Continuum Methods II

John LowengrubDept MathUC Irvine

MicrofluidicsMicroscopic drop formation and manipulation:

•DNA analysis, •Analysis of human physiological fluids•Protein crystalization

Droplets used to improve mixing efficiency•Coalescence,•Inter-drop mixing

Channel geometry is used to control hydrodynamic forces

Tan et al, Lab Chip 2004

Muradoglu, Stone PF 2005

Typical flows

•Stokes/Navier-Stokes equations

•Complex geometry

•Topology changes (pinchoff/reconnection)

•Multiple fluids

•In this talk, will focus on techniques for solving such problemson larger scales that have application to microfluidics

•Drop/Interface interactions•Coalescence cascades in polymer blends

Drop/Interface Impact

Motivation and Physical Application

•Many engineering, industrial, and biomedical applications

•Fundamental study of topological changes

•Very difficult test for numerical methods (need to resolve near contact region accurately)

Z. Mohammed-Kassim,

E. K. Longmire Phys Fluids, 2003

Experiments: Drop/Interface Impact and CoalescenceZ. Mohammed-Kassim, E. K. Longmire Phys Fluids, 2003

•Slow gap drainage•Rebound of drop•3D initiation of coalescence

0.14, / 1.17,Re 20, We 3.8, Fr 0.6

d aλ ρ ρ= == = =

, / , Re, ,d a We Frλ ρ ρCharacterized by:

0.33, / 1.19,Re 68, We 7.0, Fr 1.0

d aλ ρ ρ= == = =

•Dependence on viscosityof outer fluid

Water/glycerin Drop

Water/glycerin

oil ambient

Mathematical Model

( )

1 ( 1)Fr

0

Re

1[ ] , [ ]We

T

t

pI

ddt

ρ

µ

κΣ Σ

Σ

∂ + •∇ = ∇• − − ∂ ∇• =

= − + ∇ +∇

= − =

• = •

u u u T g

u

T u u

Tn n u 0

xn u n

S

S

Fluid 2

Fluid 1

Fluid 2

m=1

m=l

•Highly nonlinear, non-local free boundary problem

Boussinesq

1ρ =

/d aρ ρ ρ=

m=l/d aρ ρ ρ=

Multiphase Navier-Stokes System

Numerical methods for Multiphase Flows

• Boundary Integral Method: highly accurate, difficult to perform topological changes, limited physics

• Mesh-Free Methods such as Particle methods: Marker Point method, Molecular Dynamics, Dissipative Particle Dynamics

• Diffuse Interface Methods: physically based, popular in material science

• Front Tracking Methods:: sharp interface, accurate hard to do topology changes

• Volume of Fluid Method (VOF): automatic topological changes, difficult to reconstruct interfaces. Conservation of mass

• Level-set Method: automatic topological changes, easy to compute interface geometry, loss of mass

Trends in Numerical Methods:

• Hybrid method: combining the advantages of existing methods, for example, combined LS and VOF, combined MP and VOF, etc

• Adaptive Mesh: moving mesh, locally refined mesh, etc

This structured mesh has 14400 nodes. Almost the same number of nodes as in our adaptive mesh simulation.

Difficulty in simulating drop/interface impact

•Accurate evolution on large scale

•Inaccurate in near-contact region

•Unphysical coalescence

•Expensive to resolve near-contact region using uniform mesh

0.33, / 1.19,Re 68, We 7.0, Fr 1.0

d oλ ρ ρ= == = =

Uniform mesh

Adaptive Mesh Refinement

Adaptive Mesh Refinement/Multiphase Navier-Stokes

Equations/Finite-element/ Level-set/

MethodAnderson, Zheng, Cristini. J. Comp. Phys. (2005)Zheng, Lowengrub, Anderson, Cristini. J. Comp. Phys. (2005)

Adaptive mesh refinement•Unstructured meshes (our work)

-Triangles (2-D, Axisymmetric)

-Tetrahedra (3-D)

Other approaches:

•Structured mesh refinement(Provatas et al, Sussman et al, Ceniceros and Roma, Agresar et al.,…)

•Mesh mapping/moving meshes(Huang et al, Ren and Wang, Hou and Ceniceros, Wilkes et al…)

(Other 2D unstructured mesh work: Ubbink and Issa 1999; Ginzberg and Wittum 2001)

( )i eq jjj

v l l= −∑ e

eql

( )2

mesh eqE l l dΩ

= − Ω∫• Mesh energy function

• Regard mesh edges as damped springs, define local equilibrium length scale according to relevant physical quantities

Optimal mesh ⇔ Global minimum of ELocal operations

• Equilibration

• Node reconnection

• Node addition/subtraction

Anderson, Zheng, Cristini J. Comp. Phys. (2005)Cristini et al. J. Comp. Phys. 2001

eql

l

Adaptive Mesh Refinement Contd.

• Embed axisymmetric domain(red box) in a square domain where the mesh is refined

• Align the mesh to the axisymmetric boundary(red lines).

Adaptive Mesh Refinement: Axisymmetric Domain

Zheng, Lowengrub, Cristini. In preparation.•Algorithm can be used forcomplex boundaries

Before alignment After alignment

Alignment to axis of symmetry

1. First we select all the edges that intersect with the axis, and from the two endpoints of each such edge, we select the one closer to the axis to be the candidate to project to the axis. After we collect all the candidates as subset S1.

2. Then we check every triangle, if all its vertices are in S1, then we delete its node farthest from the axis from S1. After checking all triangles, we get subset S2. We project all nodes in S2 to the axis orthogonally.

3. After step 2, some intersecting edges would have no endpoints projected, then we add the crossing points of such edges with the axis into the mesh, with two additional edges added.

Distribution Boussinesq Navier-Stokes equations

| ( , ) 0 ,tφΣ = =x x

1 ( / 1)d aρ ρ ρ χ= + −

/ | |,φ φ= ∇ ∇n

Interface( Level-set representation):

0tφ φ∂

+ • ∇ =∂

u

(Osher-Sethian, Sussman…)

1 ( 1)µ λ χ= + −

ε

( ), 1O hαε α= <

Level-set evolution:

Level-set re-initialization: sgn( )(1 | |) 0, ( , 0) ( , )d d d tφ τ φ

τ∂

− − ∇ = = =∂

x x

( ) | |δ δ φ φΣ = ∇

( ) ( )

1 ( 1)Fr

01

Re WeT

t

pI I

ρ

µ δΣ

∂ + •∇ = ∇ • − − ∂ ∇ • =

= − + ∇ +∇ + −

u u u T g

u

T u u nn

Singularsurface stress(Zaleski et al.)

Uzawa-Projection Method For NSE

*, 1 2 *, 1 21/ 2, 1/ 2, 1/ 2

*, 1 1, 1 1, 1

1/ 2 1/ 2 1, 1 1, 1

1/ 2,0 1/ 2,0 1/ 2

1( ) ,Re 2

, 0,

,Re

,

l n l nn l n l n

l n l n l n+1,l+1

n n n l n l

n n n n

pt

t qtp p q q

p p

+ ++ + +

+ + + + +

+ − + + + +

+ + −

− ∇ +∇+ •∇ +∇ = +

∆= + ∆ ∇ ∇• =

∆= + − ∆

= =

u u u uu u f

u u u

u u

• Uzawa Projection scheme: use iteration to improve accuracy

21( )Re

0

p∂+ •∇ +∇ = ∇ +

∂∇• =

u u u u ft

u

• Navier-Stokes Eqns:

1. Improve accuracy for nonlinear terms;

2. Improve incompressibility of velocity, especially with singular force;

3. In adaptive mesh refinement, only need information from one earlier time step.

•Advantages:

Implementation of FE/LS Adaptive Method

Mixed Finite Element Uzawa-Projection method(P2/P1, MINI)

•Level set: Discontinuous Galerkin Method(TVD_RK2, P1)

• Navier-Stokes Eqns

•Reinitialization: Explicit Positive Coefficient Scheme(Barth and Sethian, 1998)

•Adaptive mesh:

0

( ) ( , )

min( , | ( , ) |)eql dist

h h s tφ

≈ Σ

= +

x x

x

with variable density and viscosity

smoothing εδ δΣ Σ→εχ χ→

•Surface tension term:( )I δ∑− nnwe use capillary tensor,

only normal is needed (integration by parts), which is easy to compute

Zheng, Lowengrub, Anderson, Cristini, J. Comp. Phys. (2005)

Efficiency of FE/LS Adaptive Method

h = smallest mesh size, then d.o.f.(N)=O(1/h^(n-1)) in adaptive mesh, compared toO(1/h^n) in uniform n-dimensional mesh.

Evolution Solver Cost (FEM is based MINI elements)

•Remeshing cost: ( 1)( )nO h− − very small compared to flow solver

•Example: gap 310h −∼6,000-fold reduction in CPU time

h^(-5)h^(-3)Boundary Integral

h^(-4.5)h^(-3.5)Non-adaptive FEM

h^(-3.33)h^(-2.25)Adaptive FEM3D2DMethod

5/ 4

7 / 6

/ in 2D/ in 3D

N hN h

Application to drop interface impact

Axisymmetric results

t=0

t=0.5

t=0.9

t=1.4

t=1.8

t=5

t=7

0.33, / 1.19, Re 68, We 7, Fr 1d aλ ρ ρ= = = = =

Experiment Simulation

•Excellent agreementwith experiment

Axisymmetric simulation

•Adaptive mesh follows interface

•Near contact regions accurately

resolved

•Drop rebound is captured

Largest mesh size =2,

Smallest mesh size =0.002

There are total 15890 nodes in the axisymmetric domain

10 times zoom-in of each boxed region.

The adaptive mesh

Normalized location of interface and drop surface•Solid lines are from numerical simulations•Symbols are from experiments

Quantative comparison with experiment

Minimum distance between drop and interface is 0.007, smallest mesh size h=0.002, effectively 3.6E+7 nodes in uniform mesh.

Comparison to non-adaptive mesh

Extensions•Hybrid methods

Adaptive Level-Set Volume-of-Fluid (ACLSVOF)Yang, James, Lowengrub, Zheng, Cristini JCP 2006

•Complex fluids

Viscoelastic flows-- Pillapakkam and Singh, JCP 2001

Surfactants– Xu, Li, Lowengrub, Zhao JCP 2006

Multi-drop simulation with surfactant

Ca=0.7, Pe=10, E=0.2, x=0.3, 8/ ,01.0 ],5,5[]9,9[ hth =∆=−×−=Ω

Complex drop morphologies and surfactant distributions.f(.,0)=1.

Xu, Li, Lowengrub, Zhao JCP 2006

Numerical simulation of cocontinuous polymer blends

Cocontinuous Polymer Blends

Droplet / Matrix morphology Cocontinuous morphology

Intense mixing

Immiscible polymer blends

3D sponge-like microstructureInterpenetrating

self-supporting phases

Important route to new materials(solid materials, tissue scaffolds)

• Improved processibility, Static charge control (RTP, B.F. Goodrich) ,

Packaging for moisture sensitive products (Capitol Specialty Plastics, U.S. Patent 5,911,937), Permeability applications, Tissue scaffolds, Mechanical property improvement

15

20

25

30

35

40

0 20 40 60 80 100

Polypropylene (wt. %)

Tens

ile S

treng

th (M

Pa)

15

20

25

30

35

0 20 40 60 80 100

Polyamide 6 (wt. %)

Impa

ct S

treng

th (k

J/m

2 )

in Ionomer (Xie, 1991) in PVDF (Liu, 1998)

Jeffrey A. Galloway, Matthew D. Montminy,

Christopher W. Macosko,Polymer, Vol 43 (2002)

Drops Continuous phases

NSF funded collaboration.

Features of cocontinuous flows

Continuum interface methods

Experiments:

•Fully 3D structures•Detection difficult•Optimized control parameters for formation•Stability of microstructures (non-equilibrium)

Theory/numerics:•Many topology transitions•Large number of interfaces (complex microstructures)

•Macroscopic properties depend on microstructure

Governing equations for single fluid flow

u = cos2(t)

u = -sin2(t)

fixed wall

Navier-Stokes equations

Governing equations for multi-fluid flow

Navier-Stokes equations

Laplace - Young equation

Phase-field model

• Multi component, multi phase fluid flows with deformable interfaces

• Topological changes(merging, pinch-off)

• Increasingly popular methodAnderson, McFadden, Wheeler, Shen, Liu, Feng, Glasner, Bertozzi,…

Navier-Stokes-Cahn-Hilliard system

Converges to sharp interface as approaches zero: (Liu & Shen; J. Lowengrub and L. Truskinovsky)

Phase-field modeling of multicomponent flows

ε

New improvement for phase-field models

Accurate surface tension force formulation ((Continuum Surface Force)

J.S. Kim JCP 2005.

Numerical methods

1. Projection method for the Navier-Stokes equation

2. Crank-Nicholson for the Cahn-Hilliard equation, nonlinear multigrid method

Conservative multigrid method for Cahn-Hilliard fluid, J. Comp. Phys. Kim, Kang, and Lowengrub (2004).

Convergence to sharp interface limit

Long time evolution

The growth rate is given by linear stability analysis

Convergence to sharp interface limit

Simulation of cocontinuous morphology

Numerical mixing

Ref. J.M. Ottino, The kinematic of mixing: stretching, chaos and transport, Cambridge University Press, 1989.

500 massless particles

Interface length / Area

apply shear flow

Initial morphology by spinodal decomposition

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

0 2 0 4 0 6 0 8 0 1 0 0

W e ig h t P e rc e n t P E O in B le n d

Inte

rface

/Are

a ( µ

m-1

)

50% 50%

experimental result

Interface length / Area

50% 50%

numerical result on 512x512 mesh

Annealing

30% 50% 70%50/50 PEO/PS blend morphology changes dramatically after annealing

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

0 2 0 4 0 6 0 8 0 1 0 0

W e i g h t P e r c e n t P E O in B l e n d

Inte

rface

/Are

a ( µ

m-1

) B e f o r e R h e o lo g y

e o lo g y

experimental result numerical result

Scanning electron microscopy

3-D simulationRandom shear boundary conditions on top and bottom plates

Periodic boundary conditions on side wallsRandomly distributed ellipsoids.

30%

40%

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 20 40 60 80 100

Weight Percent PEO in Blend

Inte

rface

/Are

a ( µ

m-1

)

experimental result numerical result

Future directions

•Adaptive mesh refinement

Kim, Wise, Lowengrub (in preparation)

•Multicomponent (>2) FluidsKim, Lowengrub IFB 2005

•Viscoelastic flow

•Complex domains