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Motion of thin droplets over surfaces numerical and modeling - - PowerPoint PPT Presentation

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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

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Weierstrass Institute for Applied Analysis and Stochastics

www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Dirk Peschka, Sebastian Jachalski, Barbara Wagner Stefan Bommer, Ralf Seemann, Georgy Kitavtsev, Luca Heltai

Motion of thin droplets over surfaces

ICERM Workshop „Making a Splash - Droplets, Jets and Other Singularities“

numerical and modeling techniques for moving contact lines

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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)

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  • 1. Setting
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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Setting

4 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. Setting: Newtonian liquids Viscous (Re=0) partial wetting

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Setting

5

Ω = {(x, z) : 0 < z < h(t, x)}

z

Γ

Ω

Γ = {(x, z) : z = h(t, x) > 0}

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  • 2. Droplets on solid planar surfaces

Modeling

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Modeling

7

Stokes flow with free boundary

in Ω

  • n Γ

at x± rp + µr2u = 0, r · u = 0,

  • pI + 2µD(u)
  • n = σκn,
  • u vΓ
  • · n = 0,

|rh| = tan θ,

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de Helmholtz (1869), Rayleigh (1873),…,Onsager (1929)

Modeling

8

Stokes via Helmholtz-Rayleigh variational principle

E = Z

Γ

σ

D(u, v) = Z

Ω

µ 2 D(u) : D(v) + Z

{z=0}

β−1u · v

dE dt = hdiffE, ui = D(u, u)  0

Seek u so that D(u, v) = hdiffE, vi for all v.

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Modeling

9

Thin-film limit in a nutshell: Z σ = Z σ p 1 + |rh|2 dx ⇡ Z σ

  • 1 + 1

2|rh|2

dx

Conservation of mass + kinematic condition:

D = Z D(u) : D(v) + Z

z=0

β−1uv ⇡ Z Z h (∂zu)(∂zv)dz dx + Z

z=0

β−1uv = Z Z h (∂zzu)vdz dx + Z (∂zu)vdx|h

0 + β−1uv|z=0

hdiffE, vi = Z rhr˙ hvdx = Z rh · r r · Z h vdz ! dx = Z Z h (r∆h)vdz dx

D(u, v) = hdiffE, vi

˙ h + r · Z h u dz = 0

Note: ˙ h = ∂th

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Modeling

10 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)

For given h(0, x) seek h(t, x) such that

z

For given seek such that

where |hx(t, x±)| = tan θ ˙ x± = lim

x→x±

⇣m h πx ⌘ h(t, x±(t)) ≡ 0

where

˙ h r ·

  • m(h)rπ
  • = 0

π = δE δh = ∆h, m(h) = |h|n

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  • 2. Droplets on solid planar surfaces

Contact angle: regularization vs free boundary

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Regularization vs free boundary problem

12

Driving energy E and dissipation D

D(˙ h) = Z m(h)(rπ)2

Huh, Scriven. J. Colloid. Interf. Sci. (1971)

m(h) = |h|n

Why is sliding motion singular for no-slip m(h) = |h|3? ˙ h + r · j = 0, j = mrπ = hv results in D = Z j2 m = Z h2−n|v|2 so that near a sliding contact line with velocity v0 and slope α we have D ⇡ Z x−+δ

x−

  • α(x x−)

2−nv2

0 +

Z x+

δ

...

E(h) = Z

1 2|rh|2dx + V (h)

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Regularization vs free boundary problem

13

V (h) = σµ({x : h(x) > 0})

1d

= σ|x+ − x−|

Driving energy E and dissipation D

…with regularization (disjoining pressure)

V (h) = σ Z Φ( h

ε ) dx

…or without

D(˙ h) = Z m(h)(rπ)2 E(h) = Z

1 2|rh|2dx + V (h)

s s

Γ − limit

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Regularization vs free boundary problem

14

−2 −1 1 2 0.2 0.4 0.6 0.8 1 1.2 1.4 x h precursor sharp interface

regularized (global) free boundary (supported)

+

topological transitions, standard to implement coarse meshes, stable & accurate

  • refinement near contact line

unknowns, stability no topological transitions free boundary problem

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Regularization vs free boundary problem

15

Optimize size of support vs gradients of the solution leads to contact angle but also drives motion 
 and instabilities

E(h) = Z 1 2|rh|2dx + V (h), V (h) = σ|x+ x−|

s

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P2 FEM with (heuristic) spatial adaptivity dewetting instability with mobility n=2 Plateau-Rayleigh instability with n=3

Typical instabilities on solid surfaces

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Regularization vs free boundary problem

17

180 200 220 240 260 280 300 460 480 500 520 540 560

100 120 140 160 180 200 220 240 260 280 100 200 300 400 500 600 5 10

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  • 2. Droplets on solid planar surfaces

free boundary approach - algorithm 1D

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Algorithm

19

Z x+

x−

( ˙ hφ + |h|nπxφx) dx = 0 Z x+

x−

( πϕ − τ ˙ hxϕx) dx = Z x+

x−

hxϕx dx − hxφ|x+

x−

  • 2. weak formulation (discrete in space using linear FEM)

˙ h − (|h|nπx)x = 0 π = δE δh = −hxx

  • 1. original PDE
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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Algorithm

20

Note on the handling of time-derivatives

ψt0(t, x) = x−(t) + ξ(x)

  • x+(t) − x−(t)
  • ξ(x) =

x − x−(t0) x+(t0) − x−(t0)

  • 3. define transformation

H(t, x) = h

  • t, ψt0(t, x)
  • 4. pull-back of gives

h by ψt0

ψt0(t, ·) : (x−(t0), x+(t0)) → (x−(t), x+(t))

  • h(t + τ, ·) = h(t, ·) + τ ˙

h makes no sense

  • ALE (arbitrary Lagrangian-Eulerian) transformation as post-processing
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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Algorithm

21

˙ H = ˙ h + ˙ ψhx

  • 5. time derivative uniquely decomposes

˙ h(x) 7!   ˙ ψ(x−) ˙ H ˙ ψ(x+)   xk+1 = xk + τ ˙ ψ(xk), hk+1 = hk + τ ˙ H,

  • 6. update according to time-derivatives

˙ H(t, x±(t0)) ≡ 0

t ≈ t0

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)

|

± |

˙ x± = lim

x→x±

⇣m h πx ⌘

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  • 2. Droplets on solid planar surfaces

free boundary approach - algorithm 2D

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Extension to 2D

23

  • 2. weak formulation (discrete in space using linear FEM)
  • 1. original PDE

˙ h r · (|h|nrπ) = 0 π = δE δh

Z

ω

(˙ hφ + |h|nrπ · rφ) = 0 Z

ω

(πψ τr˙ h · rψ = Z

ω

rh · rψ Z

∂ω

ψ∂νh

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Extension to 2D

24

  • 4. deformation problem
  • 3. pull-back and ALE formulation

H(t, x) = h(t, y), where y = Ψt(x) ˙ H(t, x) = ˙ h(t, y) + ˙ Ψt(x) · ryh(t, y)

Use harmonic ˙ Ψ with boundary data: normal part of mapping ˙ Ψ · ν: 0 = ˙ h(t, y) + ˙ Ψt(x) · ryh(t, y) tangential part of mapping (1 νν>) ˙ Ψ: ) Deformed meshes are more uniform.

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  • 2. Droplets on solid planar surfaces

Examples

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Examples

26

Evolution towards equilibrium h(x) =

⇣ ¯ h

  • 1 − |x−¯

x|2 R2

⌘

+

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/ 41 www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de

Examples

27 Podgorski, Flesselles, Limat
 Schwartz et al.
 Eggers/Snoeijer et al.

Gravity driven motion

˜ E(h) = E(h) + Z ρgh(αh + βx) dx

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/ 41 www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de

Examples

28

Kondic, Diez. Phys. Fluids (2004)

Energetic patterning

V (h) = Z

ω

σ(x, y) dx

s

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  • 3. Flows over liquid substrates (1D)

1 2

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/ 41 www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de

Flow over liquid substrates

30

Ω1(t) = {(x, z) : 0 < z < h1(t, x)} Ω2(t) = {(x, z) : h1(t, x) < z < h1 + h(t, x)}

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Outlook

31 Figure: composed AFM images of liquid polystyrene dewetting on top from an liquid polymethyl methacrylate substrate (R. Seemann, Univ. d. Saarlandes, Saarbrücken, Germany)

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Flow over liquid substrates

32

∂th r · (M(h)rπ) = 0 in ω(t) ∂th1 r · (m(h1)rπ1) = 0 in Ω \ ω(t)

with h = (h1, h) and π = (π1, π) with π1 = δE/δh1, π = δE/δh. M(h) = ✓ 1

3h3 1 1 2hh2 1 1 2hh2 1 µ 3 h3 + h1h2

◆ m(h1) = h3

1

3 Is also unknown ω(t) = {x ∈ Rd−1 : h(t, x) = 0}

1 2 Miksis and Kriegsmann, SIAM J. Appl. Math. (2003) Pototsky, Bestehorn, Merkt, Thiele, Phys. Rev. E (2004)

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Flow over liquid substrates

33

˙ h + ˙ x± · rh = 0 [[˙ h1 + ˙ x± · rh]] = 0

s = {(h, h1, x−, x+ : 0 < x− < x+ < L; 0 ≤ h, h1; ...}

u = {(˙ h, ˙ h1, ˙ x−, ˙ x+ : ...}

m = ✓Z h dx, Z h1 dx ◆

state velocity

Karapetsas, Craster, Matar, Phys. Fluids (2011) Huth, Jachalski, Kitavtsev, P. Gradient flow perspective on thin-film bilayer flows. J. Engr. Math. (2014)

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Flow over liquid substrates

34

energetics D,E constraints C

2 Z

ω

˙ h1φ1 + (Q11rπ1 + Q12rπ2)rφ1 dx = 0, Z

ω2

˙ hφ + (Q21rπ1 + Q22rπ2)rφ dx = 0,

˙ h + ˙ x± · rh = 0

[[˙ h1 + ˙ x± · rh]] = 0

E(h, h1) = Z σ 2 |rh1|2 + 1 2|r(h1 + h)|2dx + σ|x+ x−| D =

2

X

i,j=1

Z Qijrπi · rπjd

minimization problem via Langrange multiplier

D(u, v) + hv, C>λi = hdiffE, vi hq, Cui = 0

s

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Flow over liquid substrates

35

˙ ξ(x) := 8 > > < > > : ˙ x−

x x−

x 2 ω1 ˙ x− ⇣ 1

x−x− x+−x−

⌘ + ˙ x+ ⇣

x−x− x+−x−

⌘ x 2 ω2 ˙ x+ ⇣ 1

L−x L−x+

⌘ x 2 ω3

H(t, x) = h

  • t, ξt(x)
  • ˙

h(t, x) + ˙ ξ(x) · rh(t, x) = ˙ H(t, x) ˙ h1(t, x) + ˙ ξ(x) · rh1(t, x) = ˙ H1(t, x)

ξt+τ(x) = x + τ ˙ ξ(x), H(t + τ, x) = h(t, x) + τ ˙ H(t, x), H1(t + τ, x) = h1(t, x) + τ ˙ H1(t, x).

minimization problem via Langrange multiplier

D(u, v) + hv, C>λi = hdiffE, vi hq, Cui = 0

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Flow over liquid substrates

36

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Flow over liquid substrates

37

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Flow over liquid substrates

38

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/ 41 www.wias-berlin.de · 7ECM · July 18-22, 2016 · peschka@wias-berlin.de

Summary

39

Thin film flows as free boundary problem, where the support set is unknown and depends on time.

ω(t) = {x ∈ Rd : h(t, x) > 0}

  • contact angles naturally in variation formulation
  • analysis established (even more natural)
  • lack of practical algorithms (so-far)
  • extension to bilayer flows works, comparison promising

Outlook: Summary:

  • higher order algorithms (isoparametric FEM, w. Luca Heltai based on deal.II)
  • modeling of contact line physics and better control of contact 


line motion to better control effects such as contact line hysteresis

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de 40

Thank you!

Supported by and

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References

41

  • Bernis & Friedmann 1990 (existence weak sol.)

  • Bernis et al. 1992 (source type solutions)

  • Bertozzi & Pugh 1996 (regularity & long-time behavior)

  • Eggers 1997 (breakup & singularities)

  • Review Oron, Davis, Bankoff 1997

  • Otto 1998 (long time existence weak sol. with sharp interface with m(h)=h and nonzero angle)

  • Grün, Bertozzi & Witelski 2000 (stationary states and coarsening)

  • Grün & Rumpf 2000 (nonegativity preserving schemes)

  • Wagner, Münch & Witelski 2005 (New regimes of thin film equations)

  • Bertsch, Giacomelli & Karali 2005 (existence of arb. weak sol. with reg. contact angle)

  • Giacomelli, Knüpfer & Otto 2008 


(existence & uniqueness! in interp. spaces of weighted Sobolev spaces with zero slope) …

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Extension to 2D

42

h(t, x)

˙ H = ∂tH(t, x)

˙ h = ∂th(t, y)

H(t + τ, x) = H(t, x) + τ ˙ H(t, x) Ψ(t + τ, x) = Ψ(t, x) + τ ˙ Ψ(t, x)

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based on deal.II

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Gradient formulation in detail

44

Dissipation: D

D(u, v) = Z

Ω 1 2τ : (rv + rv>) dΩ +

Z

Γs

β1u · v dΓ = Z

Ω

(r · τ) · v dΩ + Z

Γ

(τn) · v dΓ + Z

Γs

β1u · v dΓ

where τ = µ(ru + ru>). Energy: diffE

diff E[v] = X

f

σf Z

Γf

rkid : rkv = X

f

σf (d 1) Z

Γf

κn · v Z

∂Γf

v · nΓ !

Minimization: D + diffE = 0

  • X

α

Z

Ωα

(r · τ) · v + X

f

Z

Γf

v ·

  • [[τ]] · n (d 1)σfκn
  • +

Z

Γ0

(t · τn + β1u · t)(t · v) dΓ + X

f

σf Z

∂Γf

v · nΓf = 0

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Effect of mobility on droplet shape

45

m(h) = |h|3+δ

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Regularization vs free boundary problem

46

Plateau-Rayleigh instability with cubic mobility

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Motivation

47

„A free boundary problem is a (nonlinear) PDE, 
 whose domain is part of the unknowns“

  • flows with free surfaces and interfaces (e.g. this talk: dewetting fronts & droplets)

  • geometric evolution (mean curvature flow)

  • multi-phase problems with phase transitions (Stefan problem)

  • fluid-structure interaction, obstacle problems

  • …

Examples for free boundary problems:

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/ 41 www.wias-berlin.de · Providence · March 20-25, 2017 · peschka@wias-berlin.de

Algorithm

48

Regularity of source type solutions


h(t, x) = t−

1 n+4 H(ξ),

ξ = xt−

1 n+4

y = ξ + 1 H(y) = A−ν/3yν(1 + v(y, yβ)),

ν = 3 n, A = ν(ν − 1)(2 − ν), β = √ −3ν2 + 12ν − 8 − 3ν + 4 2

Regularity near the boundary of the support is an issue!

|hx(t, x±)| = tan θ ˙ x± = lim

x→x±

⇣m h πx ⌘

m(h) = |h|n, n ∈ 3

2, 3

  • Gnann, Otto Giacomelli. Eur. J. Appl. Math. (2013)

Bernis et al. ….