mathematical analysis of models for aging non-newtonian ...€¦ · david benoit cermics - micmac...

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