! Static Equilibrium ! Static Equilibrium modulus spherical - - PowerPoint PPT Presentation

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! ! Static Equilibrium Static Equilibrium ! ! Hydrodynamic Hydrodynamic Forces and Toque Forces and Toque ! ! van van der der Waals Waals/Pull /Pull-

• off Force
• ff Force

! ! Rolling and Sliding Removal Rolling and Sliding Removal ! ! Critical Shear Velocity for Critical Shear Velocity for Detachment Detachment

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High elastic High elastic modulus spherical modulus spherical particles on particles on elastomeric elastomeric substrates. substrates.

Polystyrene on Polyurethane

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Polystyrene particles on a silicon wafer

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t

F

Po

F

l

F

t

M

O

a

d

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MOMENT DETACHMENT MOMENT DETACHMENT

a F a F ) 2 d ( F M

P L t t

≥ + α − +

SLIDING DETACHMENT SLIDING DETACHMENT

P t

kF F ≥

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V C dC f 3 F

c d t

µ π =

Drag Force Drag Force

687 . d

Re 15 . 1 C + =

( )

2 1 2 1 2 l

dy dV dy dV V d 61 . 1 F ρµ =

Lift Force Lift Force

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Hydrodynamic Hydrodynamic Torque Torque

c 2 m t

C V d f 2 M πµ = + + =

y 72 . 1 uM

Near Wall Near Wall Peak Velocity Peak Velocity

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⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + π + π + =

2 A A A 3

2 d W 3 dP W 3 2 d W 3 P K 2 d a ( ) ( )

1 2 2 2 1 2 1

E 1 E 1 3 4 K

−

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ν − + ν − =

d W 4 3 F

A JKR Po

π =

Contact Radius Contact Radius Poll Poll-

• Off Force

Off Force

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Pull Pull-

• Off Force

Off Force Contact Radius Contact Radius at Separation at Separation

d W 4 3 F

A JKR po

π =

3 1 2 A

K 8 d W 3 a ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ π =

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+ + + + + + + +

β = ≤ β − = = z y 2 w 85 . 1 y , y v y u

• 2

01085 .

• =

β

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Λ+=100

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3 / 1 2 / 1 2 2 / 3 2 / 1 2 A 2 / 3 * c

K d W C 46 . u ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ρ π =

Rolling Rolling

2 / 1 A * c

d CkW 5 . u ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ρ =

Sliding Sliding No lift, No lift, α α=0 =0

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• Rolling detachment is the dominant mechanism

Rolling detachment is the dominant mechanism for particle removal in turbulent flows. for particle removal in turbulent flows.

• Drag and hydrodynamic torque are dominant

Drag and hydrodynamic torque are dominant forces for particle detachment from the wall. forces for particle detachment from the wall.

• Turbulence near wall flow structure plays an

Turbulence near wall flow structure plays an important role in particle detachment process. important role in particle detachment process.

• There are discrepancies between theory and

There are discrepancies between theory and experimental data experimental data

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ME 437/537