A Fast Numerical Method for the Interface Motion of …elushi/msc_defense.pdfM.Sc. Thesis Defense,...

Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion

A Fast Numerical Method for the Interface Motion

of a Surfactant-Laden Bubble in Creeping Flow

Enkeleida Lushi

M.Sc. Thesis Defense,

Simon Fraser University

July 12, 2006

Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University

1 Introduction

2 Previous Work

3 The Model

4 Re-Formulation

5 Numerics

6 Convergence

7 Investigations

8 Conclusion

Studying Surfactant-Laden Bubbles: Motivation

Why do we study bubbles? They are ubiquitous & interesting.

What is a surfactant or surface active agent ?

Surfactants locally alter the surface tension.

Surface tension in uences bubble interface dynamics.

Studying Surfactant-Laden Bubbles : Goals

Study the e�ects of surfactant and non-uniform surfacetension on the bubble motion and deformation in a strainingStokes Flow (as in the �gure below)

Develop a fast and e�cient numerical solver to accuratelysimulate such interface motion in 2-D

Numerical solver should be able to handle general bubbleshapes and/or surfactant distributions

Previous Mathematical Work

Clean Stokes Flow

Tanveer & Vanconcelos ('95): analytical (polynomial) solutionsGreengard et al ('96): integral equations for Stokes FlowKropinski ('01): fast numerical solver for a single interfaceKropinski ('02): fast numerical solver for multiple interfaces

Surfactant-Laden Stokes Flow

Buckmaster & Flaherty ('73), Milliken et al ('94), Johnson &Borhan ('00), Pozrikidis ('98): boundary integral-basednumerical methods employing �nite di�erences/volumes,studies on bubble/drop stability, cusped bubble formationSiegel ('99): analytical solution, steady states, surfactant capsSiegel ('00): cusped bubble formationGilmore ('03): analytical bubble, numerical surfactant solutionothers: level-set numerics on a single bubble/drop problem

Previous Mathematical Work

Clean Stokes Flow

Tanveer & Vanconcelos ('95): analytical (polynomial) solutionsGreengard et al ('96): integral equations for Stokes FlowKropinski ('01): fast numerical solver for a single interfaceKropinski ('02): fast numerical solver for multiple interfaces

Surfactant-Laden Stokes Flow

Buckmaster & Flaherty ('73), Milliken et al ('94), Johnson &Borhan ('00), Pozrikidis ('98): boundary integral-basednumerical methods employing �nite di�erences/volumes,studies on bubble/drop stability, cusped bubble formationSiegel ('99): analytical solution, steady states, surfactant capsSiegel ('00): cusped bubble formationGilmore ('03): analytical bubble, numerical surfactant solutionothers: level-set numerics on a single bubble/drop problem

The Mathematical Model

2-D Stokes Flow/ Creeping Flow

very small (zero) Reynolds number, R = UL=�governed by the Stokes Equations r2u = rp; r � u = 0

u = velocity, p=pressure

The 2-D Domain

μ = 0

zero viscosity � inside the bubblebubble boundary C , uid domain outside the bubbledomain is evolving together with its boundary C !boundary conditions on the uid boundary/ bubble interface Cand far-�eld as jz j ! 1

The Mathematical Model

2-D Stokes Flow/ Creeping Flow

very small (zero) Reynolds number, R = UL=�governed by the Stokes Equations r2u = rp; r � u = 0

u = velocity, p=pressure

The 2-D Domain

μ = 0

zero viscosity � inside the bubblebubble boundary C , uid domain outside the bubbledomain is evolving together with its boundary C !boundary conditions on the uid boundary/ bubble interface Cand far-�eld as jz j ! 1

The Boundary Conditions

interface stress-condition on the interface C

�pn+ 2E � n = ��n�rs�

E = strain tensor, � = surface tension, � = curvature

kinematic condition, � a complex point on the interface C

dt= (u � n)n

far-�eld conditions, as jzj ! 1

u � u1 + O(1=jzj2);

p � p1

u1 = (Qx ;�Qy)=2, Q = Capillary Number, p1 t.b.d. later

�pn+ 2E � n = ��n�rs�

dt= (u � n)n

u � u1 + O(1=jzj2);

p � p1

�pn+ 2E � n = ��n�rs�

dt= (u � n)n

u � u1 + O(1=jzj2);

p � p1

The Surfactant Equation

we assume the surfactant is insoluble, i.e. no ux into the uid

surface tension � depends on the surfactant concentration �

we assume a linear dependence � = 1� ��, � is a parameter

the surfactant equation on the interface C is

dt� s

@(�S)

@s� �U� +

Pes = Peclet number, U = normal velocity component,� =the curvature, S = tangential velocity component

dt� s

@(�S)

@s� �U� +

dt� s

@(�S)

@s� �U� +

dt� s

@(�S)

@s� �U� +

The Sherman-Lauricella Integral Equations

Stokes Flow reduces to the Biharmonic Equation r4W = 0

Complex variable theory is used to �nd an integralrepresentation for this Stokes Boundary Value Problem:

!(�; t) +1

!(�; t)d ln� � �

� � �+

!(�; t)d� � �

� � �

Z!(�; t)ds = �

2p1� � i

the complex velocity u + iv has an integral representation interms of the complex weight !(�; t)

!(�; t) +1

!(�; t)d ln� � �

� � �+

!(�; t)d� � �

� � �

Z!(�; t)ds = �

2p1� � i

!(�; t) +1

!(�; t)d ln� � �

� � �+

!(�; t)d� � �

� � �

Z!(�; t)ds = �

2p1� � i

The Sherman-Lauricella Integral Equations continued

the complex weight !(�; t) on the interface is found bysolving the �rst equation

kernels in the integrals are singular, but the invertibility of the�rst equation is possible because it is a Fredholm Integral ofthe Second kind

! is used to �nd the velocity u + iv

the velocity is used next to �nd the interface position � fromthe kinematic condition and the surfactant � from its P.D.E.

The Problems that Arise...

point markers cluster in regions of high curvature

!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2

�t � C�s, with �s = min�s, �s =marker distance

inadequate interface and surfactant resolution

sti� surfactant equation, stability restriction �t � C (�s)2

not suitable for long-time, large-scale computations

an e�cient re-formulation of the problem is needed

!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2

The Re-Formulation of the Interface Problem

introduce a tangential component in the kinematic conditiond�dt

= Un that has no consequence to the curve motion itself

dt= Un+ T s

T is chosen to maintain equal arclength marker spacing

!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2

This formulation solves the point-clustering problem and easesthe stability constraint to �t = O(�s), �s is now uniformthroughout.

dt= Un+ T s

!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2

This formulation solves the point-clustering problem and easesthe stability constraint to �t = O(�s), �s is now uniformthroughout.

dt= Un+ T s

!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2

This formulation solves the point-clustering problem and easesthe stability constraint to �t = O(�s), �s is now uniformthroughout.Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University

The Re-Formulation of the other Equations

the marker-spacing gradient s� and angle � in this frame

Z 2�

U��0d�0

the velocity components U and S are

u + iv = Un+ Ss = �Ui��s�

+ S��s�

the surfactant equation in this frame of reference is

@��

@(�S)

@��

s�U�

Z 2�

U��0d�0

+ S��s�

@��

@(�S)

@��

s�U�

Z 2�

U��0d�0

+ S��s�

@��

@(�S)

@��

s�U�

The Numerical Implementation

the spatial derivatives are computed pseudo-spectrally

the spectral resolution is maintained

the integral equations have a spectrally-accurate discretization

alternating-point trapezoid rule used for the quadratures

the velocity integral equations result in dense linear systems

dense matrix-vector products are computed usingFast-Multipole Methods (FMM) in only O(N) steps

linear systems solved iteratively with GMRES, acceleratedwith FMM, in O(N) steps

The Numerical Implementation

the spatial derivatives are computed pseudo-spectrally

the spectral resolution is maintained

the integral equations have a spectrally-accurate discretization

alternating-point trapezoid rule used for the quadratures

the velocity integral equations result in dense linear systems

dense matrix-vector products are computed usingFast-Multipole Methods (FMM) in only O(N) steps

linear systems solved iteratively with GMRES, acceleratedwith FMM, in O(N) steps

The Time-Integration

explicit midpoint RK-2 for the interface � time-evolution,d�dt

= Un+ T s

explicit midpoint RK-2 for the s� time-evolutionds�dt

= 12�

R 2�

0U��0d�0

implicit-explicit midpoint RK-2 for the surfactant � evolutiond�dt

= F (�; t) + G (�; t)convective part F treated explicitly,di�usive part G treated implicitly

FFT diagonalizes G , � is updated spectrally (in Fourier space)

surfactant stability constraint is now linear, �t = O(�s)

The Time-Integration

explicit midpoint RK-2 for the interface � time-evolution,d�dt

= Un+ T s

explicit midpoint RK-2 for the s� time-evolutionds�dt

= 12�

R 2�

0U��0d�0

implicit-explicit midpoint RK-2 for the surfactant � evolutiond�dt

= F (�; t) + G (�; t)convective part F treated explicitly,di�usive part G treated implicitly

FFT diagonalizes G , � is updated spectrally (in Fourier space)

surfactant stability constraint is now linear, �t = O(�s)

Comparisons to Analytical Solutions

Siegel ('99) and later Gilmore ('03) derived analyticalsolutions for a polynomial class of surfactant-laden bubbles.

these analytical solutions can be used as test cases to checkour numerical method against

equi-parametrization of z(�; t) and interpolation of �(�; t) atthe equi-spaced markers is needed to compare to our solutionswhich are in the equal-arclength frame

next we show plots of the interface and the surfactant pro�lesby both methods and compare their di�erences

Comparisons to Analytical Solutions continue

!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!1

0 1 2 3 4 5 6!1

1x 10!5

’)|!

0 1 2 3 4 5 60

0 1 2 3 4 5 6!1

1.5x 10!5

’)|!

The di�erences between the analytical & the numerical solutions,are O(10�6) = O((�t)2), the accuracy of the integration schemes.

The Order of Convergence

bubble area ux and surfactant total ux are O((�t)2) accurate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810!12

( erro

Area Error

! t =0.01! t = 0.005! t = 0.0025! t = 0.00125

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Surfactant Total Error

! t = 0.01! t = 0.005! t = 0.0025! t = 0.00125

10!210!9

log ( ! t)

area error C(! t)2

10!210!9

log ( ! t)

surfacant total error C(! t)2

Comparisons of Diminishing Surfactant In uence

surfactant accumulating at the bubble tips lowers the surfacetension there and increases the interface motility

bubble deforms more with higher surfactant e�ects (larger �)

!1.5 !1 !0.5 0 0.5 1 1.5!1

!=0.0!=0.025!=0.05!=0.1initial bubble

!1.8 !1.75 !1.7 !1.65 !1.6 !1.55 !1.5 !1.45 !1.4!0.4

!=0.0! =0.025!=0.05! = 0.1

0 1 2 3 4 5 60

#=0.0#=0.025#=0.05#=0.1

0 1 2 3 4 5 60.5

#=0.1#=0.05#=0.025#=0.0

The Numerical Stability of the Interface

to suppress high-frequency instabilities, the velocity iscalculated at double the mesh

the eigenvalues of the Jacobian for the interface � are plotted

linear growth of eigenvalues, stable time-step is �t = O(1=N)

!20 !15 !10 !5 0 5!0.06

0.06 N=512, Unpadded

!20 !15 !10 !5 0 5!0.06

0.06 N=512, Padded

Unpadded computations seem unstable, padded ones seem stableEnkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University

The Numerical Stability of the Interface

to suppress high-frequency instabilities, the velocity iscalculated at double the mesh

the eigenvalues of the Jacobian for the interface � are plotted

linear growth of eigenvalues, stable time-step is �t = O(1=N)

!2 !1.5 !1 !0.5 0 0.5!0.05

0.05 N=512, Unpadded, Local plot

!2 !1.5 !1 !0.5 0 0.5!0.05

0.05 N=512, Padded, Local plot

The Numerical Stability of the Surfactant

to suppress high-frequency instabilities, the convective part ofthe surfactant PDE is calculated at double the mesh

the eigenvalues of the Jacobian of the surfactant are plotted

scaled by N to show linear growth, stable step �t = O(1=N)

!0.04 !0.035 !0.03 !0.025 !0.02 !0.015 !0.01 !0.005 0 0.005!0.02

!0.015

!0.005

0.02 ! Spectrum, N=512, Unpadded

!0.04 !0.035 !0.03 !0.025 !0.02 !0.015 !0.01 !0.005 0 0.005!0.02

!0.015

!0.005

0.02 ! spectrum, N=512, Padded

The Numerical Stability of the Surfactant

to suppress high-frequency instabilities, the convective part ofthe surfactant PDE is calculated at double the mesh

the eigenvalues of the Jacobian of the surfactant are plotted

scaled by N to show linear growth, stable step �t = O(1=N)

!12 !10 !8 !6 !4 !2 0 2 4x 10!3

2x 10!3 ! Spectrum, N=512, Unpadded, Local Plot

!12 !10 !8 !6 !4 !2 0 2 4x 10!3

2x 10!3 ! Spectrum, N=512, Padded, Local Plot

Interesting Phenomena: Surfactant Caps

surfactant caps: areas of non-zero surfactant on the bubbleoccur for certain parameters, e.g. Pes = 103; � = 0:1Q = 0:15 (top), Q = :30 (bottom)di�cult & time-consuming to compute with other numericalmethods due to the deformed peaks in the surfactant pro�le

!1.5 !1 !0.5 0 0.5 1 1.5!1

0 1 2 3 4 5 60

!1.5 !1 !0.5 0 0.5 1 1.5!1

0 1 2 3 4 5 60

Interesting Phenomena: Bubble Bursting

surfactant a�ects bubble stability, often facilitates bubble breakup

!4 !3 !2 !1 0 1 2 3 4!1

0 1 2 3 4 5 60

Interesting Phenomena: Bubble Bursting Comparisons

bubble breakup di�ers with higher surfactant e�ect (larger �)

bubble at time=7 for � = 0 (innermost), 0:1; 0:2 (outermost)

!15.2 !15.1 !15 !14.& !14.' !14.( !14.) !14.5 !14.4 !14.3 !14.2

Non-uniform Initial Surfactant Layer

two examples: �(�; 0) = 1 + cos(�) and �(�; 0) = 1 + sin(�)

bubble pro�le is initially circular, symmetric

!1.5 !1 !0.5 0 0.5 1 1.5!1

T=7T=0

!1.5 !1 !0.5 0 0.5 1 1.5!1

T=7T=0

interface steadies, but its position symmetry is lost

this cannot be captured by the analytical solutions

Non-uniform Initial Surfactant Layer

two examples: �(�; 0) = 1 + cos(�) and �(�; 0) = 1 + sin(�)

bubble pro�le is initially circular, symmetric

!1.5 !1 !0.5 0 0.5 1 1.5!1

T=7T=0

!1.5 !1 !0.5 0 0.5 1 1.5!1

T=7T=0

interface steadies, but its position symmetry is lost

this cannot be captured by the analytical solutions

Conclusion

Our numerical method has �ve advantages to other approaches:

it has spectral accuracy in spatial calculations,

integral equations are e�ciently solved in O(N) steps only,

the implicit-explicit scheme for surfactant equation easesstability constraint to linear w.r.t. mesh spacing,

by maintaining equal-arclength marker spacing we reinforcethis low-order stability constraint,

by computing the velocity at double the mesh, we suppressaliasing instabilities.

Conclusion

Future Research

solver for generally-shaped surfactant-laden drop interface

solver for large-scale, multiple, generally-shaped interfaces

non-linear dependencies of surface-tension to the surfactant

bubble stability, terminal shapes, tip streaming, fracturing

shrinking/expanding surfactant-laden bubbles/drops inquiescent ows

solver for soluble surfactants (there's ux into/from uid), etc.

Future Research

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