Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
1 Introduction
2 Previous Work
3 The Model
4 Re-Formulation
5 Numerics
6 Convergence
7 Investigations
8 Conclusion
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
ns
C
μ = 0
ζ
ϑ
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
ns
C
μ = 0
ζ
ϑ
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
dt=
d�
dt� s
@�
@s�
@(�S)
@s� �U� +
1
Pes
@2�
@s2
Pes = Peclet number, U = normal velocity component,� =the curvature, S = tangential velocity component
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
dt=
d�
dt� s
@�
@s�
@(�S)
@s� �U� +
1
Pes
@2�
@s2
Pes = Peclet number, U = normal velocity component,� =the curvature, S = tangential velocity component
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
dt=
d�
dt� s
@�
@s�
@(�S)
@s� �U� +
1
Pes
@2�
@s2
Pes = Peclet number, U = normal velocity component,� =the curvature, S = tangential velocity component
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
dt=
d�
dt� s
@�
@s�
@(�S)
@s� �U� +
1
Pes
@2�
@s2
Pes = Peclet number, U = normal velocity component,� =the curvature, S = tangential velocity component
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
2�i
IC
!(�; t)d ln� � �
� � �+
1
2�i
IC
!(�; t)d� � �
� � �
+
Z!(�; t)ds = �
�
2
@�
@s+
i
2p1� � i
Q
2�
the complex velocity u + iv has an integral representation interms of the complex weight !(�; t)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
2�i
IC
!(�; t)d ln� � �
� � �+
1
2�i
IC
!(�; t)d� � �
� � �
+
Z!(�; t)ds = �
�
2
@�
@s+
i
2p1� � i
Q
2�
the complex velocity u + iv has an integral representation interms of the complex weight !(�; t)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
2�i
IC
!(�; t)d ln� � �
� � �+
1
2�i
IC
!(�; t)d� � �
� � �
+
Z!(�; t)ds = �
�
2
@�
@s+
i
2p1� � i
Q
2�
the complex velocity u + iv has an integral representation interms of the complex weight !(�; t)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
�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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
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
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
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
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
d�
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
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
!1.8 !1.6 !1.4 !1.2 !1 !0.8 !0.6 !0.4 !0.2
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
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
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
The Re-Formulation of the other Equations
the marker-spacing gradient s� and angle � in this frame
ds�
dt=
1
2�
Z 2�
0
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
d�
dt=
T
s�
@�
@��
1
s�
@(�S)
@��
1
s�U�
@�
@�+
1
Pes
1
s2�
@2�
@�2
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
The Re-Formulation of the other Equations
the marker-spacing gradient s� and angle � in this frame
ds�
dt=
1
2�
Z 2�
0
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
d�
dt=
T
s�
@�
@��
1
s�
@(�S)
@��
1
s�U�
@�
@�+
1
Pes
1
s2�
@2�
@�2
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
The Re-Formulation of the other Equations
the marker-spacing gradient s� and angle � in this frame
ds�
dt=
1
2�
Z 2�
0
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
d�
dt=
T
s�
@�
@��
1
s�
@(�S)
@��
1
s�U�
@�
@�+
1
Pes
1
s2�
@2�
@�2
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
Comparisons to Analytical Solutions continue
!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!1
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x(!)
y(!
)
0 1 2 3 4 5 6!1
!0.5
0
0.5
1x 10!5
!
|z(!
’)|!
|z(!
)|
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
!
"(!
)
0 1 2 3 4 5 6!1
!0.5
0
0.5
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
10!11
10!10
10!9
10!8
10!7
10!6
time
log
( erro
r )
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
10!10
10!9
10!8
10!7
10!6
time
log(
erro
r )
Surfactant Total Error
! t = 0.01! t = 0.005! t = 0.0025! t = 0.00125
10!210!9
10!8
10!7
10!6
log ( ! t)
log
( err
or )
area error C(! t)2
10!210!9
10!8
10!7
10!6
log ( ! t)
log
( err
or )
surfacant total error C(! t)2
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x
y
!=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
!0.3
!0.2
!0.1
0
0.1
0.2
0.3
0.4
!=0.0! =0.025!=0.05! = 0.1
0 1 2 3 4 5 60
1
2
3
4
5
6
7
8
9
10
!
" ( !
)
#=0.0#=0.025#=0.05#=0.1
0 1 2 3 4 5 60.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
!
" ( !
)
#=0.1#=0.05#=0.025#=0.0
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.04
!0.02
0
0.02
0.04
0.06 N=512, Unpadded
!20 !15 !10 !5 0 5!0.06
!0.04
!0.02
0
0.02
0.04
0.06 N=512, Padded
Unpadded computations seem unstable, padded ones seem stableEnkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.04
!0.03
!0.02
!0.01
0
0.01
0.02
0.03
0.04
0.05 N=512, Unpadded, Local plot
!2 !1.5 !1 !0.5 0 0.5!0.05
!0.04
!0.03
!0.02
!0.01
0
0.01
0.02
0.03
0.04
0.05 N=512, Padded, Local plot
Unpadded computations seem unstable, padded ones seem stableEnkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.01
!0.005
0
0.005
0.01
0.015
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
!0.01
!0.005
0
0.005
0.01
0.015
0.02 ! spectrum, N=512, Padded
Unpadded computations seem unstable, padded ones seem stableEnkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2x 10!3 ! Spectrum, N=512, Unpadded, Local Plot
!12 !10 !8 !6 !4 !2 0 2 4x 10!3
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2x 10!3 ! Spectrum, N=512, Padded, Local Plot
Unpadded computations seem unstable, padded ones seem stableEnkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.5
0
0.5
1
x
y
0 1 2 3 4 5 60
1
2
3
!
"
!1.5 !1 !0.5 0 0.5 1 1.5!1
!0.5
0
0.5
1
x
y
0 1 2 3 4 5 60
1
2
3
4
5
!
"
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
Interesting Phenomena: Bubble Bursting
surfactant a�ects bubble stability, often facilitates bubble breakup
!4 !3 !2 !1 0 1 2 3 4!1
!0.5
0
0.5
1
x
y
0 1 2 3 4 5 60
1
2
3
4
5
!
" ( !
, t)
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.1
!0.05
0
0.05
0.1
0.15
x
y
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x
y
T=7T=0
!1.5 !1 !0.5 0 0.5 1 1.5!1
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x y
T=7T=0
interface steadies, but its position symmetry is lost
this cannot be captured by the analytical solutions
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x
y
T=7T=0
!1.5 !1 !0.5 0 0.5 1 1.5!1
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
x y
T=7T=0
interface steadies, but its position symmetry is lost
this cannot be captured by the analytical solutions
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations Conclusion
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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University
Outline Introduction Previous Work The Model Re-Formulation Numerics Convergence Investigations 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.
Enkeleida Lushi M.Sc. Thesis Defense, Simon Fraser University