Sharp Interface Limit for the Navier Stokes Korteweg Equations

Transcript

1 Sharp Interface Limit for the Navier Stokes Korteweg Equations Albert-Ludwigs-Universität Freiburg Johannes Daube (Joint work with H. Abels, C. Kraus and D. Kröner) 11th DFG-CNRS Workshop Micro-Macro Modeling and Simulations of Liquid-Vapour Flows Paris, March 3, 2016

2 Organisation Introduction The Diffuse Interface Model The Sharp Interface Model The Sharp Interface Limit March 3, 2016 Johannes Daube SIL for the NSK Equations 2 / 21

3 Diffuse and Sharp Interface Models Diffuse Interface Models interface of non-zero thickness (diffuse interface) system of PDEs on whole domain describes motion of fluid smooth order parameter (here: density) indicates different phases Sharp Interface Models infinitely thin (sharp) interface separates different fluid phases system of PDEs in each bulk phase describes motion of fluid coupling by boundary conditions at interface jump discontinuities at interface β 1 ε 0 Sharp Interface Limits relates diffuse to sharp interface models letting in diffuse models the interface thickness tend to zero leads to corresponding sharp interface models β 1 β 2 March 3, 2016 Johannes Daube SIL for the NSK Equations 3 / 21

4 Navier Stokes Korteweg Equation (Diffuse Interface Model) extension of the compressible Navier Stokes equations (Korteweg 1901) diffuse interface model for liquid vapour flows which allows for phase transitions a small parameter ε > 0 represents the thickness of an interfacial area, where phase transitions occur phase field like scaling (Hermsdörfer, Kraus and Kröner 2011) (DIM) ε t ρ ε + div(ρ ε v ε ) = 0 in [0,T], t (ρ ε v ε ) + div(ρ ε v ε v ε ) + 1 ε p(ρ ε) = 2div(µ(ρ ε )Dv ε ) + εγρ ε ρ ε in [0,T], where ρ ε density, v ε velocity, p(ρ ε ) (non-monotone) pressure, µ(ρ ε ) viscosity. Boundary conditions: ρ ε ν = 0 and v ε = 0 on [0,T]. Initial conditions: ρ ε (,0) = ρ ε (i) and v ε (,0) = v ε (i) in. March 3, 2016 Johannes Daube SIL for the NSK Equations 4 / 21

5 Pressure and Double Well Potential non-monotone pressure function p(ρ) = ρ W (ρ) W(ρ) (normalized) double-well potential W(ρ) = W(ρ) l(ρ), where l Maxwell line and 0 < β 1 < β 2 Maxwell points Note: p (ρ) = ρ W (ρ) = ρw (ρ) = p(ρ) = p (ρ) ρ = ρw (ρ) ρ p Ρ W Ρ W Ρ vapour Α1 Α2 liquid Ρ Β1 Α1 Α2 Β2 Ρ Β1 Α1 Α2 Β2 Ρ Growth of the Double-Well Potential. For a = β 1 +β 2 2 and b = β 2 β 1 2, there exist constants C 1 > 0, C 2 (0,b) and p > 2 such that W (z) C 1 z a p 2 for all z R with z a b C 2. Prototypical example (p = 4): W(z) = (z β 1 ) 2 (z β 2 ) 2. March 3, 2016 Johannes Daube SIL for the NSK Equations 5 / 21

6 Static Case Motivation for phase field like scaling of the Navier Stokes Korteweg equations. Static case: velocity v ε 0, time independent; γ = 1 for simplicity. In the static case (DIM) ε reduces to 1 ε p(ρ ε) = ερ ε ρ ε (p (ρ) = ρw (ρ)) = ερ ε ρ ε = 1 ε p (ρ ε ) ρ ε = 1 ε ρ ε W (ρ ε ) ( ) 1 (assume ρ ε > 0) = ε W (ρ ε ) ε ρ ε = 0. Hence, there is a constant l ε R such that 1 ε W (ρ ε ) ε ρ ε = l ε. Euler Lagrange equation with Lagrange multiplier l ε of { } min E ε (ρ) : ρ H 1 1 (), ρ(x)dx = m, E ε (ρ) = ε W(ρ) + ε 2 ρ 2 dx, where m (β 1,β 2 ) (prescribed total mass). March 3, 2016 Johannes Daube SIL for the NSK Equations 6 / 21

7 Static Case: Heuristics Heuristic picture from minimization problem { 1 min ε W(ρ) + ε } 2 ρ 2 dx : ρ H 1 (), ρ(x)dx = m. 1 ε W(ρ)dx favours configurations close to pure phases β 1 and β 2 due to the double-well shape of W. minimizing 1 ε W(ρ)dx leads to phase separation. 1 2 ε ρ 2 dx penalizes occurrence of large interfaces, i.e., rapid transitions between {ρ β 1 } and {ρ β 2 }. March 3, 2016 Johannes Daube SIL for the NSK Equations 7 / 21

8 Aim (DIM) ε t ρ ε + div(ρ ε v ε ) = 0 in [0,T], t (ρ ε v ε ) + div(ρ ε v ε v ε ) + 1 ε p(ρ ε) = 2div(µ(ρ ε )Dv ε ) + εγρ ε ρ ε in [0,T], Sharp interface limit for (DIM) ε : (DIM) ε ε 0 Sharp interface model Main steps 1 Convergence/compactness results for solutions (ρ ε,v ε ) of (DIM) ε 2 Introduction/Justification of the sharp interface model 3 Limit functions of (ρ ε,v ε ) are solutions of the sharp interface model March 3, 2016 Johannes Daube SIL for the NSK Equations 8 / 21

9 (DIM) ε : Notion of Weak Solutions A couple (ρ ε,v ε ) ( L (0,T;H 1 ()) L (0,T;L p ()) ) L 2 (0,T;H0 1()n ) is a weak solution to (DIM) ε w.r.t. prescribed initial values (ρ ε (i),v ε (i) ), if Conservation of mass: For all ϕ C(0) ( [0,T)), it holds ρ ε t ϕ + ρ ε v ε ϕ dx dt + ρ ε (i) ϕ(0)dx = 0. Conservation of momentum: For all ψ C (0) ( [0,T))n, it holds ρ ε v ε t ψ + ρ ε v ε v ε : ψ + 1 ε p(ρ ε)div(ψ) 2µ(ρ ε )Dv ε : Dψ dx dt + ρ ε (i) v ε (i) ψ(0)dx = γε ρ ε ρ ε : ψ ρ ε 2 div(ψ) + ρ ε ρ ε div(ψ)dx dt. Energy inequality: Let Eε tot 1 (t) = ε W(ρ ε (t)) + ε 2 γ ρ ε(t) ρ ε(t) v ε (t) 2 dx. For almost every t (0,T), it holds t Eε tot (t) + 2 µ(ρ ε ) Dv ε 2 dx ds Eε tot (0) assumption C. March 3, 2016 Johannes Daube SIL for the NSK Equations 9 / 21

10 (DIM) ε : Compactness of Weak Solutions Family of weak solutions (ρ ε,ε ε ) ε (0,1) to (DIM) ε with uniformly bounded initial energy. Theorem (Compactness of weak solutions to (DIM) ε ) There are functions ρ 0 C 0, 1 29 ([0,T];L 2 ()) L (0,T;BV(,{β 1,β 2 })) and v 0 L 2 (0,T;H 1 0 ()n ) such that (up to subsequences) ρ ε ρ 0 in C 0, 1 29 ([0,T];L 2 ()) and v ε v 0 weakly in L 2 (0,T;H 1 () n ). Limiting sharp interface: ρ 0 (t) BV(,{β 1,β 2 }) indicates phases (t) = {ρ 0 (t) = β 1 } and + (t) = {ρ 0 (t) = β 2 } of finite perimeter, interface Γ(t) = ( (t)). Convergence to a reasonable sharp interface model? March 3, 2016 Johannes Daube SIL for the NSK Equations 10 / 21

11 (SIM) 0 : Two-phase Incompressible Navier Stokes Equation with Surface Tension Sharp interface model is free boundary problem with unknown interface Γ : [0,T] R n such that = (t) Γ(t) + (t), velocity function v(t) : \ Γ(t) R n, pressure function p(t) : \ Γ(t) R satisfying β 1 t v + β 1 (v )v + p = 2div(µ(β 1 )Dv) in (t), β 2 t v + β 2 (v )v + p = 2div(µ(β 2 )Dv) in + (t), (Navier Stokes) div(v) = 0 in \ Γ(t), (incompressible phases) (SIM) 0 [v] + = 0 on Γ(t), (continuous velocity) (Navier Stokes) V = v ν on Γ(t), (pure transport of interface) [T] + ν = 2σκν on Γ(t). (Young Laplace law) T (v,p) = 2µ(β 1 )Dv pi and T + (v,p) = 2µ(β 2 )Dv pi stress tensors, [ ] + jump across Γ(t), V normal velocity of interface, σ surface tension constant, κ mean curvature. + suitable boundary conditions on and initial conditions March 3, 2016 Johannes Daube SIL for the NSK Equations 11 / 21

12 (SIM) 0 : Weak Solutions to Sharp Interface Model For (ρ,v) L (0,T;BV(,{β 1,β 2 })) ( L (0,T;L 2 σ ()) L 2 (0,T;H 1 0 ()n ) ) let (t) = {ρ(t) = β 1 } and Γ(t) = ( (t)) (density determines interface). Then (ρ,v) is a weak solution of (SIM) 0, if Transport equation: characteristic function χ(t) = ρ(t) β 2 β 1 β 2 of t χ + v χ = 0 in (0,T). of (t) is a distributional solution Energy inequality: For almost every t (0,T), it holds 1 t ρ(t) v(t) 2 dx + 2σH n 1 (Γ(t)) + 2 µ(ρ) Dv 2 dx ds 2 1 ρ (i) v (i) 2 dx + 2σH n 1 (Γ(0)). 2 Variational formulation: For all divergence free test functions ψ C0 ([0,T);C 0,σ ()), it holds ρv t ψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt + ρ (i) v (i) ψ(0)dx = 2σ ν ν : ψ dh n 1 (x)dt. 0 Γ(t) March 3, 2016 Johannes Daube SIL for the NSK Equations 12 / 21

13 (SIM) 0 : Divergence Free Test Functions Variational formulation ρv t ψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt + ρ (i) v (i) ψ(0)dx = 2σ ν ν : ψ dh n 1 (x)dt 0 Γ(t) uses divergence free test functions ψ C 0 ([0,T);C 0,σ ()) = { ψ [0,T) : ψ C 0 ( ( 1,T))n, div(ψ) = 0 }. +: Removes pressure function from weak formulation. Consequence : No passage to the limit of pressure term. -: Reconstruction of pressure function from weak formulation. March 3, 2016 Johannes Daube SIL for the NSK Equations 13 / 21

14 (SIM) 0 : Reconstruction of Pressure (for Smooth Solutions) Additional smoothness assumptions: interface: Γ = Γ(t) is space-time interface of class C 3,1. velocity: all derivatives of v exist in L 2 ( ) and L 2 ( + ), where ( (t) {t} ) and ( + (t) {t}). t (0,T) t (0,T) Note: No additional regularity across interface is assumed! Theorem (Reconstruction of pressure) For a sufficiently regular weak solution (ρ,v) to (SIM) 0, there exists a unique p L 2 ( (0,T)) with the following properties. 1 p ± L 2 (0,T;W 1,2 ( ± (t))). 2 For almost all t [0,T) it holds p(t)dx = 0. 3 p = β 1 t v + 2µ(β 1 )div(dv) β 1 (v )v a.e. in. 4 p = β 2 t v + 2µ(β 2 )div(dv) β 2 (v )v a.e. in +. 5 [p] + = 2[ µ(ρ)dvν ] + ν + 2σκ in L 2 (0,T;H 1 2 (Γ(t))). Jusitfies notion of weak solutions to (SIM) 0 March 3, 2016 Johannes Daube SIL for the NSK Equations 14 / 21

15 (DIM) ε (SIM) 0 : The Sharp Interface Limit Aim Weak solutions (ρ ε,v ε ) ε (0,1) of (DIM) ε ε 0 weak solution (ρ 0,v 0 ) of (SIM) 0. Recall: Compactness properties of weak solutions: ρ ε ρ 0 in C 0, 1 29 ([0,T],L 2 ()) with ρ 0 C 0, 1 29 ([0,T];L p ()) L (0,T;BV(,{β 1,β 2 })) v ε v 0 weakly in L 2 (0,T;H 1 () n ) with v 0 L 2 (0,T;H 1 0 ()n ) Question: (ρ 0,v 0 ) weak solution of (SIM) 0? March 3, 2016 Johannes Daube SIL for the NSK Equations 15 / 21

16 (DIM) ε (SIM) 0 : Mathematical Main Difficulties Task: Passing to the limit in the weak formulation. Momentum balance: Recall variational formulation ρ ε v ε t ψ + ρ ε v ε v ε : ψ + 1 ε p(ρ ε)div(ψ) 2µ(ρ ε )Dv ε : Dψ dx dt + ρ ε (i) v ε (i) ψ(,0)dx = γε ρ ε ρ ε : ψ ρ ε 2 div(ψ) + ρ ε ρ ε div(ψ)dx dt. for any ψ C (0) ( [0,T))n. Challenging terms in the passage to the limit ε 0: 1 T p(ρ ε )div(ψ)dx dt and ε ρ ε ρ ε : ψ dx dt. ε March 3, 2016 Johannes Daube SIL for the NSK Equations 16 / 21

17 (DIM) ε (SIM) 0 : Treatment of Pressure Term Weak formulation for (SIM) 0 takes into account divergence-free test functions ψ C0 ([0,T);C 0,σ ()) = 1 p(ρ ε )div(ψ)dx dt = 0. ε Resolves problem of passing pressure term to the limit! Variational formulation simplifies to for any ψ C 0 ([0,T);C 0,σ ()). ρ ε v ε t ψ + ρ ε v ε v ε : ψ dx dt + ρ ε (i) v ε (i) ψ(,0)dx = 2µ(ρ ε )Dv ε : Dψ γε ρ ε ρ ε : ψ dx dt March 3, 2016 Johannes Daube SIL for the NSK Equations 17 / 21

18 (DIM) ε (SIM) 0 : Treatment of Capillary Term Due to energy inequality, 1 ε W(ρ ε) + ε 2 γ ρ ε 2 weak- pre-compact in L 1 (0,T;C 0 ()). Problem: Identification of limit function. Solution: If 1 ε W(ρ ε) + ε 2 γ ρ ε 2 converges to appropriate surface measure. Then control of εγ ρ ε ρ ε : ψ dx dt. March 3, 2016 Johannes Daube SIL for the NSK Equations 18 / 21

19 (DIM) ε (SIM) 0 : The Sharp Interface Limit (Solving the Sharp Interface Model) Assumption: For every ϕ L 1 (0,T;C 0 ()) it holds ( lim 1ε W(ρ ε ) + ε ε 0 2 γ ρ ε 2) ϕ dx dt = 2σ ϕ dh n 1 (x)dt 0 Γ(t) with σ = β 2 γw(r) β 1 2 dr and Γ(t). Theorem (Convergence of capillary term) For every ψ C0 ([0,T);C 0,σ ()) it holds lim εγ ρ ε ρ ε : ψ dx dt = 2σ ν ν : ψ dh n 1 (x)dt. ε 0 0 Γ(t) Theorem (Solving the sharp interface model) The couple (ρ 0,v 0 ) is a weak solution of (SIM) 0. March 3, 2016 Johannes Daube SIL for the NSK Equations 19 / 21

20 Conclusion Navier Stokes Korteweg equations (diffuse interface model) notion of weak solution compactness properties Two-phase incompressible Navier Stokes equations with surface tension (sharp interface model) notion of weak solution consistency result Sharp interface limit sufficient condition: convergence of energy to associated surface measure open question: possible to weaken/remove additional assumption? March 3, 2016 Johannes Daube SIL for the NSK Equations 20 / 21

21 References Abels, H.: On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9, No. 1, (2007). Chen, X.: Global asymptotic limit of solutions of the Cahn Hilliard equation. J. Differ. Geom. 44, No. 2, (1996). Hermsdörfer, K., Kraus, C., Kröner, D.: Interface conditions for limits of the Navier Stokes Korteweg model. Interfaces Free Bound. 13, No. 2, (2011). Korteweg, D. J.: Sur la forme que prennent les équations du mouvement des fluids si l on tient compte des forces capillaires causées par les variations de densite mais continues et sur la théorie de la capillarité dans l hypothèse d une variation continue de la densité. Arch. Néerl. (2) 6, 1 24 (1901). Thank you for your attention! March 3, 2016 Johannes Daube SIL for the NSK Equations 21 / 21