Weierstrass Institute for Applied Analysis and Stochastics Motion of thin droplets over surfaces numerical and modeling techniques for moving contact lines Dirk Peschka, Sebastian Jachalski, Barbara Wagner Stefan Bommer, Ralf Seemann, Georgy Kitavtsev, Luca Heltai ICERM Workshop „Making a Splash - Droplets, Jets and Other Singularities“ www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de
Outline 1. Setting 2. Droplets on solid planar surfaces (solid substrates) • Modeling • Contact angle: regularization vs free boundary • Free boundary approach: numerical algorithm • Examples 3. Flows over liquid substrates (discuss extension of the method and compare with experiments)
1. Setting
Setting Setting : Newtonian liquids Viscous (Re=0) partial wetting Droplet motion is a free boundary problem with contact lines. Droplet shape is a graph h , but h has finite support. Support is unknown and can depend on time. www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 4 / 41
Setting z Γ Ω Ω = { ( x, z ) : 0 < z < h ( t, x ) } Γ = { ( x, z ) : z = h ( t, x ) > 0 } www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 5 / 41
2. Droplets on solid planar surfaces Modeling
Modeling Stokes flow with free boundary �r p + µ r 2 u = 0 , in Ω r · u = 0 , � � � p I + 2 µ D ( u ) n = σκ n , on Γ � � · n = 0 , u � v Γ | r h | = tan θ , at x ± www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 7 / 41
Modeling Stokes via Helmholtz-Rayleigh variational principle Seek u so that D ( u , v ) = �h di ff E, v i for all v . d E d t = h di ff E, u i = � D ( u , u ) 0 Z µ Z β − 1 u · v D ( u , v ) = 2 D ( u ) : D ( v ) + { z =0 } Ω Z E = σ Γ Helmholtz (1869), Rayleigh (1873),…,Onsager (1929) www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 8 / 41
Modeling Thin-film limit in a nutshell: D ( u , v ) = �h di ff E, v i Z Z β − 1 uv D = D ( u ) : D ( v ) + z =0 Z Z h Z β − 1 uv ⇡ ( ∂ z u )( ∂ z v )d z d x + z =0 0 Z Z h Z ( ∂ z u ) v d x | h 0 + β − 1 uv | z =0 = ( � ∂ zz u ) v d z d x + 0 Z Z Z Z r h r ˙ 1 + | r h | 2 d x ⇡ p � 1 + 1 2 | r h | 2 � h di ff E, v i = h v d x σ = d x σ σ Z h ! Z = � r h · r r · v d z d x 0 Z Z h = ( �r ∆ h ) v d z d x Note: ˙ h = ∂ t h 0 Z h ˙ h + r · u d z = 0 Conservation of mass + kinematic condition: 0 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 9 / 41
Modeling For given h (0 , x ) seek h ( t, x ) such that For given seek such that ˙ � � h � r · m ( h ) r π = 0 π = δ E δ h = � ∆ h, m ( h ) = | h | n z where h ( t, x ± ( t )) ≡ 0 where | h x ( t, x ± ) | = tan θ ⇣ m ⌘ x ± = lim ˙ h π x x → x ± Existence weak solutions: Bernis & Friedmann (1990) Positivitiy-preserving schemes: Zhornitskaya and Bertozzi (1999); Grün and Rumpf (2000) Existence of classical solution in weighted Sobolev spaces: Giacomelli & Knüpfer (2010), Bertsch et al. (2005) www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 10 / 41
2. Droplets on solid planar surfaces Contact angle: regularization vs free boundary
Regularization vs free boundary problem Driving energy E and dissipation D Z Z D (˙ 2 | r h | 2 d x + V ( h ) 1 m ( h )( r π ) 2 E ( h ) = h ) = m ( h ) = | h | n Why is sliding motion singular for no-slip m ( h ) = | h | 3 ? ˙ h + r · j = 0 , j = � m r π = h v results in Z j 2 Z h 2 − n | v | 2 D = m = so that near a sliding contact line with velocity v 0 and slope α we have Z x − + δ Z x + � 2 − n v 2 � α ( x � x − ) 0 + D ⇡ ... δ x − Huh, Scriven. J. Colloid. Interf. Sci. (1971) www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 12 / 41
Regularization vs free boundary problem Driving energy E and dissipation D Z Z D (˙ 2 | r h | 2 d x + V ( h ) 1 m ( h )( r π ) 2 E ( h ) = h ) = …with regularization (disjoining pressure) Z Φ ( h V ( h ) = σ ε ) d x s Γ − limit …or without 1 d V ( h ) = σ µ ( { x : h ( x ) > 0 } ) = σ | x + − x − | s www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 13 / 41
Regularization vs free boundary problem 1.4 precursor sharp interface 1.2 1 0.8 h 0.6 0.4 0.2 0 − 2 − 1 0 1 2 x regularized (global) free boundary (supported) topological transitions, + coarse meshes, stable & accurate standard to implement refinement near contact line no topological transitions - unknowns, stability free boundary problem www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 14 / 41
Regularization vs free boundary problem Z 1 2 | r h | 2 d x + V ( h ) , E ( h ) = V ( h ) = σ | x + � x − | s Optimize size of support vs gradients of the solution leads to contact angle but also drives motion and instabilities www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 15 / 41
Typical instabilities on solid surfaces P2 FEM with (heuristic) spatial adaptivity dewetting instability with mobility n=2 Plateau-Rayleigh instability with n=3
Regularization vs free boundary problem 10 560 5 540 0 600 520 500 400 500 300 480 200 100 460 0 260 280 240 220 180 200 160 120 140 100 180 200 220 240 260 280 300 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 17 / 41
2. Droplets on solid planar surfaces free boundary approach - algorithm 1D
Algorithm 1. original PDE ˙ h − ( | h | n π x ) x = 0 π = δ E δ h = − h xx 2. weak formulation (discrete in space using linear FEM) Z x + ( ˙ h φ + | h | n π x φ x ) d x = 0 x − Z x + Z x + ( πϕ − τ ˙ h x ϕ x d x − h x φ | x + h x ϕ x ) d x = x − x − x − www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 19 / 41
Algorithm Note on the handling of time-derivatives • h ( t + τ , · ) = h ( t, · ) + τ ˙ h makes no sense • ALE (arbitrary Lagrangian-Eulerian) transformation as post-processing 3. define transformation ψ t 0 ( t, · ) : ( x − ( t 0 ) , x + ( t 0 )) → ( x − ( t ) , x + ( t )) � � ψ t 0 ( t, x ) = x − ( t ) + ξ ( x ) x + ( t ) − x − ( t ) x − x − ( t 0 ) ξ ( x ) = x + ( t 0 ) − x − ( t 0 ) h by ψ t 0 4. pull-back of gives � � H ( t, x ) = h t, ψ t 0 ( t, x ) www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 20 / 41
Algorithm 5. time derivative uniquely decomposes ˙ ψ ( x − ) ˙ H ( t, x ± ( t 0 )) ≡ 0 ˙ ˙ H = ˙ h + ˙ ˙ h ( x ) 7! ψ h x H | ± | ˙ t ≈ t 0 ψ ( x + ) ⇣ m ⌘ x ± = lim ˙ h π x x → x ± 6. update according to time-derivatives x k +1 = x k + τ ˙ ψ ( x k ) , h k +1 = h k + τ ˙ H, https://github.com/dpeschka/thinfilm-freeboundary.git (about 120 lines MATLAB proof-of-concept 1D code) P. Thin-film free boundary problems for partial wetting. J. Comp. Phys. (2015) P. Numerics of contact line motion for thin films, IFAC PapersOnline (2015) www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 21 / 41
2. Droplets on solid planar surfaces free boundary approach - algorithm 2D
Extension to 2D 1. original PDE ˙ h � r · ( | h | n r π ) = 0 π = δ E δ h 2. weak formulation (discrete in space using linear FEM) Z (˙ h φ + | h | n r π · r φ ) = 0 ω Z Z Z ( πψ � τ r ˙ h · r ψ = r h · r ψ � ψ∂ ν h ω ω ∂ω www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 23 / 41
Extension to 2D 3. pull-back and ALE formulation H ( t, x ) = h ( t, y ) , where y = Ψ t ( x ) H ( t, x ) = ˙ ˙ h ( t, y ) + ˙ Ψ t ( x ) · r y h ( t, y ) 4. deformation problem Use harmonic ˙ Ψ with boundary data: normal part of mapping ˙ Ψ · ν : 0 = ˙ h ( t, y ) + ˙ Ψ t ( x ) · r y h ( t, y ) tangential part of mapping (1 � νν > ) ˙ Ψ : ) Deformed meshes are more uniform. www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 24 / 41
2. Droplets on solid planar surfaces Examples
Examples ⇣ �⌘ x | 2 ¯ 1 − | x − ¯ � Evolution towards equilibrium h ( x ) = h R 2 + www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 26 / 41
Examples Z ˜ E ( h ) = E ( h ) + ρ gh ( α h + β x ) d x Gravity driven motion Podgorski, Flesselles, Limat Schwartz et al. Eggers/Snoeijer et al. www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de 27 / 41
Examples Z V ( h ) = σ ( x, y ) d x Energetic patterning s ω Kondic, Diez. Phys. Fluids (2004) www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de 28 / 41
3. Flows over liquid substrates (1D) 2 1
Flow over liquid substrates Ω 1 ( t ) = { ( x, z ) : 0 < z < h 1 ( t, x ) } Ω 2 ( t ) = { ( x, z ) : h 1 ( t, x ) < z < h 1 + h ( t, x ) } www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de 30 / 41
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