mathematical analysis of models for aging non-newtonian ...€¦ · david benoit cermics - micmac...
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Mathematical analysis of models for agingnon-Newtonian fluid: macroscopic and
mesoscopic approaches
David BENOITCERMICS - MICMAC
March 26, 2013
Joint works with Lingbing He, Claude Le Bris and Tony Lelievre
March 26, 2013 1 / 22
March 26, 2013 2 / 22
1. Various kinds of fluids
2. Macroscopic description
3. Mesoscopic description
4. Link between both descriptions
Various kinds of fluids March 26, 2013 3 / 22
Newtonian fluids
I linear response : stress proportional to the strain rate - the rate ofchange of its deformation
Various kinds of fluids March 26, 2013 4 / 22
Non-Newtonian fluids
(a) gels (b) polymer solution
I some non-linearity
I presence of some microstructures in the fluid
Various kinds of fluids March 26, 2013 5 / 22
Specific kind of non-Newtonian fluid
(a) ketchup (b) concrete
I aging
I flow induced rejuvenation
Various kinds of fluids March 26, 2013 6 / 22
1. Various kinds of fluids
2. Macroscopic description
3. Mesoscopic description
4. Link between both descriptions
Macroscopic description March 26, 2013 7 / 22
A test case (2D): flow past a cylinder
I variables from fluid mechanics function of space x and time t
I a.e. velocity u
I additional variable for aging : fluidity f
Macroscopic description March 26, 2013 8 / 22
Another testcase (1D) : Couette flow
x
ya
a
00
Newtonian
ρ∂u
∂t=
∂
∂y
(η∂u
∂y
) non-Newtonianρ∂u
∂t=
∂
∂y
(η∂u
∂y+ τ
),
λ∂τ
∂t= G
∂u
∂y− τ,
I boundary conditions u(t, 0) = 0 and u(t, 1) = a ≥ 0 for all t ∈ [0,T ]
Macroscopic description March 26, 2013 9 / 22
Macroscopic equations for Couette flow
ρ∂u
∂t= η
∂2u
∂y2+∂τ
∂y,
λ∂τ
∂t= G
∂u
∂y− f τ,
∂f
∂t= −f 2 − νf 3︸ ︷︷ ︸
aging
+ ξ|τ |f 2︸ ︷︷ ︸rejuvenation
(1a)
(1b)
(1c)
I C. Derec, F. Lequeux... : λ/f relaxation time of the extra-stress τ :f = 0 solid
I empirical equation on f
I limitations with f = 0
C. Derec, A. Ajdari, and F. Lequeux.
Rheology and aging: a simple approach, Eur. Phys. J. E, 4(3):355–361,2001.
Macroscopic description March 26, 2013 10 / 22
Theoretical results
I Existence and uniqueness of a global-in-time strong solution inappropriate functional spaces.Ideas of the existence proof
I formal energy estimatesI estimates on a sequence of approximate solutionsI passage to the limit
I Longtime behaviour depends on the boundary condition a and thesize where the fluidity f vanishes.
Macroscopic description March 26, 2013 11 / 22
Numerical results: evolution of velocity u and fluidity f
x
ya
a
00
Macroscopic description March 26, 2013 12 / 22
Transition
macro
∂u
∂t= A(u, τ, f )
∂τ
∂t= B(u, τ, f )
∂f
∂t= C(u, τ, f )
non-linearities in the macroscopicequations
meso
∂u
∂t= A(u, τ, f )
∂p
∂t= D(u, p)
τ = E(p)
f = F(p)
average middle-sized description of themicrostructures:p probability of the stress σ
τ and f are ’moments’ of p
Macroscopic description March 26, 2013 13 / 22
1. Various kinds of fluids
2. Macroscopic description
3. Mesoscopic description
4. Link between both descriptions
Mesoscopic description March 26, 2013 14 / 22
Initial models
redistribution in the phase space σ:
P. Hebraud and F. Lequeux,Mode-coupling theory for the pasty rheology of soft glassy materials,Phys. Rev. Lett., 81(14):2934–2937, Oct 1998.
redistribution in x and σ:
L. Bocquet, A. Colin, and A. Ajdari,Kinetic theory of plastic flow in soft glassy materials,Phys. Rev. Lett., 103(3):036001, Jul 2009.
Mesoscopic description March 26, 2013 15 / 22
Our model : no redistribution
p(t, σ) probability to find a stress σ at a time t
∂p
∂t
= −1l|σ|>σcp +
(∫|σ′|>σc
p(t, σ′)dσ′
)δ0(σ)︸ ︷︷ ︸
return to 0 above a threshold σc
− γ(t)∂p
∂σ︸ ︷︷ ︸elastic deformation γ =
∂u
∂x
Theoretical resultsI Existence and uniqueness of a solution in appropriate functional
spacesI Longtime behaviour : return to equilibrium at exponential rate
Mesoscopic description March 26, 2013 16 / 22
1. Various kinds of fluids
2. Macroscopic description
3. Mesoscopic description
4. Link between both descriptions
Link between both descriptions March 26, 2013 17 / 22
Averaging
Mesoscopic equation
∂p
∂t= −1l|σ|>σc
p +
(∫|σ′|>σc
p(t, σ′)dσ′
)δ0(σ)− γ(t)
∂p
∂σ(2)
is ’averaged’ :
∫(2) φ(σ)dσ
∂
∂t
∫pφ =
∫p(t, σ) {φ(0)− φ(σ)} 1l|σ|>σc
dσ + γ(t)
∫p(t, σ)
∂φ
∂σ(σ)dσ.
Link between both descriptions March 26, 2013 18 / 22
First link
The ’moments’ (φ(σ) = σ, 1l|σ|>σc)
τ(t) =
∫σp(t, σ)dσ, f (t) =
∫1l|σ|>σc
p(t, σ)dσ
satisfy ∂τ
∂t= −
∫|σ|>σc
σp(·, σ)dσ + γ,
∂f
∂t= −f + γ(p(·, σc)− p(·,−σc)).
I We need a closure.
Link between both descriptions March 26, 2013 19 / 22
Second link
Use ε→ 0 and a macroscopic time θ.Consider pε the solution of the mesoscopic equation with drift γ∞(εt)and its corresponding moments τε and fε.Consider (τ∗ε , f
∗ε ) solution of
εdτ∗εdθ
= −ϑf ∗ε τ∗ε + γ∞
εdf ∗εdθ
= −f ∗ε +γ∞
σc + γ∞.
Then,
∣∣∣∣τε(θε)− τ∗ε (θ)
∣∣∣∣+
∣∣∣∣fε(θε)− f ∗ε (θ)
∣∣∣∣ = O(ε).
Link between both descriptions March 26, 2013 20 / 22
Numerical results with γ∞(s) = s
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
t
τ
τε
τ∗ε2
τ∗ε1
Link between both descriptions March 26, 2013 21 / 22
Conclusion
I from a mesoscopic equation on a probability density,
I derivation of equations on macroscopic variables which are’moments’ of the probability density.
D. Benoit, L. He, C. Le Bris, and T. Lelievre.
Mathematical analysis of a one-dimensional model for an aging fluid, M3AS, inpress.
arXiv:1203.0928.
Link between both descriptions March 26, 2013 22 / 22