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) ~ 10μm size

Peculiar features of Dilatant Fluid

Ebata, Tatsumi and Sano, PRE(2009)

Persistent or expanding hall

Peculiar features of Dilatant Fluid

• A. Fall, N. Huang, F. Bertrand, G. Ovarlez, D Bonn, PRL(2008)

Jamming Transition

0.01 0.1 1 10 100 1000

Viscosity (Pa·s)

Shear rate (s

• 1)

41wt% cornstarch suspension

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.

Fluid dynamics model of dilatant fluid Simulation of simple shear flows Experiment of Taylor-Couette flow

The present model reproduce basic nature of dilatant fluid and predicts shear thickening oscillation We observed clear oscillations.

Outline

Fluid dynamics model for dilatant fluid

Modeling the dynamics of dilatant Fluid

φ = 0 φ = 1

under low stress under high stress

1) Phenomenological description for shear thickening

Introduce a state variable:

Modeling the dynamics of dilatant Fluid

2) Viscosity is strongly increase func. of

η(φ) = η0 exp

• φ

1 − φ

• 10

100 1000 1

φ(r, t)

We assume Vogel-Fulcher type divergence:

Modeling the dynamics of dilatant Fluid

3) State variable , in turn, depends on stress

φ(r, t)

2 3 4

S0 S0 S0 S0

S

φ∗(S) = φM (S/S0)2 1 + (S/S0)2

where,

S =

• 1

2Tr(ˆ ˙ σˆ ˙ σ)

Steady value

Model Equations

ρDvi Dt = ∂ ∂xj (−Pδij + σij)

Incompressible Navier-Stokes eq.

σij = η(φ) ∂vi ∂xj + ∂vj ∂xi

• Relaxation is driven by deformation (athermal)

1 τ = 1 r |˙ γ|

|˙ γ| : local shear rate r : dimansionless parameter

|ˆ ˙ γ| =

• 1

2Tr(ˆ ˙ γˆ ˙ γ)

0.01 0.1 1 10 100 1000

Viscosity (Pa·s)

Shear rate (s

• 1)

41wt% cornstarch suspension

Fall, Huang, Bertrand, Ovarlez, Bonn, PRL (2008).

η0 10Pa · s

S0 50Pa

Length and time scale

η(φ) = η0 exp

• φ

1 − φ

• φ∗(S) = φM

(S/S0)2 1 + (S/S0)2

Time Scale Length Scale

τ0 = η0 S0

⌅0 = ⇥ ⇤0

Parameters and scales

S0 ≈ 50Pa η0 ≈ 10Pa · sec. ρ ≈ 103kg/m3

0 ≈ 5cm

τ0 ≈ 0.2sec.

For 41wt% cornstarch suspension

Relaxed state viscosity: Thickening stress: Density:

Simple Shear Flow of Dilatant Fluid

Se Se

Boundary condition S(z, t)

• z=±h = Se

Simple Shear Flow of Dilatant Fluid

Steady State Solution of the Model Equation

low viscosity high viscosity

Shear flow in the unstable branch

φ parameter shear rate

Flow oscillates spontaneously under constant stress

φM = 1.0, h = 3.0, Se = 1.0, r = 0.1

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

1 2 3 4 5 6 0.5 1 1.5 2 flow width h shear stress Se

• scillate

steady

h* ~5cm for

41wt% suspension.

Experiment with starch-water mixture

Time Scale Length Scale

τ0 = η0 S0

⌅0 = ⇥ ⇤0

Parameters and scales

S0 ≈ 50Pa η0 ≈ 10Pa · sec. ρ ≈ 103kg/m3

0 ≈ 5cm

τ0 ≈ 0.2sec.

For 41wt% cornstarch suspension

Relaxed state viscosity: Thickening stress: Density:

Experimental Setup

encoder 22cm weight starch-water mixture spring & dumper 1 ~ 5cm

• 55wt% CsCl solution and potato-starch

mixture

• Volume fraction: 41~42.5%
• Flow thickness: 1~5cm
• weight: 0.5 ~ 10kg (0.1~2.3kPa)

Oscillation: 1000fps movie

8.0 8.5 9.0 10 20 30

time(s)

angular speed(rad/s)

Density=42.5wt%, h=4cm

9.0 8.8 8.6

0.2 0.4 0.6 2 4 6 8 10 12 14 Average Shear Rate Time A =1,

M=1, r =0.1, Se=1.1

h=1.3 2.0 3.0

Angular speed of the center rod

A State diagram of the flow

1 2 3 4 5 0.5 1 1.5 2 2.5 flow width h [cm] external stress Se [kPa] no oscillation intermit or noisy steady oscillation idle

1 2 3 4 5 6 0.5 1 1.5 2 flow width h shear stress Se

• scillate

steady

42.5wt% suspension

Se

∗ 0.1kPa

Threshold stress.

Oscillatory

Stress dependence of freq. and amplitude

5 10 15 20 0.0 0.5 1.0 amplitude[rad/sec.] external stress [kPa] density=42.5wt% h=2cm h=3cm h=4cm h=5cm 5 10 15 20 25 30 35 frequency[Hz] density=42.5wt%

*Frequency stays almost constant near threshold

Frequency vs flow thickness

• No systematic dependence either on the thickness and shear stress
• Frequencies are always around 20Hz (twice the predicted value)

5 10 15 20 25 30 35 1 2 3 4 5 6

Frequency (Hz)

Flow thickness h(cm)

(b)42.5wt% 0.2kPa 0.3kPa 0.6kPa 1.0kPa (b)42.5wt%

Experimental observation

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

2D Simulations

φ(r, t = 0) = ξi

Initial noise:

x

z

L = 10h

|ξi| = 10−4

Initial condition:

vi(r, t = 0) = 0

–– Maker and Cell (MAC) method

φM=0.85, Se=1.0

Inhomogeneous Oscillation

50 100 1.0 2.0 3.0 4.0 time

Small noise is given to initial φ

Jamming caused by instability

time 5 10 101 102 103 104 105 time maximum viscosity 10 5 10 0.01 0.1 1 10

φM = 1.0

Inhomogeneous Oscillation

φM=0.85, rin=1.0, rout=3.0, Se=2.0

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

Inhomogeneous Oscillation

h and Se independent Frequency

summary and remarks

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

and next...