15 - interfaces between continua rheologybiomechanics.stanford.edu/me338_13/me338_s15.pdf · 15 - interfaces between continua 1 15 - interfaces between continua 15 - interfaces between

1 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.


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 .


newtonian and non-newtonian fluids

4

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

◆



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 · &)



6

newtonian fluid (boring)

solid (boring)

rheology

non-newtonian fluid (cool)


complex fluid interfaces

7

key examples: •  emulsions and foams in industrial processes



8

key examples: •  membranes of living cells



9 15 - interfaces between continua

interfacial rheology

10

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



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)



12

Interfacial shear rheology: •  experimental setup

�ei(!t+�(!)) = G⇤(!)✏ei!t

Fuller and Vermant (2012)


interfaces between continua

13

•  fluid-solid interface

•  fluid-fluid interface

•  fluid-membrane-fluid interface



14

fluid-solid interaction



15


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



16


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



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

⇢0@V

@t�rX · P = B0

rx

· v = 0

⇢@v

@t+ ⇢(v � v) ·r

x

v �rx

· � = b



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



19

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



20

fluid-fluid interface: surface tension



21


⌦f10

⌦f20

�ff0

⌦f10

⌦f20

�ff0

⌦f10

⌦f20

�ff0

⌦f1t

⌦f2t

�fft

⌦f1t

⌦f2t

�fft⌦f1

t

⌦f2t

�fft

�



22


�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



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)



24

fluid-fluid interface: surface tension numerical examples

Saksono and Peric (2006)



25

fluid-membrane-fluid interface



26


⌦f10

⌦f20

�ff0

⌦f10

⌦f20

�ff0

⌦f1t

⌦f2t

�fft⌦f1

t

⌦f2t

�fft

�

⌦m0

⌦mt

⌦m0

⌦mt



27


bulk

surface



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



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� ]



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



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


continua with boundary energies

32

fluid-membrane-fluid interface coupled problem •  standard problem inside the volume •  include membrane contribution on the

boundary volume

surface



33

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

•  Local equilibrium for the surface



34

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

•  global equilibrium for the closed surface



35

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

•  constitutive equation for the surface

Documents

15 - interfaces between continua rheologybiomechanics.stanford.edu/me338_13/me338_s15.pdf · 15 - interfaces between continua 1 15 - interfaces between continua 15 - interfaces between