A simulation study on shear thickening in wide-gap Couette geometry. Ryohei Seto, Romain Mari Jeffery Morris, Morton Denn, Eliot Fried

Transcription

1 A simulation study on shear thickening in wide-gap Couette geometry Ryohei Seto, Romain Mari Jeffery Morris, Morton Denn, Eliot Fried

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3 Stokes flow: Zero-Reynolds number fluid mechanics Repulsive Attractive

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5 The e ective shear viscosity of a colloidal suspension of rigid spherical particles in a Newtonian fluid can vary by orders of magnitude depending on how rapidly it is sheared, as characterized by the applied shear rate. Laun

6 Jamming transition fluid solid viscosity particle volume fraction

7 Coupled SD-DEM Seto, Mari, Morris & Denn (2013) and Mari, Seto, Morris & Denn (2014, 2015) fluid mechanics granular physics Stokesian Dynamics Hydrodynamic interactions far field + lubrication Overdamped dynamics (Repulsive interparticle forces) (Brownian motion) Discrete Element Method Contact forces sti sphere + friction Inertia of particles

8 SD-DEM simulation Friction!

9 Particle simulations for local rheology Stress controlled simulation vs rate controlled vs experiment / = / 0 PRE 2015 PNAS 2015 S-shaped rheology curves Brownian force (+ repulsive force)

10 Where does the rate-dependence come from? Low High

11 higher µ =1 ü Ë Ë ü ü ü ü üüü Ë Ë Ë lower Ë Ë Ë Ë Ë µ =0 ( ) hydrodyn. contact repulsive

12 If macroscopic rheology = local rheology, our simulation can reproduce rheology measurements. Rheometer is macroscopic ~1mm

13 However, does macroscopic rheology = local rheology? de Cagny 2015 (J. Rheol) R in =4.1cm R out =6cm R out R in =1.9 cm d = 40 µm polystyrene beads Magnetic Resonance Imaging =0.59 =0.59

14 urements, oscillation measurements and more detailed measurements of the variatio the gap of the plate plate cell under an imposed normal stress. In order for these data to be comprehensible, we do have to repeat some of the earlier data and discuss In this way, we obtain a more complete picture of the shear-thickening behavior. Macroscopic Discontinuous Shear Thickening Fall 2015 (PRL) PHYSICAL REVIEW LETTERS PRL 114, (2015) DST is observed only when the flow separates into a (a) Ω ðϕ Þ, the first measurable velocity localized. The flow subsequently rem low-density flowing and a high-density jammed region thousands units of strains. DST is thus cle II. MATERIALS AND METHODS The cornstarch particles (from Sigma Aldrich) are relatively monodisperse parti with, however, irregular shapes (Fig. 1). Suspensions are prepared by mixing the c starch with a 55 wt. % solution of CsCl in demineralized water. The CsCl allows on perfectly match the solvent and particle DST 0 densities [Merkt et al. (2004)]. We study susp sions of volume fraction ranging between 38% and 46%, and focus here mainly on Rin = 3 cm Rout = 5 cm Rout Rin = 2 cm cornstarch suspension PHYSICAL = PRL 114, (2015) with shear localization. As Ω increases further, the velocity prof extend to the right (i.e., towards the oute cases, the system remains separated into a the inner cylinder and a jammed region n The fraction of the gap that is jammed is rep it jumps at ΩDST ðϕ0 Þ and then slowly dec Comparing these velocity profiles [Fig density data [Fig. 2(b)], we find that, qu flow localization at Ω ¼ 10 rpm ΩDST R E V I E W FIG. L1.EMicrograph TT E Remergence Sparticles. with the sudden of the cornstarch P H Y SofI Cdensity AL PRL 114, (2015) Namely, the volume fraction decreases in (a) while it increases in the jammed!region, a X conservation of particle number. AsdΩ Ω γ_ ðrsubject T of th ; TSOR0license Þ¼ 2see Downloaded 09 May 2013 to Ω Redistribution or copyright; progressive extension DST ðϕ 0iÞ, tothe dt accompanied by a broadening n¼0 of the lowhigh strain rates, the density saturates, in at a packing fraction ϕmin 33% 35% =at0.439 It cannot be used D the jammed region, around ϕ ϕrcpthe. It is no profile can achieve multiple for becausedensity the apparent singularity shear history [24]. resolved Letexperimentally. us emphasize that the Neverth change of d the DST transition istwo irreversible. Indeed, that using the first terms of profile ϕðr; Ω1 Þ had been produced by qualitative 1(b) some behavior arbitrary Ω1 as > Ωin, we found DSTFig. profile remained the same under any sub below that the flow does remain ho of Ω. This irreversibility shows up in which rheometry allows us (3)cell to setupto (theuse smalleq. Couette Magnetic Resonance Imaging (b) =

15 Our original simulation Lees-Edwards periodic b.c. (mainly 3D) To reproduce bulk rheology New simulation Wide-gap rotary Couette cell (2D) To mimic a non-uniform macroscopic b.c. periodic image no wall periodic image imposing simple shear flow Walls

16 Overdamped dynamics with constrained particles (F f rct) (j) (F f P) (ji) j known to find i (F m P ) (ij) k (U m ) (k) force balance equations F m H F m FH f + P 0 FP f + Frct f F m H F f H = R mm FU R fm FU RFU mf RFU = 0 0 U m U f step1 dynamics step2 used in rheology

17 Frame invariant model v (R in )=R in = v (R in )=0 v (R out )=0 v (R out )= R out - Density matched suspension: particle = liquid - Overdamped dynamics, i.e., no inertia (no centrifugal force) - Hydrodynamics interaction is only lubrication: - No background flow is imposed F H R Lub U cf. previous model

18 v (R in )=R in v (R in )=0 v (R out )=0 v (R out )= R out

19 The present model is frame invariant Rotating inner wheel Rotating outer wheel v (R in )=R in v (R out )=0 v (R in )=0 v (R out )= R out area r R in R out R in Area fraction profile v (r) r out Rin r R in R out R in Velocity profile

20 The present model can reproduce DST in simple shear both by Lees-Edwards and by walls periodic boundary sheared by wall particles / / 0

21 Macroscopic apparent rheology torque shear stress rate in R in R out R in apparent viscosity by the inner wheel area =0.78 n = 6000 rate dependence due to repulsive force 0 =6 0 a 2 /F R

22 area =0.78 n = 9000

23 shear thickening + shear banding area Rout Rin 30 2a 0.01 in / v (r ) Rin area (r ) = 0.78 n = / r Rin Rout Rin r Rin Rout Rin

24 system size 1000 n = / in / area v (r) R in r R in r R in R out R in R out R in

25 Frictional particles vs. Frictionless particles µ =1 µ = in cf. simple shear simulation higher µ =1 ü Ë Ë ü ü ü ü üüü Ë Ë Ë lower Ë Ë Ë Ë Ë µ =

26 Frictionless µ =0 Frictional µ = area µ =1 µ =0 v (r) R in µ =1 µ = r R in r R in R out R in R out R in

27 Migration by particle pressure gradient (Morris and Boulay 1999) normal stresses Nin Nout (Nin + Nout)/ µ = in µ = shear stress ~ torque balance j /r P N in R 2 in in R 2 out out 0 µ =1 µ = in N out

28 Frictional jamming achieved by migration area v (r) R in µ =1 µ = r R in R out R in r R in R out R in

29 Conclusion We usually assume uniform simple shear flows in our DST simulation. Some experimentalists reported macroscopic DST. Macroscopic rheology local rheology? We extended our DST simulation for a wide-gap Couette geometry, and we also obtained similar macroscopic behavior Migration + Shear banding + Jamming + Shear thickening This work may provide some insights for macroscopic description of DST suspension flows.