LECTURE 4: GEOMETRIC PROBLEMS

Transcription

1 LECTURE 4: GEOMETRIC PROBLEMS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and Related Topics Isaac Newton Institute, January 2014

2 LECTURE 4: GEOMETRIC PROBLEMS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and Related Topics Isaac Newton Institute, January 2014

3 OUTLINE Shape Differential Calculus Geometric Gradient Flows Example 1: Mean Curvature Flow Example 2: Surface Diffusion Example 3: Optimal Shape Design Example 4: Biomembranes Example 5: Director Fields on Flexible Surfaces Electrowetting with Moving Contact Line

4 Shape Differential Calculus Ω U ad : J(Ω, y(ω )) = inf Ω U ad J(Ω, y(ω)), y(ω) : Ly(Ω) = f on Ω. Goal: Construct a geometric gradient flow of J(Ω) that will lead to a local minimum. This represents a continuous descent direction and is dictated by a geometric PDE. Deformation of Domains (Hadamard, Murat, Delfour, olesio) Given (Lipschitz) vector velocity V Consider mapping X(, t) : x R d X(x, t) R d defined by dx dt = V(X(x, t)), X(x, 0) = x Ω, (Ω := Ω0) Deformed domain Ω t = {X(x, t) : x Ω 0}

5 Shape Derivatives Eulerian derivative of J(Ω) in the direction V: 1 δ ΩJ(Ω; V) = lim J(Ω t 0 t) J(Ω 0) t 1 Material Derivative: ẏ(ω; V) := lim t 0 y(ω t t) X(, t) y(ω 0) Shape Derivative: y (Ω; V) := ẏ(ω; V) {z } y(ω), V {z } material derivative space derivative Notation: V := V ν, := Ω, h := κ 1 + κ 2 (total curvature) Reynolds Theorems: J(Ω) := R φ(, Ω)dx δωj(ω; V) = R Ω Ω φ (Ω; V) + R φv ds J(Ω) := R φ(, Ω)ds δj(ω; V) = R φ (Ω; V) + R νφ + φh V ds

6 Shape Derivatives Eulerian derivative of J(Ω) in the direction V: 1 δ ΩJ(Ω; V) = lim J(Ω t 0 t) J(Ω 0) t 1 Material Derivative: ẏ(ω; V) := lim t 0 y(ω t t) X(, t) y(ω 0) Shape Derivative: y (Ω; V) := ẏ(ω; V) {z } y(ω), V {z } material derivative space derivative Notation: V := V ν, := Ω, h := κ 1 + κ 2 (total curvature) Reynolds Theorems: J(Ω) := R φ(, Ω)dx δωj(ω; V) = R Ω Ω φ (Ω; V) + R φv ds J(Ω) := R φ(, Ω)ds δj(ω; V) = R φ (Ω; V) + R νφ + φh V ds

7 Shape Derivatives Eulerian derivative of J(Ω) in the direction V: 1 δ ΩJ(Ω; V) = lim J(Ω t 0 t) J(Ω 0) t 1 Material Derivative: ẏ(ω; V) := lim t 0 y(ω t t) X(, t) y(ω 0) Shape Derivative: y (Ω; V) := ẏ(ω; V) {z } y(ω), V {z } material derivative space derivative Notation: V := V ν, := Ω, h := κ 1 + κ 2 (total curvature) Reynolds Theorems: J(Ω) := R φ(, Ω)dx δωj(ω; V) = R Ω Ω φ (Ω; V) + R φv ds J(Ω) := R φ(, Ω)ds δj(ω; V) = R φ (Ω; V) + R νφ + φh V ds

8 Examples of Shape Derivatives Example 1: Volume J(Ω) = Example 2: Area J() = Ω 1 δj(ω; V ) = 1 δj(; V ) = Example 3: Bending Energy 1 J()) := 2 h2 ds δj(; V ) = because shape calculus gives V dσ HV dσ h V 1 2 h3 V + 2hκV ν (, V) = V, h (, V) = V, νh = ν 2 = (h 2 2κ) whence δ W (, φ) = = hh + (h νh h3 )V V h h ν 2 V h 3 V.

9 Geometric Gradient Flows Derivative of J(Ω) in the direction V satisfies: δj(ω; V ) = R GV for explicit functions g, f. G(x, Ω) = g(x, Ω)H + f(x, Ω) Scalar product B(, ) on and corresponding norm. Gradient flow: B(V, W ) = R GW δj(ω; V ) = B(V, V ) V 2 < 0 Energy decreasing flow! Gradient flow in operator form: BV = G

10 Implicit Time Discretizaion Implicit Euler: X n+1 = X n + τ V n+1 (Ω n Ω n+1 ) Minimization: argmin Ω n+1 J(Ω n+1 ) + 1 2τ d2 (Ω n+1, Ω n ) Distance between Domains: d 2 (Ω n+1, Ω n ) = τv n+1 2 n+1 = τ 2 V n+1 2 n+1 = τ 2 B(V n+1, V n+1 ) Implicit Minimization: argmin W n+1 J(Ω n+1 ) + τ2 W n+1 2 n+1 Optimality Condition: (Euler-Lagrange equation) B n+1 (V n+1, W ) = δj(ω n+1 ; W ) = G n+1 W, W n+1 Energy Decrease Property: Ω n+1 = Ω n + τv n+1 J(Ω n+1 ) J(Ω n + τv n+1 ) + τ 2 V n+1 2 J(Ω n ).

11 Implicit Time Discretizaion Implicit Euler: X n+1 = X n + τ V n+1 (Ω n Ω n+1 ) Minimization: argmin Ω n+1 J(Ω n+1 ) + 1 2τ d2 (Ω n+1, Ω n ) Distance between Domains: d 2 (Ω n+1, Ω n ) = τv n+1 2 n+1 = τ 2 V n+1 2 n+1 = τ 2 B(V n+1, V n+1 ) Implicit Minimization: argmin W n+1 J(Ω n+1 ) + τ2 W n+1 2 n+1 Optimality Condition: (Euler-Lagrange equation) B n+1 (V n+1, W ) = δj(ω n+1 ; W ) = G n+1 W, W n+1 Energy Decrease Property: Ω n+1 = Ω n + τv n+1 J(Ω n+1 ) J(Ω n + τv n+1 ) + τ 2 V n+1 2 J(Ω n ).

12 Linearization: Semi-Implicit Time Discretization Treat geometry explicitly: replacement in Optimality Condition Ω n+1 Ω n, ν n+1 ν n, G n+1 Ĝn Ĝn := g(x, Ω n )H n+1 + f(x, Ω n ) [Dziuk], [Bänsch, Morin, N], [Barrett, Garcke, Nürnberg], [Dziuk, Elliott] The Riesz representative Ĝn of shape gradient handles curvature implicitly. Gradient flow equation: B n+1 (V n+1, W ) = R G n+1 W, W n+1 B n (V n+1, W ) = Ĝ n W, W n B n V n+1 + g n H n+1 = f n

13 Operator Splitting Issue: How to compute H, H Differential Geometry Identity [G. Dziuk]: X = H = H ν KEY IDEA: write the problem in terms of scalar and vector quantities H, H, V, V H = H ν, V = V ν X n X n+1 = X n + τ n V n+1 H n+1 + τ n n V n+1 = n X n H n+1 H n+1 ν n = 0 B n V n+1 + g n H n+1 = f n V n+1 V n+1 ν n = 0 (Operator Splitting)

14 Operator Splitting Issue: How to compute H, H Differential Geometry Identity [G. Dziuk]: X = H = H ν KEY IDEA: write the problem in terms of scalar and vector quantities H, H, V, V H = H ν, V = V ν X n X n+1 = X n + τ n V n+1 H n+1 + τ n n V n+1 = n X n H n+1 H n+1 ν n = 0 B n V n+1 + g n H n+1 = f n V n+1 V n+1 ν n = 0 (Operator Splitting)

15 Operator Splitting Issue: How to compute H, H Differential Geometry Identity [G. Dziuk]: X = H = H ν KEY IDEA: write the problem in terms of scalar and vector quantities H, H, V, V H = H ν, V = V ν X n X n+1 = X n + τ n V n+1 H n+1 + τ n n V n+1 = n X n H n+1 H n+1 ν n = 0 B n V n+1 + g n H n+1 = f n V n+1 V n+1 ν n = 0 (Operator Splitting)

16 Semi-Implicit FEM Geometry: ν = ν n h, = n h Discrete spaces (C 0 piecewise linears): V h () H 1 (), Vh () H 1 () Fully discrete problem: seek V n+1, H n+1 V h (), V n+1, H n+1 V h () such that D H n+1, φ E E E h τ D V n+1, φh = D X n, φh φ h V h () D E H n+1, φ h H n+1 ν, φ h = 0 φ h V h () B n (V n+1, φ h ) + g n H n+1, φ h = f n, φ h φ h V h () D V n+1, φ E h DV n+1, ν φ E h = 0 φ h V h ()

17 Mean Curvature Flow: Parametric Approach (G. Dziuk) L 2 - Gradient flow: J() = R V W = δj(; W ) V = H Energy law: R V 2 + d dt = 0 Vector form: V = V ν = H ν = H = X t X X = 0 (vector heat equation on ) Semi-implicit time discretization: X n+1 τ n n X n+1 = X n

18 Mean Curvature Flow: Level Set Approach (Osher, Sethian) Levet Set Function φ: Unit normal, mean curvature and normal velocity read ν = φ tφ, H = div ν, V = φ φ Motion by Mean Curvature: V = H tφ φ div φ φ tφ nx i,j=1 δ ij x i φ x φ j φ 2 xi x j φ = 0 Viscosity Solutions: [Evans, Spruck], [Chen, Giga, Goto] Numerical Methods: [Osher-Sethian], [Walkington], [Chambolle], etc Error estimate for Regularized Flow: [Deckelnick], [Deckelnick, Dziuk, Elliott].

19 Mean Curvature Flow: Phase Field Approach (Allen-Cahn) Allen-Cahn Equation for the Phase Variable ϕ: ɛ tϕ ɛ ɛ ϕ ɛ + 1 ɛ f(ϕɛ) = 0 f(ϕ) = ϕ(ϕ2 1) Interface Convergence: Error bounds of order ɛ 2 log ɛ 2 for Haussdorff distance between zero level set of ϕ ɛ and surface moving by mean curvature in the smooth regime [Bellettini, Paolini] Viscosity Solutions: Convergence of ϕ ɛ to viscosity solution to MCF pass singularities [Evans, Soner, Souganidis]. Double Obstacle Phase Field Model: The nonlinearity f becomes f(ϕ) = ϕ + I [,1,1] (ϕ). Interface Convergence: Estimates of order ɛ 2 for the smooth regime [N, Paolini, Verdi], and of order ɛ pass singularities [N, Verdi]. Techniques: Comparison principles, barrier arguments, non-degeneracy.

20 Surface Diffusion (w. E. Bänsch and P. Morin) Epitaxially stressed solid due to mismatch with the rigid substrate. Simplified model: solid governed by the Laplace equation Dynamics of free surface (t): Rearrangement of particles on (t) so as to minimize surface tension (plastic deformation) V = `κ + ε S Ω S where ε = y 2 ( elastic energy ), and 8 y = 0 in Ω >< y = x on D y = x + periodic on S >: νy = 0 on D

21 Surface Diffusion (Continued) Domain functional: J(Ω) := y 2 + γ ds Ω {z } {z 1 } elastic energy surface energy y = 0 Ω, νy = 0 y = g D (fixed). Shape derivative y = y (Ω; V ) satisfies y = 0 Ω, νy = div (V y), y = 0 D. Shape derivative of J(Ω) in the direction V satisfies dj(ω; V ) = R y 2 + γh V ds {z } =G=f+gH

22 Surface Diffusion (Continued) Domain functional: J(Ω) := y 2 + γ ds Ω {z } {z 1 } elastic energy surface energy y = 0 Ω, νy = 0 y = g D (fixed). Shape derivative y = y (Ω; V ) satisfies y = 0 Ω, νy = div (V y), y = 0 D. Shape derivative of J(Ω) in the direction V satisfies dj(ω; V ) = R y 2 + γh V ds {z } =G=f+gH

23 Optimal Shape Design (w. G. Dogan, M. Morin, and M. Verani) Domain functional: J(Ω) = 1 2 D `y(ω) zg 2 y = 0 Ω, νy = 1 1, y = 0 2 (fixed). Shape derivative y = y (Ω; V ) satisfies y = 0 Ω, νy = div (V y) + HV 1, y = 0 2. Dual (or adjoint) solution p accounts for L 2 -norm in J(Ω) and satisfies p = χ D(y z g) Ω, νp = 0 1, p = 0 2. Derivative of J(Ω) in the direction V satisfies dj(ω; V ) = R 1 y p + Hp V ds {z } =G=f+Hg

24 Optimal Shape Design (w. G. Dogan, M. Morin, and M. Verani) Domain functional: J(Ω) = 1 2 D `y(ω) zg 2 y = 0 Ω, νy = 1 1, y = 0 2 (fixed). Shape derivative y = y (Ω; V ) satisfies y = 0 Ω, νy = div (V y) + HV 1, y = 0 2. Dual (or adjoint) solution p accounts for L 2 -norm in J(Ω) and satisfies p = χ D(y z g) Ω, νp = 0 1, p = 0 2. Derivative of J(Ω) in the direction V satisfies dj(ω; V ) = R 1 y p + Hp V ds {z } =G=f+Hg

25 L 2 - Gradient Flow k = 3 J = k = 6 J = k = 9 J = k = 22 J = L 2 -gradient flow from a non-centered ellipse to a centered circle. The exact solution is a circle of radius one centered at the origin, and the initial configuration is a small ellipse centered at (0.5, 0.7). The L 2 -dynamics is locally stable where p > 0 (upper right) and locally unstable where p < 0 (lower left). The bottom row is a zoom of the unstable region exhibiting oscillations at the mesh level.

26 (Stable) Weighted H 1 - Gradient Flow k = 0 J = k = 20 J = k = 40 J = k = 60 J = k = 80 J = k = 100 J = k = 120 J = k = 140 J = Stable dynamics: Weighted H 1 flow from a non-centered ellipse to a centered circle.

27 Biomembranes (w. A. Bonito and M.S. Pauletti) Bending energy: W () = 1 H 2 + λ 1 + p 1 2 Ω δ W = H 1 «2 H3 + 2KH ν + λhν + pν Equivalent Form of Derivative of Willmore Energy δ W : δ W (, V ) = V H H ν 2 V H 3 V. Vector Form of Derivative of Willmore Energy δ W : V H = V : H ( x+ x T ) V : H δ W (; V) = H ν V = H ν 2 1 H 2 V 2 V : H ( x+ x T ) V : H+ 1 2 Related methods: [Rusu], [Dziuk], [Barrett, Garcke, Nürnberg] H ν V H V

28 Biomembranes (w. A. Bonito and M.S. Pauletti) Bending energy: J() = 1 H 2 + λ 1 + p 1 2 Ω δ J = H 1 «2 H3 + 2KH ν + λhν + pν Equivalent Form of Derivative of Willmore Energy δ W : δ W (, V ) = V H H ν 2 V H 3 V. Vector Form of Derivative of Willmore Energy δ W : V H = V : H ( x+ x T ) V : H δ W (; V) = H ν V = H ν 2 1 H 2 V 2 V : H ( x+ x T ) V : H+ 1 2 Related methods: [Rusu], [Dziuk], [Barrett, Garcke, Nürnberg] H ν V H V

29 Biomembranes (w. A. Bonito and M.S. Pauletti) Bending energy: J() = 1 H 2 + λ 1 + p 1 2 Ω δ J = H 1 «2 H3 + 2KH ν + λhν + pν Equivalent Form of Derivative of Willmore Energy δ W : δ W (, V ) = V H H ν 2 V H 3 V. Vector Form of Derivative of Willmore Energy δ W : V H = V : H ( x+ x T ) V : H δ W (; V) = H ν V = H ν 2 1 H 2 V 2 V : H ( x+ x T ) V : H+ 1 2 Related methods: [Rusu], [Dziuk], [Barrett, Garcke, Nürnberg] H ν V H V

30 Biomembranes (w. A. Bonito and M.S. Pauletti) Bending energy: J() = 1 H 2 + λ 1 + p 1 2 Ω δ J = H 1 «2 H3 + 2KH ν + λhν + pν Equivalent Form of Derivative of Willmore Energy δ W : δ W (, V ) = V H H ν 2 V H 3 V. Vector Form of Derivative of Willmore Energy δ W : V H = V : H ( x+ x T ) V : H δ W (; V) = H ν V = H ν 2 1 H 2 V 2 V : H ( x+ x T ) V : H+ 1 2 Related methods: [Rusu], [Dziuk], [Barrett, Garcke, Nürnberg] H ν V H V

31 Biomembranes (w. A. Bonito and M.S. Pauletti) Bending energy: J() = 1 H 2 + λ 1 + p 1 2 Ω δ J = H 1 «2 H3 + 2KH ν + λhν + pν Equivalent Form of Derivative of Willmore Energy δ W : δ W (, V ) = V H H ν 2 V H 3 V. Vector Form of Derivative of Willmore Energy δ W : V H = V : H ( x+ x T ) V : H δ W (; V) = H ν V = H ν 2 1 H 2 V 2 V : H ( x+ x T ) V : H+ 1 2 Related methods: [Rusu], [Dziuk], [Barrett, Garcke, Nürnberg] H ν V H V

32 Fluid-Membrane Interaction (w. A. Bonito and M.S. Pauletti) Refs: Coutand-Shkoller (Incompressible) Navier-Stokes Equations Ω ρ v div ( pi + µd(v) ) = b in Ω t, {z } Σ div v = 0 in Ω t, [Σ]ν = δ J on t, v = ϑ on t, v(, 0) = v 0 in Ω 0. D Ω in g Ω out Membrane Force: Bending δ J = H + 1 «2 H3 2KH ν + λhν.

33 Fluid-Membrane Interaction (w. A. Bonito and M.S. Pauletti) Refs: Coutand-Shkoller (Incompressible) Navier-Stokes Equations Ω ρ v div ( pi + µd(v) ) = b in Ω t, {z } Σ div v = 0 in Ω t, [Σ]ν = δ J on t, v = ϑ on t, v(, 0) = v 0 in Ω 0. D Ω in g Ω out Membrane Force: Bending δ J = H + 1 «2 H3 2KH ν + λhν.

34 Fluid-Membrane Interaction (w. A. Bonito and M.S. Pauletti) Refs: Coutand-Shkoller (Incompressible) Navier-Stokes Equations Ω ρ v div ( pi + µd(v) ) = b in Ω t, {z } Σ div v = 0 in Ω t, [Σ]ν = δ J on t, v = ϑ on t, v(, 0) = v 0 in Ω 0. D Ω in g Ω out Membrane Force: Bending δ J = H + 1 «2 H3 2KH ν + λhν.

35 Geometric vs Fluid Red Cell: 5x5x1 Ellipsoid play

36 Fluid Red Blood Cell: Ellipsoid 5x5x1 Streamlines

37 Comparison of Geometric with Fluid Red Cell: Physical Parameters Comparison: Physical Parameters

38 Prototype Model and Applications Basic setting: Surface R 3 and director field n : S 2 n Physics: is the surface constituted by some molecules and n is the orientation of those or other molecules Question: How do shape and orientational order interact? Basic Model:, n minimize Helfrich and Frank energy with coupling F (, n) E[, n] = 1 H n 2 + F (, n) 2 2 for in fixed topology class and n a unit-length vector-field (H = div ν).

39 Prototype Model and Applications Basic setting: Surface R 3 and director field n : S 2 n Physics: is the surface constituted by some molecules and n is the orientation of those or other molecules Question: How do shape and orientational order interact? Basic Model:, n minimize Helfrich and Frank energy with coupling F (, n) E[, n] = 1 H n 2 + F (, n) 2 2 for in fixed topology class and n a unit-length vector-field (H = div ν).

40 Nonlinear Model: First Variation of the Energy E(, n) E(, n) := 1 (div ν δdiv n) 2 + λ n 2 + µ` n f(n ν) 2 2 2ε 2 Extend n constant along the normal ν, which makes sense for biomembranes. The first variation of E(, n) with respect to is (H = div ν) D δe E δ, φ = H, φ + H ν 2, φ δ H( n : ν), φ + δ H(ν T n), φ δ (div n), φ + δ div n ν 2, φ δ 2 div n( n : ν), φ + δ 2 div n(ν T n), φ 1 H(H + δdiv n) 2, φ λ ( n) T : Dν( n), φ 2 λ H n 2, φ 1 f (n ν)n, φ 1 Hf(n ν), φ, 2 ε ε for all φ C (). The variation of E(, n) with respect to n is D δe E δn, φ = δ H+δdiv n, div m +λ n, m + 2µn, m + 1 f (n ν), m ν, ε for all m C (; R 3 ). Expression simplifies if m is perpendicular to n.

41 Nonlinear Model: First Variation of the Energy E(, n) E(, n) := 1 (div ν δdiv n) 2 + λ n 2 + µ` n f(n ν) 2 2 2ε 2 Extend n constant along the normal ν, which makes sense for biomembranes. The first variation of E(, n) with respect to is (H = div ν) D δe E δ, φ = H, φ + H ν 2, φ δ H( n : ν), φ + δ H(ν T n), φ δ (div n), φ + δ div n ν 2, φ δ 2 div n( n : ν), φ + δ 2 div n(ν T n), φ 1 H(H + δdiv n) 2, φ λ ( n) T : Dν( n), φ 2 λ H n 2, φ 1 f (n ν)n, φ 1 Hf(n ν), φ, 2 ε ε for all φ C (). The variation of E(, n) with respect to n is D δe E δn, φ = δ H+δdiv n, div m +λ n, m + 2µn, m + 1 f (n ν), m ν, ε for all m C (; R 3 ). Expression simplifies if m is perpendicular to n.

42 Relaxation Dynamics: L 2 -Gradient Flow If v denotes the normal velocity of, we have to solve the system of PDE on D δe E v, φ = δ, φ for all φ C (), D δe E tn, m = δn, m for all m C (; T ns 2 ), subject to the constraint that n(t, x) S 2 for almost every (t, x). Semi-implicit Time Discretization v j 1 `Xj X j 1 ν j 1 τ with ν j 1 : j 1 S 2 the outer unit normal to j 1, and j 1X j = H j ν j 1 Refs: Dziuk, Dziuk-Elliott, Bänsch-Morin-Nochetto, Barrett-Garcke-Nürnberg.

43 Relaxation Dynamics: L 2 -Gradient Flow If v denotes the normal velocity of, we have to solve the system of PDE on D δe E v, φ = δ, φ for all φ C (), D δe E tn, m = δn, m for all m C (; T ns 2 ), subject to the constraint that n(t, x) S 2 for almost every (t, x). Semi-implicit Time Discretization v j 1 `Xj X j 1 ν j 1 τ with ν j 1 : j 1 S 2 the outer unit normal to j 1, and j 1X j = H j ν j 1 Refs: Dziuk, Dziuk-Elliott, Bänsch-Morin-Nochetto, Barrett-Garcke-Nürnberg.

44 Simulations for Surfactants Evolution from a prolate ellipsoid to a discocyte, with (δ = 1) and without (δ = 0) surfactant, and with area and volume constraints. Snapshots of the evolution after n = 100, 200, 400 and 1200 time-steps. The upper plots show the evolution in the presence of surfactants (δ = 1) while the second row shows the Helfrich-flow (δ = 0). The coupling of the surface and the director field decelerates the evolution and leads to a less pronounced biconcave shape.

45 Simulations for Biomembranes: Two Positive Degree-One Defects Two degree +1 defects pointing outwards at the poles develop a cone-like structure near poles (stomatocyte shape) 2 Defects +1

46 Simulations for Biomembranes: Two Positive Degree-One Defects Two inward and outward pointing at the poles of degree +1 develop a heart-like shape (echinocytes) 2 Defects +1

47 Simulations for Biomembranes: Two Negative Degree-One Defects Biomembrane develops saddle-shapes at the poles, where the negative degree-one defects are located (top row) and cone shapes in neighborhoods of the four positive degree-one defects located on the equator (see the cut through the equator in the lower left picture). The surface and the director field are colored by div n.

48 Phase Field Model for Electrowetting Two-Phase Flow: Phase variable φ satisfies a Cahn-Hilliard type equation tφ + u φ = div `M(φ) µ 1 µ = γ δ f(φ) δ φ 1 2 ɛ (φ) V ρ (φ) u 2 where u is fluid velocity, M(φ) mobility, µ chemical potential, f(φ) cubic nonlinearlity, ɛ(φ) electric permitivity, V electric potential, ρ(φ) density. Variable Density Navier-Stokes: Fluid velocity and pressure (u, p) satisfy D t`ρ(φ)u div `η(φ)σ(u) + p = µ φ q (V + λq) div u = 0, where q is the electric charge. Navier boundary condition, electrostatic equation for V, transport equation for q, and boundary conditions. Formulation and Analysis: [N, Salgado, Walker] [Grün et al.], [Garcke et al.]

49 Electrowetting Simulation: Droplet Splitting 2d Configuration: Some Parameters: ρ 1 = 1, ρ 2 = 100, δ = , γ = 50, λ = 0.5, M = 10 2, V = 20, η 1 = 1, η 2 = 10,