Chris Macosko Department of Chemical Engineering and Materials Science NSF- MRSEC (National Science Foundation sponsored Materials Research Science and Engineering Center) IPRIME (Industrial Partnership for Research in Interfacial and Materials Engineering) IMA Annual Program Year Tutorial Introduction to Funny (Complex) Fluids: Rheology, Modeling and The September 12-13, 2009 erstanding silly putty, snail s and other funny fluids
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Chris Macosko Department of Chemical Engineering and Materials Science NSF- MRSEC (National Science Foundation sponsored Materials Research Science and.
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Chris Macosko
Department of Chemical Engineering and Materials Science
NSF- MRSEC(National Science Foundation sponsored Materials Research Science and Engineering Center)
IPRIME
(Industrial Partnership for Research in Interfacial and Materials Engineering)
IMA Annual Program Year TutorialAn Introduction to Funny (Complex) Fluids: Rheology, Modeling and Theorems
September 12-13, 2009
Understanding silly putty, snail slime and other funny fluids
What is rheology? (Greek) =
=
rheology =
honey and mayonnaise
rate of deformation
stress= f/area
to flow
every thing flows
study of flow?, i.e. fluid mechanics?
honey and mayo
rate of deformation
viscosity = stress/rate
What is rheology? (Greek) =
=
rheology =
rubber band and silly putty
time of deformation
modulus = f/area
to flow
every thing flows
study of flow?, i.e. fluid mechanics?
honey and mayo
rate of deformation
viscosity
4 key rheological phenomena
fluid mechanician: simple fluids complex flows
rheologist: complex fluids simple flows
materials chemist: complex fluids complex flows
rheology = study of deformation of complex materials
rheologist fits data to constitutive equations which
- can be solved by fluid mechanician for complex flows - have a microstructural basis
from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994).
ad majorem Dei gloriam
Goal: Understand Principles of Rheology: (stress, strain, constitutive equations)
• shear thinning (thickening) • time dependent modulus G(t)
• normal stresses in shear N1
• extensional > shear stress u>
Key Rheological Phenomena
Gτ B - I
VISCOUS LIQUID2The resistance which arises
From the lack of slipperiness
Originating in a fluid, other
Things being equal, is
Proportional to the velocity
by which the parts of the
fluids are being separated
from each other.
Isaac S. Newton (1687)
yx d x
dy
measured in shear1856 capillary (Poiseuille)1880’s concentric cylinders
(Perry, Mallock, Couette, Schwedoff)
dv
dy x
yx
dv
dy
Newton, 1687 Stokes-Navier, 1845
Bernoulli
Familiar materials have a wide range in viscosity
Adapted from Barnes et al. (1989).
measured in shear1856 capillary (Poiseuille)1880’s concentric cylinders
(Perry, Mallock, Couette, Schwedoff)
measured in extension1906 Trouton
u = 3
dv
dy x
yx
dv
dy
Newton, 1687 Stokes-Navier, 1845
Bernoulli
To hold his viscous pitch samples, Trouton forced a thickened end into a small metal box. A hook was attached to the box from which weights were hung.
“A variety of pitch which gave by the traction method = 4.3 x 1010 (poise) was found by the torsion method to have a viscosity = 1.4 x 1010 (poise).” F.T. Trouton (1906)
polystyrene 160°CMünstedt (1980)
Goal1.Put Newton’s Law in 3 dimensions
• rate of strain tensor 2D• show u = 3
Separation and displacement of point Q from P
( , t)
w y F s
s = w - y = F s
x
s = w - ys′ = w′ - y′
w′
y′
s′
P
Q
recall Deformation Gradient Tensor, F
s'
P
Q
Motions
P
x3
x2
x1
Rest or past state at t' Deformed or present state at t
Q
ˆ
ˆ
ˆ
w
y
s
P
Q
rate of separation
t
s F s
s F ss F
t t
Alternate notation:v
ˆ ˆ jTi j
i j ix
v L x x
dt
d
d d d
sv = F s F x
v F Lx x
is the velocity gradient tensor.
or
lim lim x x x xort t
d d d d
d d d
L
vv v L
v F L
F L F
F I F L
x xx
x x
Velocity Gradient Tensor
Viscosity is “proportional to the velocity by which the parts of the fluids are being separated from each other.” —Newton
θ
r z
v Ωr
v v 0
0 Ω 0
Ω 0 0
0 0 0
Lij
lim lim ( )
lim +
x x x x
x x
t t
• • •
• • •
F V R
F = V R + V R
V R I
F L V R
( )
is anti-symmetric
T T • • • •
•
L V R V R
R
Can we write Newton’s Law for viscosity as = L?
solid body rotation
2 ( ) + 2•
DV L L v vT T Rate of Deformation Tensor D
≠
2 •
D
Other notation:Vorticity Tensor W 2 2
+ ( )
•
R W L L
L D W v
T
T
t t lim 2
BShow D
d
dt
Example 2.2.4 Rate of Deformation Tensor is a Time Derivative of B.
t t
t t
lim +
recall that lim
lim + 2
Thus
B F F
F L
B L L D
T
x x
T
0 0 0 0 0 0
2 0 0 0 0 0 2 - 2 0 0
0 0 0 0 0 0 0 0 0
T TD L L W L L
Show that 2D = 0 for solid body rotation
t t t t
lim lim
BB F F F F F F
F F I
T T T
T
d
dt
Here planes of fluid slide over each other like cards in a deck.
Steady simple shear
Newtonian Liquid
1 1 2 2 2 3 3x x γx x = x x = x
= 2D or T = -pI + 2D
Time derivatives of the displacement functions for simple, shear
2 2
11 2
x
12 3
2
lim v 0 v
vv and v v 0 (2.2.10)
22
x
1 2 2
dx dγ dxx
dt dt dt
dx x
dx
12 12 21 T
0 0 0 0 0 0 0
0 0 0 0 0 2 0 0
0 0 0 0 0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0 0
ij ji ij
ij
L L D
T p
1 1
1 1 11 1 1 1
2 2 2 3 3 3
x x
α or
t
1 1 1
dx dα d x α x x v x x
d dt dt
v x v x
time derivatives of the displacement functionsat
Similarly
1
ij 2
3
0 0
L 0 0
0 0
1
1
1
0 0 0 0
L 0 2 0 0 2 0
0 0 2 0 0 2
2
2 (L L )
ij
ij ij ji
0 0
D 0 0
0 0
1 2 3
2 3
12 3 2 3
(1.7.9)
0
0
υ υ
2
incompressible fluid
or
symmetric, and thus
v
Steady Uniaxial Extension
Newtonian Liquid
Apply to Uniaxial Extension = 2D
11
22 33
2
From definition of extensional viscosity
11 22 3u
2
2 ij
0 0
D 0 0
0 0
Newton’s Law in 3 Dimensions•predicts 0 low shear rate•predicts u0 = 30
but many materials show large deviation
Newtonian Liquid
melt ePolystyren
T11T11
n1. stress at point on plane
Summary of Fundamentals
simple T - extension and shear
T = pressure + extra stress = -pI + .
symmetric T = TT i.e. T12=T21
2. area change around a point on planeTB F Fsymmetric, eliminates rotation
gives Hooke’s Law in 3D, E=3G
3. rate of separation of particles 2 ( ) + T T D L L v v
symmetric, eliminates rotation
gives Newton’s Law in 3D, 3u
Course Goal: Understand Principles of Rheology:
stress = f (deformation, time) NeoHookean: Newtonian:
• shear thinning (thickening) • time dependent modulus G(t)