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1 15 - interfaces between continua
15 - interfaces between continua
15 - interfaces between continua
rheology
2
rheology /ri���ləd�i/ is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force rheology generally accounts for the behavior of non-newtonian fluids, by characterizing the minimum number of functions that are needed to relate stresses with rate of change of strains or strain rates.
15 - interfaces between continua
rheology
3
rheology is the science of deformation and flow. this definition of was due to bingham, who, together with scott blair and reiner, helped form the society of rheology in 1929. the idea that everything has a time scale, and that if we are prepared to wait long enough then everything will flow was known to the greek philosopher heraclitus, and prior to him, to the prophetess deborah – the mountains flowed before the lord. not surprisingly, the motto of the society of rheology is παντα ρει (everything flows), a saying attributed to heraclitus .
15 - interfaces between continua
newtonian and non-newtonian fluids
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newtonian • characterized by constant viscosity
non-newtonian • any fluid that does not obey the previous equation
� = �pI + 2µrs
x
v
� = �pI + &
�1Dt& +1
2�3(A1 · & + & ·A1) +
1
2�5(tr(&))A1 +
1
2�6(tr(& ·A1))I =
µ
✓A1 + �2DtA1 + �4A
21 +
1
2�7(tr(A
21))I
◆
15 - interfaces between continua
newtonian and non-newtonian fluids
5
non-newtonian • example: Oldroyd 8 constant model
� = �pI + &
�1Dt& +1
2�3(A1 · & + & ·A1) +
1
2�5(tr(&))A1 +
1
2�6(tr(& ·A1))I =
µ
✓A1 + �2DtA1 + �4A
21 +
1
2�7(tr(A
21))I
◆� = �pI + &
�1Dt& +1
2�3(A1 · & + & ·A1) +
1
2�5(tr(&))A1 +
1
2�6(tr(& ·A1))I =
µ
✓A1 + �2DtA1 + �4A
21 +
1
2�7(tr(A
21))I
◆
where A1 = 2rs
x
v
Dt
& =@&
@t+ v ·r
x
& + (& ·rx
v �rT
x
v · &)
15 - interfaces between continua
newtonian and non-newtonian fluids
6
newtonian fluid (boring)
solid (boring)
rheology
non-newtonian fluid (cool)
15 - interfaces between continua
complex fluid interfaces
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key examples: • emulsions and foams in industrial processes
15 - interfaces between continua
complex fluid interfaces
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key examples: • membranes of living cells
15 - interfaces between continua
complex fluid interfaces
9 15 - interfaces between continua
interfacial rheology
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Interfacial tension (quasi-static) • Laplace-Young equation
Interfacial dilatational rheology
Interfacial shear rheology
�n = pint
n = pext
n+ 2�Hn
2H = �rs
· n = (I � n⌦ n)rx
n
�ei(!t+�(!)) = E⇤(!)�A
A0ei!t
�ei(!t+�(!)) = G⇤(!)✏ei!t
15 - interfaces between continua
interfacial rheology
11
Interfacial tension (quasi-static)
Interfacial dilatational rheology • experimental setup
�n = pint
n = pext
n+ 2�Hn
2H = �rs
· n = (I � n⌦ n)rx
n
�ei(!t+�(!)) = E⇤(!)�A
A0ei!t
Xu et al. (2005)
15 - interfaces between continua
interfacial rheology
12
Interfacial shear rheology: • experimental setup
�ei(!t+�(!)) = G⇤(!)✏ei!t
Fuller and Vermant (2012)
15 - interfaces between continua
interfaces between continua
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• fluid-solid interface
• fluid-fluid interface
• fluid-membrane-fluid interface
15 - interfaces between continua
interfaces between continua
14
fluid-solid interaction
15 - interfaces between continua
interfaces between continua
15
fluid-solid interaction
recall: master balance law in spatial setting
d
dt
Z
⌦x
↵d⌦x
=
Z
@⌦x
�T
x
nx
d@⌦x
+
Z
@⌦x
�d@⌦x
d
dt
Z
⌦x
↵d⌦x
=
Z
@⌦x
�T
x
nx
d@⌦x
+
Z
@⌦x
�d@⌦x
scalar
vector
15 - interfaces between continua
interfaces between continua
16
fluid-solid interaction
Arbitrary Lagrangian Eulerian (ALE) methods • balance equations in advective form • used for moving domains
Z
⌦x
@↵
@t+ (v � v) ·r
x
↵+ ↵rx
· v �rx
· � � �
�d⌦
x
= 0
Z
⌦x
@↵
@t+ (v � v) ·r
x
↵+↵rx
· v �rx
· �� �
�d⌦
x
= 0
where is the velocity of the domain Z
⌦x
@↵
@t+ (v � v) ·r
x
↵+ ↵rx
· v �rx
· � � �
�d⌦
x
= 0
15 - interfaces between continua
interfaces between continua
17
fluid-solid interaction Solid problem • balance equations in Lagrangian description • balance of linear momentum
Fluid problem • balance equations in ALE description • continuity equation • balance of linear momentum
@t�rX · P = B0
rx
· v = 0
⇢@v
@t+ ⇢(v � v) ·r
x
v �rx
· � = b
15 - interfaces between continua
interfaces between continua
18
fluid-solid interaction coupled problem
Bazilevs et al. 2008 reference configuration
current configuration
ALE mapping (moving domain)
uv, pu
solid displacements fluid velocities and pressure domain displacements
15 - interfaces between continua
interfaces between continua
19
fluid-solid interaction: interface coupled problem compatibility conditions
compatibility of kinematics
compatibility of stresses
v|�fst
= @[email protected] � ��1
����fst
(�fnfst + �snfs
t )����fst
= 0
u|�fst
= u � ��1����fst
compatibility of ALE map
15 - interfaces between continua
interfaces between continua
20
fluid-fluid interface: surface tension
15 - interfaces between continua
interfaces between continua
21
fluid-fluid interface: surface tension
⌦f10
⌦f20
�ff0
⌦f10
⌦f20
�ff0
⌦f10
⌦f20
�ff0
⌦f1t
⌦f2t
�fft
⌦f1t
⌦f2t
�fft⌦f1
t
⌦f2t
�fft
�
15 - interfaces between continua
interfaces between continua
22
fluid-fluid interface: surface tension
�n = pint
n = pext
n+ 2�Hn
Young-Laplace equation at the interface
fluid problem in eulerian description inside the domain r
x
· � + ⇢b = 0
• determine the motion of only one fluid • example of application: pendant drop experiments
15 - interfaces between continua
interfaces between continua
23
fluid-fluid interface: surface tension incremental kinematics to use Lagrangian description and to avoid ALE mapping
incremental constitutive equation for nearly incompressible newtonian fluid
� = �K(J � 1)I + 2µd
xt+1 = �(xt)
�(xt) = xt + u
d = 1�t
Pi ln(�(i))n(i) ⌦ n(i)
15 - interfaces between continua
interfaces between continua
24
fluid-fluid interface: surface tension numerical examples
Saksono and Peric (2006)
15 - interfaces between continua
interfaces between continua
25
fluid-membrane-fluid interface
15 - interfaces between continua
interfaces between continua
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fluid-membrane-fluid interface
⌦f10
⌦f20
�ff0
⌦f10
⌦f20
�ff0
⌦f1t
⌦f2t
�fft⌦f1
t
⌦f2t
�fft
�
⌦m0
⌦mt
⌦m0
⌦mt
15 - interfaces between continua
interfaces between continua
27
fluid-membrane-fluid interface
bulk
surface
15 - interfaces between continua
interfaces between continua
28
fluid-membrane-fluid interface membrane kinematics
A1
A2
a1
a2
A1
A2
a1
a2
A1
A2
a1
a2
A1
A2
a1
a2
i1
i2
i3
i1
i2
i3
i1
i2
i3
15 - interfaces between continua
interfaces between continua
29
fluid-membrane-fluid interface membrane kinematics: differential geometry of surfaces
A↵ ·A� = ��↵
A↵ ·A� = A↵�
bF = a↵ ⌦A↵
JA =pdetA↵�
Ja =pdeta↵�
J = JaJA
da = JdA
A↵ ·A� = ��↵
A↵ ·A� = A↵�
bF = a↵ ⌦A↵
JA =pdetA↵�
Ja =pdeta↵�
J = JaJA
da = JdA
A↵ ·A� = ��↵
A↵ ·A� = A↵�
bF = a↵ ⌦A↵
JA =pdetA↵�
Ja =pdeta↵�
J = JaJA
da = JdA
dual basis
metric tensor
area defined by basis vectors
covariant derivative V ↵;� = A� ·rV ↵ = DV ↵[A� ]
15 - interfaces between continua
interfaces between continua
30
fluid-membrane-fluid interface membrane kinematics
A↵ ·A� = ��↵
A↵ ·A� = A↵�
bF = a↵ ⌦A↵
JA =pdetA↵�
Ja =pdeta↵�
J = JaJA
da = JdA
surface deformation gradient
identity tensor on the surface
pseudo-inverse of the deformation gradient
15 - interfaces between continua
interfaces between continua
31
fluid-membrane-fluid interface membrane balance equations
dDiv bP = bP↵�;�
bP = @ b @cF
linear momentum balance
divergence on the surface
surface strain-energy function b = b (bF )
first Piola-Kirchhoff stress tensor
15 - interfaces between continua
continua with boundary energies
32
fluid-membrane-fluid interface coupled problem • standard problem inside the volume • include membrane contribution on the
boundary volume
surface
15 - interfaces between continua
continua with boundary energies
33
fluid-membrane-fluid interface coupled problem • Local equilibrium for the volume
• Local equilibrium for the surface
15 - interfaces between continua
continua with boundary energies
34
fluid-membrane-fluid interface coupled problem • global equilibrium for the volume
• global equilibrium for the closed surface
15 - interfaces between continua
continua with boundary energies
35
fluid-membrane-fluid interface coupled problem • constitutive equation for the volume
• constitutive equation for the surface