NON-MONOTONIC FLOW CURVES AND VORTICITY BANDING INSHEAR THICKENING SUSPENSIONS
Romain MariDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge
Laboratoire Interdisciplinaire de Physique, CNRS-Université Grenoble-Alpes
Rahul ChackoDept. of Physics, Durham
Suzanne Fielding Mike CatesDAMTP, Cambridge
Ryohei SetoOkinawa (OIST)
Jeff MorrisLevich Institute, City College of NY
Morton Denn
SHEAR THICKENING[Cwalina & Wagner, JOR 2014]
260nm silica + polymer brush in PEG 200
~500nm calcium carbonate + polymer brush in PEG 200
[Egres & Wagner, JOR 2005]
SHEAR THICKENING
Transition between two (roughly) Newtonian branchesNewtonian suspending �uidHard particlesSize Brownian motion not necessaryInertia is not involved (Stokes �ow)Stabilized (=short-range repulsion)
100nm − 100μm
SUSPENSIONS OF HARD PARTICLES
Rate independent rheology
SUSPENSIONS OF HARD PARTICLES
Rate independent rheology
But it depends on friction
no contacts contacts
THICKENING SCENARIO (1)[Fernandez et al, PRL 2013]
[Seto et al, PRL 2013][Heussinger, PRE 2013]
[Wyart and Cates PRL 2013][Mari et al, JOR 2014]
STABILIZATION = SOFT REPULSION
Small stresses , no contacts
Large stresses , many contacts
σ ≪ /F ∗ a2
σ ≫ /F ∗ a2
NUMERICAL SIMULATIONS, RATE CONTROLLED[Seto, Mari, Morris & Denn, PRL 2013][Mari, Seto, Morris & Denn, JOR 2014]
THICKENING SCENARIO (2)[Wyart & Cates, PRL 2014]
Continuous Shear Thickening
THICKENING SCENARIO (2)[Wyart & Cates, PRL 2014]
Continuous Shear Thickening
THICKENING SCENARIO (2)[Wyart & Cates, PRL 2014]
Discontinuous Shear Thickening
THICKENING SCENARIO (2)[Wyart & Cates, PRL 2014]
Discontinuous Shear Thickening
THICKENING SCENARIO (2)[Wyart & Cates, PRL 2014]
Shear Jamming
STRESS-CONTROLLED SIMULATIONS[Mari, Seto, Morris & Denn, PRE 2015]
Non-monotonic �ow curves:S-shaped (discontinuous thickening)Arches (shear jamming)
WYART-CATES MODEL[Wyart & Cates, PRL 2014]
In practice,
"Minimal constitutive model" with qualitative features of ST:
σ = ηγ̇η(ϕ, f) = ( (f) − ϕη0 ϕJ )−2
(f) = f + (1 − f)ϕJ ϕμ
J ϕ0J
f = f(σ)
: "fraction of frictional contacts": only lubricated contacts: only frictional contacts
ff = 0f = 1
f(σ) ≈ exp(− /σ)σ0
FLOW INSTABILITIES
Stress-controlleduniform �ow curves
Uniform �ow unstable
[Hermes et al, 2015]
NORMAL STRESSES ACROSS SHEAR THICKENING[Mari, Seto, Morris and Denn, JOR 2014]
Normal stresses almost proportional to shear stress
STEADY GRADIENT BANDINGAt the interface:
=σ(1)xy σ
(2)xy
=p(1)yy p
(2)yy
STEADY GRADIENT BANDINGAt the interface:
=σ(1)xy σ
(2)xy
=p(1)yy p
(2)yy
Impossible!
STEADY VORTICITY BANDINGAt the interface:
=p(1)zz p
(2)zz
=γ̇(1) γ̇(2)
Impossible!
BANDING AND PARTICLE MIGRATIONSuspension balance model [Nott & Brady, JFM 1994]:
∇ ⋅ = ϕR(ϕ)( − )Σp vp vp+f
Reducing the problem to 1d
Conservation relationsMass conservation:
Momentum balance:
Stress control:
Constitutive model:Wyart-Cates + linear response:
Σ → ≡ σσzz
v → ≡ vvz
ϕ + (ϕv) = 0∂t ∂z
σ = −Rϕv∂z
⟨σ⟩ = ⟨η(ϕ, f)⟩γ̇
η(ϕ, f) = ( (f) − ϕη0 ϕJ )−2
(f) = f + (1 − f)ϕJ ϕμJ ϕ0
Jf = − [f − (σ)]∂t γ̇γ−1
0 f ∗
(σ) = exp(− /σ)f ∗ σ0
VORTICITY INSTABILITY MODEL
VORTICITY INSTABILITY MODELLinear stability analysis:
Unstable when
X = + δXX0 eikz+λt
η < − η∂σγ̇0k2γ0
ϕR∂ϕ
Hopf bifurcation and
Instability towards travelingbands
Reλ > 0 Imλ ≠ 0
TRAVELING BANDS
0:00 / 0:37
Fields snapshot
Strain-rate vs strain
TRAVELING BANDS
TRAVELING BANDS
Higher imposed stress
Very similar to Hermes et al.
"STOKESIAN DYNAMICS" SIMULATIONS
Instability for: η < − η∂σγ̇0k2 γ0
ϕR∂ϕ
Need /a ≳ 60Lz
Simulations with very large aspect ratio in favor of the vorticity
0:00 / 0:14
"STOKESIAN DYNAMICS" SIMULATIONSUniform �ow curve
σ/ = 0.5σ00:00 / 0:09
σ/ = 1σ00:00 / 0:15
σ/ = 2σ00:00 / 0:04
σ/ = 4σ00:00 / 0:01
COMPARISON WITH MODELSimulation Model
Snapshots from simulation
Flow curve when bandedFinite size effectsPhase diagram? Need to explore moreparameter spaceExperiments are not controlling volume,vorticity normal stress bounded
NEAR FUTURE
0:00 / 0:21
