continuum methods ii - university of california,...
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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