1 15 - interfaces between continua

15 - interfaces between continua

15 - interfaces between continua

rheology

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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

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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

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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

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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

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complex fluid interfaces

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key examples: •  membranes of living cells

15 - interfaces between continua

complex fluid interfaces

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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

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interfacial rheology

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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

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Interfacial shear rheology: •  experimental setup

�ei(!t+�(!)) = G⇤(!)✏ei!t

Fuller and Vermant (2012)

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interfaces between continua

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•  fluid-solid interface

•  fluid-fluid interface

•  fluid-membrane-fluid interface

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interfaces between continua

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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

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interfaces between continua

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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

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interfaces between continua

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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

⇢0@V

@t�rX · P = B0

rx

· v = 0

⇢@v

@t+ ⇢(v � v) ·r

x

v �rx

· � = b

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interfaces between continua

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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

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interfaces between continua

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fluid-solid interaction: interface coupled problem compatibility conditions

compatibility of kinematics

compatibility of stresses

v|�fst

= @u@t � ��1

����fst

(�fnfst + �snfs

t )����fst

= 0

u|�fst

= u � ��1����fst

compatibility of ALE map

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interfaces between continua

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fluid-fluid interface: surface tension

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interfaces between continua

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fluid-fluid interface: surface tension

⌦f10

⌦f20

�ff0

⌦f10

⌦f20

�ff0

⌦f10

⌦f20

�ff0

⌦f1t

⌦f2t

�fft

⌦f1t

⌦f2t

�fft⌦f1

t

⌦f2t

�fft

�

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interfaces between continua

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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

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interfaces between continua

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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)

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interfaces between continua

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fluid-fluid interface: surface tension numerical examples

Saksono and Peric (2006)

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interfaces between continua

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fluid-membrane-fluid interface

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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

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interfaces between continua

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fluid-membrane-fluid interface

bulk

surface

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interfaces between continua

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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

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interfaces between continua

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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� ]

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interfaces between continua

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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

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interfaces between continua

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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

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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

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fluid-membrane-fluid interface coupled problem •  Local equilibrium for the volume

•  Local equilibrium for the surface

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continua with boundary energies

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fluid-membrane-fluid interface coupled problem •  global equilibrium for the volume

•  global equilibrium for the closed surface

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continua with boundary energies

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fluid-membrane-fluid interface coupled problem •  constitutive equation for the volume

•  constitutive equation for the surface