Experimental Observation of Shear Thickening Oscillation in - PowerPoint PPT Presentation

Experimental Observation of Shear Thickening Oscillation in Dilatant Fluid S. Nagahiro (Sendai National College of Technology, Miyagi), H. Nakanishi (Kyushu University, Fukuoka) & N. Mitarai (Niels Bohr institute, Copenhagen)

What is Dilatant Fluid? A typical example: Dense mixture of starch and water. (starch particles) ~ 10 μ m size

Peculiar features of Dilatant Fluid Persistent or expanding hall Ebata, Tatsumi and Sano, PRE(2009)

Peculiar features of Dilatant Fluid Jamming Transition 41wt% cornstarch suspension Viscosity (Pa·s) 1000 100 10 0.01 0.1 1 -1 ) Shear rate (s A. Fall, N. Huang, F. Bertrand, G. Ovarlez, D Bonn, PRL(2008)

Why it shear thickens? A possible explanation ✦ Densely packed sand dilate upon deformation ✦ Coffee beans in vacuum bag is rigid because it cannot dilate due to the pressure. ✦ In the mixture, interstitial water surface could have particle size curvature. Pressure decreases due to the surface tension.

1. thickening is severe and instantaneous 2. relaxation after removal of the external stress is fast but not instantaneous. 3. thickened state is almost rigid and does not allow much elastic deformation 4. viscosity shows hysteresis 5. spontaneous oscillation due to shear thickening is observed.

Outline Fluid dynamics model of dilatant fluid Simulation of simple shear flows The present model reproduce basic nature of dilatant fluid and predicts shear thickening oscillation Experiment of Taylor-Couette flow We observed clear oscillations.

Fluid dynamics model for dilatant fluid

Modeling the dynamics of dilatant Fluid 1) Phenomenological description for shear thickening Introduce a state variable: φ = 0 φ = 1 under high stress under low stress

1 Modeling the dynamics of dilatant Fluid φ ( r, t ) 2) Viscosity is strongly increase func. of We assume Vogel-Fulcher type divergence: � � φ η ( φ ) = η 0 exp 1 − φ 1000 100 10 0

S 0 S S 0 S 0 S 0 Modeling the dynamics of dilatant Fluid 3) State variable , in turn, depends on stress φ ( r, t ) ( S/S 0 ) 2 φ ∗ ( S ) = φ M Steady value 1 + ( S/S 0 ) 2 where, � 1 2Tr(ˆ σ ˆ S = ˙ σ ) ˙ 0 2 3 4

Model Equations Relaxation is driven by deformation (athermal) � 1 | ˆ 2Tr(ˆ γ ˆ γ | = ˙ ˙ γ ) ˙ τ = 1 1 | ˙ γ | : local shear rate r | ˙ γ | r : dimansionless parameter Incompressible Navier-Stokes eq. � ∂ v i ρ Dv i ∂ � + ∂ v j Dt = ( − P δ ij + σ ij ) σ ij = η ( φ ) ∂ x j ∂ x j ∂ x i

Length and time scale ( S/S 0 ) 2 � � φ φ ∗ ( S ) = φ M η ( φ ) = η 0 exp 1 − φ 1 + ( S/S 0 ) 2 41wt% cornstarch suspension Viscosity (Pa·s) 1000 100 η 0 � 10Pa · s S 0 � 50Pa 10 0.01 0.1 1 -1 ) Shear rate (s Fall, Huang, Bertrand, Ovarlez, Bonn, PRL (2008).

Parameters and scales S 0 ≈ 50Pa Thickening stress: η 0 ≈ 10Pa · sec . Relaxed state viscosity: ρ ≈ 10 3 kg / m 3 Density: � � 0 τ 0 = η 0 Length Scale Time Scale ⌅ 0 = ⇥ ⇤ 0 S 0 For 41wt% cornstarch suspension � 0 ≈ 5cm τ 0 ≈ 0 . 2sec .

Simple Shear Flow of Dilatant Fluid S e S e � Boundary condition S ( z, t ) z = ± h = S e �

Simple Shear Flow of Dilatant Fluid Steady State Solution of the Model Equation high viscosity low viscosity

Shear flow in the unstable branch Flow oscillates spontaneously under constant stress φ M = 1 . 0 , h = 3 . 0 , S e = 1 . 0 , r = 0 . 1 φ parameter shear rate

Shear flow in the unstable branch Saw-tooth like wave --- moderately increases and suddenly drops ✦ moderate increase and sudden drop

Shear flow in the unstable branch a State Diagram for steady and oscillatory region 6 5 oscillate 4 flow width h 3 2 steady h* ~5cm for 1 41wt% suspension. 0 0 0.5 1 1.5 2 shear stress Se

Experiment with starch-water mixture

Parameters and scales S 0 ≈ 50Pa Thickening stress: η 0 ≈ 10Pa · sec . Relaxed state viscosity: ρ ≈ 10 3 kg / m 3 Density: � � 0 τ 0 = η 0 Length Scale Time Scale ⌅ 0 = ⇥ ⇤ 0 S 0 For 41wt% cornstarch suspension � 0 ≈ 5cm τ 0 ≈ 0 . 2sec .

Experimental Setup encoder ‣ 55wt% CsCl solution and potato-starch mixture starch-water mixture ‣ Volume fraction: 41~42.5% ‣ Flow thickness: 1~5cm 22cm ‣ weight: 0.5 ~ 10kg (0.1~2.3kPa) 1 ~ 5cm weight spring & dumper

Oscillation: 1000fps movie

Angular speed of the center rod 0.6 h =1.3 M =1, r =0.1, S e =1.1 A =1, Average Shear Rate 2.0 3.0 0.4 Density=42.5wt%, h=4cm 0.2 0 0 2 4 6 8 10 12 14 Time 30 angular speed(rad/s) 20 10 8.6 8.8 9.0 0 8.0 8.5 9.0 time(s)

A State diagram of the flow 42.5wt% suspension 6 5 oscillate 5 Oscillatory 4 flow width h 4 3 flow width h [cm] 3 2 steady 1 2 0 0 0.5 1 1.5 2 1 no oscillation intermit or noisy shear stress Se steady oscillation idle 0 0 0.5 1 1.5 2 2.5 external stress Se [kPa] Threshold stress. ∗ � 0 . 1kPa S e

Stress dependence of freq. and amplitude density=42.5wt% density=42.5wt% 35 frequency[Hz] 30 25 20 15 10 5 0 h=2cm amplitude[rad/sec.] 20 h=3cm *Frequency stays almost h=4cm h=5cm constant near threshold 15 10 5 0 0.0 0.5 1.0 external stress [kPa]

Frequency vs flow thickness • No systematic dependence either on the thickness and shear stress • Frequencies are always around 20 Hz (twice the predicted value) 35 (b)42.5wt% (b)42.5wt% 30 Frequency (Hz) 25 20 15 0.2kPa 10 0.3kPa 5 0.6kPa 1.0kPa 0 0 1 2 3 4 5 6 Flow thickness h(cm)

Experimental observation • About 20Hz frequency. • Oscillation starts with Hopf bifurcation . • Frequency does not depend on both Se and h

2D Simulations –– Maker and Cell (MAC) method z x L = 10 h Initial condition: v i ( r , t = 0) = 0 φ ( r , t = 0) = ξ i | ξ i | = 10 − 4 Initial noise:

Inhomogeneous Oscillation Small noise is given to initial φ φ M =0.85, Se=1.0 4.0 3.0 2.0 1.0 0 50 100 time

Jamming caused by instability φ M = 1 . 0 10 5 10 maximum viscosity 10 4 1 10 3 10 2 0.1 10 1 0 0.01 10 0 5 10 0 5 10 time time

Inhomogeneous Oscillation φ M =0.85, r in =1.0, r out =3.0, S e =2.0

Inhomogeneous Oscillation h and S e independent Frequency 0.5 (a) h=2.0, Se=3.0 angular speed h=1.0, Se=3.0 0.4 0.3 0.2 0.1 0 10 20 30 40 50 time 0.4 (b) Se=3 (c) h=1.6 0.3 frequency 0.2 0.1 0.0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 h Se

summary and remarks • We proposed phenomenological model • the model predicts spontaneous oscillation • the oscillation is also observed experimentally and next... • We’d like to confirm if the thickening band governs the oscillation. • measure pressure of the fluid (?) • measure off-center force acts on the rod (?)