For the mesh refinement or coarsening, we use the a posteriori error estimates. A natural choice of error indicator is the one based on the gradient jump of the finite element solutions on the interface of adjacent element. To test the efficiency and accuracy of our adaptive method, we then carried out several three dimensional simulations. The test examples include: coalescence of two droplets, droplet spreading on stationary surface, droplet motion on moving surface and droplet spreading on chemically patterned surface. These examples have demonstrated reduction in computational costs and saving in storage. The paper is organized as follows: In section 2, we briefly introduce the phase field model with GNBC; In section 3, we describe the finite element discretization and our adaptive technique; In section 4, some numerical results are presented using the method given in section 3, and the paper is finally concluded in section 5. 2. The phase field model with GNBC. Following [1], the moving contact line problem can be modeled by the Cahn-Hilliard Navier-Stokes system ∂φ (1) + u · ∇φ = M ∆µ, in Ω ∂t 1 (2) µ = −ε∆φ − (φ − φ3 ), in Ω ε ∂u Re( (3) + (u · ∇)u) + ∇p = ∆u + λµ∇φ + f , in Ω ∂t (4) ∇ · u = 0, in Ω with the relaxational boundary condition for φ √ 2 πφ (5) φt + uτ ∂τ φ = −Γ(ε∂n φ − πcosθs cos( )), on ∂Ω 6 2 and the GNBC for the tangential velocity uτ along the boundary √ uslip 2 πφ τ (6) = −(∂n uτ + ∂τ un ) + λ(ε∂n φ − πcosθs cos( ))∂τ φ, on ∂Ω Ls 6 2 together with the non-penetration boundary conditions (7)

un = 0, ∂n µ = 0, on ∂Ω

In the above system, u is the velocity, p is the pressure, φ is the phase field, and µ is the chemical potential. n and τ denote the normal and tangential directions respectively. Here the parameter ε denotes the interface thickness, M is the Inverse Problems and Imaging

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phenomenological mobility coefficient, Re is the Reynolds number, λ is inversely proportional to the capillary number. Γ is a positive phenomenological parameter, θs is the static contact angle and Ls is the slip length. The above system is then completed by the initial conditions (8)

u(0) = u0 , φ(0) = φ0

3. The adaptive finite element method for the system. In this section, we first derive the finite element discretization for our Cahn-Hilliard Navier-Stokes system. Then we describe our adaptive technique for this system. 3.1. The finite element discretization. For simplicity, we assume that Ω is a polyhedron domain. Let Th be a regular partition of Ω into tetrahedrons with h be the mesh size of an element T . For each element T ∈ Th , let Pk (T ) = {v| v is a polynomial of degree at most k on T }, where k is a positive integer. We use P2 − P1 finite element spaces to discretize the Navier-Stokes equations and P1 − P1 element to discretize the Cahn-Hilliard equation. Define the finite element spaces Uh and Vh by Uh = {vh | vh ∈ C 0 (Ω); vh |T ∈ P2 (T ), ∀T ∈ Th }, Vh = {qh | qh ∈ C 0 (Ω); qh |T ∈ P1 (T ), ∀T ∈ Th }. To guarantee the uniqueness of the numerical solution of pressure p, we define V0,h by Z V0,h = {qh | qh ∈ Vh , qh dx = 0}. Ω

For the temporal discretization, the Cahn-Hilliard and Navier-Stokes equations are decoupled to simplify the simulation. We employ a semi-implicit convex splitting scheme [7] for the Cahn-Hilliard equations and a linearized semi-implicit scheme for the Navier-Stokes equations to avoid nonlinear iterations. The finite element discretization for the Cahn-Hilliard equations is formulated as follows: find (φh , µh ) ∈ Vh × Vh , such that (9)

(

φn+1 − φnh h , wh ) + (unh · ∇φn+1 , wh ) = −M (∇µn+1 , ∇wh ), ∀wh ∈ Vh , h h ∆t

and (10)

S n+1 S (φ , wh ) − (φnh , wh ) ε h√ ε 1 n 2 πφn − (φh − (φnh )3 , wh ) + h− πcosθs cos h , wh i ε 6 2 n − φ 1 φn+1 h + [h h , wh i + hunτ,h ∂τ φnh , wh i], ∀wh ∈ Vh Γ ∆t

(µn+1 , wh ) = ε(∇φn+1 , ∇wh ) + h h

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After solving φn+1 and µn+1 , the Navier-Stokes equations is then solved by the h h following finite element discretization: find (uh , ph ) ∈ (Uh )2 × V0,h , such that (11)

(

n+1 uh − un n+1 h , vh ) + ν(∇un+1 , ∇vh ) + (un , vh ) h · ∇uh h ∆t

−

(pn+1 ,∇ h

· vh ) −

λ(µn+1 ∇φn+1 , vh ) h h

= (f , vh ) + ν[h−

√ + λh(ε∂n φn+1 − h

un+1 τ,h Ls

, vτ,h i

πφn+1 2 πcosθs cos h )∂τ φn+1 , vτ,h i], ∀vh ∈ (Uh )2 h 6 2

and (∇ · un+1 , qh ) = 0, ∀qh ∈ Vh h

(12)

Here, ( , ) denotes the L2 inner product in Ω, and h , i denotes the L2 inner product on ∂Ω. For the system (3.1)-(3.4), the Cahn-Hilliard and Navier-Stokes equations are solved in a decoupled way, which introduce only mild CFL time constraints. For numerical stability, we find that the largest size of time step can be the same order of the smallest mesh size in the numerical simulations. Two linear systems are obtained for the Cahn-Hilliard and Navier-Stokes equations respectively which can be solved efficiently by the preconditioned generalized minimum residual method(GMRES). Remark 1. It is easy to see that the P2 −P1 pair of finite element space for NavierStokes equations and P1 −P1 pair of finite element space for Cahn-Hilliard equations satisfy the inf-sup stable condition(also known as the LBB condition)[13, 14]. Remark 2. In [7], it is proved that the convex splitting scheme is unconditionally 2 stable when S ≥ 3B2−1 where B = supx∈Ω {|φ|} is about 1.0. In this paper, S = 2.5 is chosen for our simulations, which can satisfy the inequality condition above. 3.2. The mesh adaptive strategy. Here, we use the a posteriori error estimates for our adaptive strategy. A reasonable error indicator is important for the mesh adaption. One natural choice is based on the gradient jump of the finite element solutions on the interface of adjacent element. In our computations, the jumps of finite element solutions on the interfaces are used for both φh and uh . The a posteriori error indicators can be chosen as X Z X Z 1 1 ∂φh 2 ∂uh ∂uh ] de) 2 , ηT (uh ) = ( h3 [ ]·[ ]de) 2 . (13) ηT (φh ) = ( h3 [ ∂ne ∂ne ∂ne e e e∈∂T

e∈∂T

where [·] denotes the jump on the element boundary, and h is the length of edge e. Another important aspect in the adaptive mesh algorithm is the strategy for mesh refinement and coarsening. By the fact that the change of velocity near the interface is more moderate than that of phase field variable and the domain needed to be refined should be wider for the velocity. We use the indicator ηT (φh ) to refine and coarse the mesh, and then use ηT (uh ) to refine the mesh only to satisfy some given tolerances. This could avoid the complexity of using multi-mesh techniques, which is much more complicated for the 3D problems. Given parameters 0 < θ1 , θ2 , θ3 < 1, and prescribed tolerances tol1 and tol2 , our adaptive mesh refine and coarsen strategy include the following two steps. Inverse Problems and Imaging

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Step 1. While max {ηT (φh )} > tol1 , refine the mesh which satisfies T ∈Th

ηT (φh ) > θ1 · max {ηT (φh )},

(14)

T ∈Th

and coarsen the mesh which satisfies ηT (φh ) < θ2 · max {ηT (φh )}.

(15)

T ∈Th

Step 2. While max {ηT (uh )} > tol2 , only refine the mesh which satisfies T ∈Th

ηT (uh ) > θ3 · max {ηT (uh )}.

(16)

T ∈Th

The parameters θ1 and θ3 denote the proportions to be refined by φh and uh , and θ2 denotes the proportion to be coarsened by φh . The choice of tolerances tol1 and tol2 will affect the density of elements at the interface. Thus one can control the minimal element size near the interface by adjusting the tolerances to certain values. In our simulations, we choose θ1 = 0.25, θ2 = 0.05, θ3 = 0.25, tol1 = 0.004 and tol2 = 0.001. The adaptive algorithm mentioned above is found to be effective by our numerical experiments, though there is no theoretical verification for the convergence of solutions. Remark 3. We observe that the choice of parameters is not so sensitive to the convergence of the adaptive algorithm. The numerical tests show that the tolerance between 0.001 and 0.01 is reasonable for both variables φh and uh . 4. Numerical examples. In this section, we present several numerical examples to validate our 3D adaptive method. The test examples include: coalescence of two droplets, droplet spreading on stationary surface, droplet motion on moving surface and droplet spreading on chemically striped surface. The numerical results are obtained by using the adaptive finite element toolbox PHG [23], which uses bisection method for the adaptive local mesh refinement. The parameters we use in the examples are listed as follows, Ls = 0.0025, ν = 1.0, λ = 12.0, ε = 0.02, M = 0.001, Γ = 100. 4.1. Coalescence of two interacting bubbles. Let the computational domain be Ω = [0, 1] × [0, 1] × [0, 1]. In this example, the coalescence of two interacting bubbles due to the effect of surface tension is considered. Let the time step be dt = 0.0005. Figure 1 shows the shape evolution of the two bubbles at time t=0.0005, 0.025, 0.1 and 0.425. As time evolves, the two bubbles coalesce into one larger bubble due to surface tension. Figure 2 shows the mesh at t=0.025 and 0.1. It can be seen that most of the mesh nodes are clustered in the region of interface, which attain our objective of adaptive method. We compare the adaptive results with the results obtained by using 68,705 nodes of uniform grid. Figure 3 shows the comparison of iso-surfaces(φ = 0) obtained by using 68,705 uniform grids and adaptive grids in the slice x = 0.5 when t=0.025 and 0.1. The red line denotes the interface obtained using adaptive mesh and blue line denotes the interface obtained by uniform mesh. We can see that a good agreement between the two results is achieved. Furthermore, the computational time is also significantly reduced by using the adaptive method. If we solve the equation to time t = 1, the computational time of using adaptive mesh is about 1, 822 CPU minutes, while it needs 4, 813 minutes for using the uniform mesh. Inverse Problems and Imaging

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(a) t=0.0005

(b) t=0.025

(c) t=0.1

(d) t=0.425

Figure 1. Coalescence of two interacting bubbles at time t=0.0005, 0.025, 0.1 and 0.425.

(a) Mesh at t=0.025

(b) Mesh at t=0.1

Figure 2. Mesh distribution at time t=0.025 and 0.1.

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(b) Interfaces at t=0.1

Figure 3. Comparison of interfaces in a slice x=0.5 at time t=0.025 and 0.1. 4.2. Droplet spreading on stationary surface. In this section, the evolution of a droplet on a stationary flat surface is simulated. We set the static contact angle θs to be 60o and the time step dt = 0.001. As shown in figure 4(a), a droplet of radius 0.22 is initially placed on the surface with contact angle of 90o . The droplet will then spread and eventually reaches a steady state with static contact angle θs = 60o . Figure 4 shows the spreading of the droplet at time t = 0, 0.05, 0.2 and 0.6. When t = 0.6, it has reached the steady state. The velocity field with the iso-surfaces(φ = 0) in a slice x = 0.5 at t = 0.05 and 0.2 are shown in figure 5. Note that contact line slip is clearly observed. 4.3. Droplet motion on moving surface. In this example, we simulate the motion of droplet on a moving surface. The droplet size, initial condition, and static contact angle is the same as the above example. But we let the bottom wall move at a constant speed uw = 1 in y-direction. GNBC is used for the bottom and top boundaries, and we use the homogeneous Neumann boundary condition for the surrounding boundaries. As shown in figure 6, the downstream contact line moves faster than the upstream contact line. The velocity field with the iso-surfaces (φ = 0) in the slice x = 0.5 at t = 0.05 and 0.2 are shown in figure 7. The grid distributions at t = 0.05 and 0.2 are shown in figure 8. It is clear that grids are concentrated in the the interfacial region, and gradually coarsens away from the interface, which indicates that our adaptive method is effective. Again the comparison of interfaces in the slice x = 0.5 with results from uniform grids are shown in the figure 9 at time t = 0.05 and 0.2. 4.4. Droplet spreading on chemically striped surface. For the last example, we employ the adaptive method to study the problem of droplet spreading on chemically patterned surfaces. As shown in figure 10, the bottom surface is patterned by hydrophilic and hydrophobic stripes with contact angles 60o and 120o respectively. The width of the middle stripe is 0.24. Two situations are considered: the first is that the middle stripe is hydrophobic (Figure 10(a)) and the other is that the middle stripe is hydrophilic(Figure 10(b)). Initially, a droplet of radius 0.22 and contact angle of 90o is placed in the middle of the bottom surface. In the Inverse Problems and Imaging

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(a) initial position

(b) t=0.05

(c) t=0.2

(d) t=0.6

Figure 4. Droplet spreading with static contact of 60o at time t=0, 0.05, 0.2 and 0.6.

(a) Velocity field at t=0.05

(b) Velocity field at t=0.2

Figure 5. Velocity field with the interface in a slice x=0.5 at time t=0.05 and 0.2.

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(a) t=0.05

(b) t=0.1

(c) t=0.2

(d) t=0.3

Figure 6. Droplet motion on moving surface at time t=0.05, 0.1, 0.2 and 0.3.

(a) Velocity field at t=0.05

(b) Velocity field at t=0.2

Figure 7. Velocity field with the interface in a slice x=0.5 at time t=0.05 and 0.2.

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(a) Mesh at t=0.05

(b) Mesh at t=0.2

Figure 8. Mesh distribution at time t=0.05 and 0.2.

(a) Interfaces at t=0.05

(b) Interfaces at t=0.2

Figure 9. Comparison of interfaces in a slice x=0.5 at time t=0.05 and 0.2.

(a) Surface 1

(b) Surface 2

Figure 10. Chemically striped bottom surfaces with contact angles 60o and 120o .

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(a) t=0.01

(b) t=0.1

(c) t=0.3

(d) t=0.8

Figure 11. Droplet spreading on chemically striped surface 1 at time t=0.01, 0.1, 0.3 and 0.8.

(a) t=0.01

(b) t=0.1

(c) t=0.3

(d) t=0.8

Figure 12. Droplet spreading on chemically striped surface 2 at time t=0.01, 0.1, 0.3 and 0.8.

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first case, the droplet in the hydrophobic part retracts, and it spreads into the two hydrophilic parts, as shown in figure 11. Viewed form top, the final shape of the droplet is a butterfly-like configuration, which is located over two hydrophilic and one hydrophobic stripes. In the second case, where the middle stripe of the bottom surface is hydrophilic. Figure 12 shows that the parts of droplet in the hydrophobic stripe retract back to the middle hydrophilic stripe gradually. The droplet seems to try to avoid the hydrophobic surface, and covers solely on the hydrophilic region eventually. Our numerical results agree quite well with the results presented in [24, 25, 26, 27] using the lattice Boltzmann method. 5. Conclusions. We have developed a three-dimensional adaptive finite element method for a phase field model for the moving contact line problem. Our method is based on a gradient stable semi-implicit scheme with mesh adaptation based on a posteriori error estimates. A two step adaptive strategy is employed in order to take into account of the behavior differences of phase field and velocity field near the interface. Our three dimensional numerical results have shown that the method is both efficient and reliable. Acknowledgments. This publication was based on work supported in part by Award No. SA-C0040/UK-C0016, made by King Abdullah University of Science and Technology (KAUST), Hong Kong RGC-GRF Grants 605311, 604209 and the National Basic Research Program Project of China under project 2009CB623200. REFERENCES [1] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306, 15 pp. [2] Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59–78. [3] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun. Comput. Phys., 1 (2006), 1–52. [4] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, A variational approach to the moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333–360. [5] Yana Di and Xiao-Ping Wang, Precursor simulations in spreading using a multi-mesh adaptive finite elment method, J. Comput. Phys., 228 (2009), 1380–1390. [6] Qiaolin He, R. Glowinski and Xiao-Ping Wang, A least square/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line, J. Comput. Phys., 230 (2011), 4991–5009. [7] Min Gao and Xiao-Ping Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372–1386. [8] Xiongping Luo, Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line over undulating surfaces, Solid. Stat. Commun., 139 (2006), 623–629. [9] Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310–1322. [10] Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293–317. [11] Pengtao Yue, Chunfeng Zhou, James J. Feng, Carl F. Ollivier-Gooch and Howard H. Hu, Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys., 219 (2006), 47–67. [12] Chunfeng Zhou, Pengtao Yue and James J. Feng, 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids, J. Comput. Phys., 229 (2010), 498–511. [13] Xiaobing Feng, Fully discrete finite element approximations of the Navier-Stokes-CahnHilliard diffuse interface model for two phase flows, SIAM J. Numer. Anal., 44 (2006), 1049–1072. Inverse Problems and Imaging

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[14] Xiaobing Feng and A. Prohl, Error analysis of a mixed finite element method for the CahnHilliard equation, Numer. Math., 99 (2004), 47–84. [15] V. Girault and P.-A. Raviart, “Finite Element Method for Navier-Stokes Equations. Theory and Algorithms,” Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. [16] Xianliang Hu, Ruo Li and Tao Tang, A multi-mesh adaptive finite element approximation to phase field models, Commun. Comput. Phys., 5 (2009), 1012–1029. [17] H. D.Ceniceros, Rudimar L. Nos and Alexandre M. Roma, Theree-dimensional, fully adaptive simulations of phase-field fluid models, J. Comput. Phys., 229 (2010), 6135–6155. [18] Yana Di, Ruo Li and Tao Tang, A general moving mesh framework in 3D and its application for simulating the mixture of multi-phase flows, Commun. Comput. Phys., 3 (2008), 582–602. [19] Nikolas Provatas, Nigel Goldenfeld and Jonathan Dantzig, Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys., 148 (1999), 265–290. [20] Jie Shen and Xiaofeng Yang, A phase-field model and its numerical approximation for twophase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159–1179. [21] Qiang Du and Jian Zhang, Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations, SIAM J. Sci. Comput., 30 (2008), 1634–1657. [22] Yinnian He, Yunxian Liu and Tao Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616–628. [23] Linbo Zhang, Parallel hierarchical grid. Available from: http://lsec.cc.ac.cn/phg/index_ en.htm. [24] J. M. Yeomans and H.Kusumaatmaja, Modelling drop dynamics on patterned surfaces, Bull. Pol. Acad. Sci.: Tech. Sci., 55 (2007), 203–210. [25] J. L´ eopold` es, A. Dupuis, D. G. Bucknall and J. M. Yeomans, Jetting micron-scale droplets onto chemically heterogeneous surfaces, Langmuir, 19 (2003), 9818–9822. [26] A. Dupuis and J. M. Yeomans, Droplet dynamics on patterned substrates, Pramana J. Phys., 64 (2005), 1019–1027. [27] Qingming Chang and J. I. D. Alexander, Analysis of single droplet dynamics on striped surface domains using a lattice Boltzmann method, Microfluid Nanofluid, 2 (2006), 309–326. [28] R. Verfurth, “A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques,” Wiley-Teubner, 1996.

Received July 2012; revised December 2012. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]

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