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Les Houches School of Foam:Rheology of Complex Fluids
Andrew Belmonte
The W. G. Pritchard Laboratories
Department of Mathematics, Penn State University
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Fluid Dynamics
(tossing a coin)
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The Variety of Fluid Behavior
So how can we measure the properties of a fluid in a controlled way when it canswirl and splash?
Somehow separate the study of:
* what velocity fields occur in a fluid flow
* what are the material properties of the fluid
This is what rheology is meant to do!
Viscometric flows vs Non-viscometric flows(→ control the velocity field through confined geometry)
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Overall Plan for Two Lectures
Today (Complex Fluids I):
• Tools to study the rheology of complex fluids
• Constitutive equations, stress tensor
• What is seen in rheology of polymers and other fluids
Connections to foamy issues: talks of Francois (today) and Sylvie (Monday?)
Monday (Complex Fluids II - Le Retour):
• Possible rheology continuation
• Hydrodynamic flows - compare Newtonian, polymeric, other
• Flows of other non-foam systems (brief)
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Steady shear rheology
Consider a fluid between two plates, where the top plate moves at a speedV0, and a steady state has been reached:
In this simple geometry the velocity field with be linearly dependent on thedistance y from the stationary plate:
V (y) =V0
hy
It is important that h be small enough for this to occur
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Steady shear rheology
Consider a fluid between two plates, where the top plate moves at a speedV0, and a steady state has been reached:
For certain materials the force per unit area σ on the moving plate will beproportional to V0, and inversely proportional to h:
σ = ηV0
h
(= η
dV
dy= ηγ̇
)where η is the viscosity. These fluids are called Newtonian.
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What is a Newtonian Fluid?
Most fluids with simple structure at the microscale (point-like atoms ormolecules) are Newtonian - though we are not yet to the point of predictingthis a priori... It’s better to measure first (shear stress σ vs shear rate γ̇):
shear rate
stres
sNewtonian
Newtonian case is linear, slope is the viscosity η
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What is a Newtonian Fluid?
Most fluids with simple structure at the microscale (point-like atoms ormolecules) are Newtonian - though we are not yet to the point of predictingthis a priori... It’s better to measure first (shear stress σ vs shear rate γ̇):
shear rate
shear thinning
stres
sNewtonian
More complex, structured fluids can be nonlinear in shear.
Most polymer fluids are shear-thinning,
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What is a Newtonian Fluid?
Most fluids with simple structure at the microscale (point-like atoms ormolecules) are Newtonian - though we are not yet to the point of predictingthis a priori... It’s better to measure first (shear stress σ vs shear rate γ̇):
shear rate
shear thinning
shear thickening
stres
sNewtonian
though shear-thickening is also possible.
As we will see later, there are several other ways to be non-Newtonian...
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How to Measure this in Practice?
A closed fluid volume is preferable, which usually implies rotational flow.
Commercial rheometer - fits on a (strong) table, includes fluid cell attach-ments (usually temperature controlled), motors and transducers:
Rheometrics (TA Inst) RFS-III controlled rate of strain rheometer
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Rheological cell: Cone-and-Plate geometry
The problem with rotation is that the outer fluid moves the fastest, whereas nearthe center it hardly moves at all. A Cone-and-Plate geometry compensates forthis, with the cone surface leading to a homogenous shear rate (see Blackboard)
· usually bottom rotates, torquemeasured at top
· angle β must be small for visco-metric flow
· note open edge - requires viscousmaterial; fast enough rotation spinsout fluid
· normal forces can also be mea-sured (pushing up on cone)→ Sylvie’s talk
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Rheological cell: Cone-and-Plate geometry
Cone-and-Plate rheometer for foams (Sylvie’s talk):
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Rheological cell: Couette geometry
The Couette or concentric cylinder geometry is more convenient for less viscousfluids (they cannot flow out), and can also have a greater surface area. Moreoverlaminar Couette flow is an exact solution to the Navier-Stokes equations.
However...
· inhomogeneous shear rate, butapprox. constant for h/R << 1
· typically h ∼ 1 mm, R ∼ 30 mm
· rod climbing instability can drawfluid up out of cylinder
· cannot measure normal forces
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Survey of Steady Shear Rheology in Complex Fluids
Shear thinning polymer solution:
1
10
100
1000
0.01 0.1 1 10
σ [
dyn/
cm2 ]
shear rate [s-1]
T = 25.2°C
1
10
100
1000
0.01 0.1 1 10
η a [P
oise
]
shear rate [s-1]
T = 25.2°C
(0.2% xanthan gum in 80:20 glycerol/water)
The ‘zero shear viscosity’ η0 ' 500 Poise.
Estimated relaxation time λ ' 40 s.
This is a ‘power law fluid’, with ηa ∼ γ̇β, and β = −0.63
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Survey of Steady Shear Rheology in Complex Fluids
Experimentally obtained rheological quantities:
1. Apparent or effective viscosity ηa - simply defined as
ηa =σ
γ̇
2. Zero shear viscosity η0 - for small enough γ̇, it seems that every fluid isNewtonian. This is the low shear rate plateau of ηa.
3. (Longest) relaxation time λ - estimated in steady shear rheology usingthe shear rate γ̇c at the edge of the plateau in effective viscosity:
λ =1γ̇c
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Survey of Steady Shear Rheology in Complex Fluids
Shear thinning Wormlike Micellar Fluid∗
10
100
1000
0.01 0.1 1 10 100
stre
ss
[dyn
/cm
2 ]
shear rate [s-1]
a)
80 mM CPCl / 60 mM NaSal
From this data we obtain
• relaxation time λ ' 2 s
• zero shear viscosity η0 ' 430 Poise.
∗briefly defined at the blackboard
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Survey of Steady Shear Rheology in Complex Fluids
Certain fluids have a stress plateau in steady shear, or a shear rate gap (canbe rheometer dependent - discuss stress or shear rate controlled):
Porte, Berret, & Harden, J. Physique II 7 (1997), 459.
Cappelaere, Berret, Decruppe, Cressely, & Lindner, Phys. Rev. E 56 (1997), 1869.
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Shear Banding in Wormlike Micellar Fluids
The plateau region corresponds to a nonhomogeneous state: the coexistenceof two or more shear bands:
Cappelaere, Berret, Decruppe, Cressely, & Lindner, Phys. Rev. E 56 (1997), 1869.
Shear bands also seen in foams (→ Sylvie’s talk)
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What Equation to Use for Complex Fluids?
A difficult question, still open in general; in many ways we were lucky to besurrounded with Newtonian fluids...
Recall Nikolai’s slide:
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What Equation to Use for Complex Fluids?
The question of the Constitutive Equation for a given fluid is a problem inMaterial Science.
Here we will discuss in a little detail:
• what we can expect
• some of what has been done
• an example of where these equations can come from
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The Flow of a Fluid in General
Your beloved Navier-Stokes equation (momentum equation)
ρ (∂t~u + (~u · ∇)~u) = −∇p + η∇2~u
is actually two equations!
Momentum (Cauchy) equation:
ρ (∂t~u + (~u · ∇)~u) = −∇p +∇ ·T
where T is the extra stress tensor, and
Constitutive equation:T = 2ηD,
where
D =12(∇u +∇uT )
is the deformation rate tensor....
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The Flow of a Fluid in General
Before we talk about this tensor T, note that:
Any other constitutive equation is non-Newtonian...
Besides shear thinning / thickening, what can happen?
• memory / relaxation effects
• elastic effects (related)
• nonlinearities in deformation
• tensor product ‘mixing’ (e.g. normal stresses)
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Physical Reasons Why Stress Must Be a Tensor
Consider an internal region of a flowing fluid (below). As this material movespast a particular face (say with normal n2), it may exert a force F1 in the 1direction:
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Physical Reasons Why Stress Must Be a Tensor
Consider an internal region of a flowing fluid (below). As this material movespast a particular face (say with normal n2), it may exert a force F1 in the 1direction:
- and additionally a force F2 in the 2 direction
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Physical Reasons Why Stress Must Be a Tensor
Consider an internal region of a flowing fluid (below). As this material movespast a particular face (say with normal n2), it may exert a force F1 in the 1direction:
- and additionally a force F3 in the 3 direction
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Physical Reasons Why Stress Must Be a Tensor
All of these forces act on the face with normal n2. But then there are theother faces...
We would like a quantity to describe the forces that flowing fluid can exert,which we can dot product with n2 to get a vector force:
~F = n2 · T
this is a tensor (2-tensor: 2 directions, discuss?)
Similarly we will use not just the shear rate velocity gradient γ̇, but thesymmetric tensor of all velocity gradients, which represents the deformation:
Dij =12
(∂ui
∂xj+
∂uj
∂xi
)
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Physical Reasons Why Stress Must Be a Tensor
Thus whatever constitutive equation we derive, propose, or find should bein terms of these tensors.
We will briefly look at some of the most well-known in the non-Newtonian fluidscommunity.
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Memory / Relaxation E�ectsMaxwell model
The first non-Newtonian constitutive equation is due to J. C. Maxwell (gastheory, 1867):
λ∂T∂t
+ T = 2ηD
In addition to the viscosity, this gives one more material property, therelaxation time λ.
Solve this using the integrating factor et/λ:
T(x, t) = T(x, 0)e−t/λ +∫ t
0
2ηD(s)(
e−(t−s)/λ
λ
)ds
(fading memory) (memory kernel)
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Two Material Properties → Two Nondimensional Parameters
• viscous fluids: viscosity → Reynolds number
diffusion time
advection time∼ L2/ν
L/U=
UL
ν= Re
• viscoELASTIC fluids: stress relaxation time → Deborah number
relaxation time
advection time∼ λ
L/U=
λU
L= De
Note: in the limit De → 0, one gets Navier-Stokes.
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Normal Stress E�ects
The Rod Climbing or Weissenberg effect is driven by extra stresses whichare exerted radially inward:
(Gareth McKinley’s lab - MIT, USA)
→ will be discussed later for foams (Sylvie)
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Constitutive List: of Increasing Mathematical Horrors
T = 0(Euler 1748)
T = 2ηD(Newton / Navier-Stokes 1660 - 1845)
λ∂T∂t
+ T = 2ηD
(Maxwell 1867)
λ
(∂T∂t
+ (~u · ∇)T)
+ T = 2ηD
(convected + Maxwell)
λ
(∂T∂t
+ (~u · ∇)T−∇uT ·T−T · ∇u)
+ T = 2ηD
(Oldroyd 1950)
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Pause - Back to Experiments for Inspiration
(we stopped here, continue in Complex Fluids II)
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Summary
Today: viscometric flows of Newtonian and non-Newtonian fluids
1. Steady rheology
2. Rheometers: cone-and-plate, couette
3. Rheology of complex fluids: shear thinning, shear banding,
normal stresses (more to come)
4. Constitutive equations
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