1

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!! van van derder WaalsWaals/Pull/Pull--off Force for off Force for Rough SurfacesRough Surfaces

!! Rolling and Sliding Removal of Rolling and Sliding Removal of Rough ParticlesRough Particles

!! Hydrodynamic Forces and TorqueHydrodynamic Forces and Torque

!! Critical Shear Velocity for Critical Shear Velocity for DetachmentDetachment

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PullPull--Off Force Off Force

Contact RadiusContact Radius

at Separationat Separation

dW43F A

JKRpo π=

31

2A

K8dW3a ⎟⎟

⎠

⎞⎜⎜⎝

⎛ π=

31

Po

K2dFa ⎟⎠⎞

⎜⎝⎛=( ) ( ) 1

2

22

1

21

E1

E1

34K

−

⎥⎦

⎤⎢⎣

⎡ ν−+

ν−=

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PullPull--Off Off Force Force

Contact Contact RadiusRadius

c/6.0Po

2M eNfaF ∆−π=

31

M

K2dFa ⎟⎠⎞

⎜⎝⎛=

σδ

=∆ cc

Number of Number of Asperities per Asperities per

Unit AreaUnit Area

PullPull--Off Force Off Force for each for each AsperityAsperity

Max Asperity Max Asperity ExtensionExtension

Roughness Height Roughness Height Standard deviationStandard deviation

Estimated

2

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3/1

2

2Po

c K3f

⎥⎦

⎤⎢⎣

⎡β

=δ

0.1N ≈σβ

JKR JKR ModelModel

GreenwoodGreenwood--WilliamsonWilliamson

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σ= 5.9kr r0.53ke =Average Roughness Height Average Roughness Height Displaced Origin of Velocity Profile Displaced Origin of Velocity Profile

Equilibrium Separation Distance Equilibrium Separation Distance

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VC

df3Fc

tµπ

=Drag ForceDrag Force

( ) 2/12/12

l dy/dVdy/dVVd61.1F ρµ=Lift ForceLift Force

Hydrodynamic Hydrodynamic Torque Torque

c

2m

t CVdf2M πµ

=

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+++

+++

++

β=

≤β−=

=

zy2w

85.1y,yv

yu

o

o

2

01085.0o =β

3

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+++++

+ =α−+σ+= LH76.22du 0

)H76.22d(

2w 0

+++++

+ α−+σ+Λ

β=

α−+σ+= 0H76.22dL

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CLdu8.5F

2*

tπρ

=

νρ

=Lud95.1F

3*2

L

CLdu14.2M

22*

tπρ

=

Drag ForceDrag Force

Lift ForceLift Force

Hydrodynamic Hydrodynamic Torque Torque

AhmadiME 437/537

tF

MF

lFtM

O

a

d

Lmg

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tF

aF

lFtM

a

2/d

mg

4

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MOMENT DETACHMENTMOMENT DETACHMENT

aFa)mgF()2d(FM ML0tt ≥−+α−+

SLIDING DETACHMENTSLIDING DETACHMENT

)FmgF(kF LMt −+≥

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RollingRolling

SlidingSliding

2/1

*c2

2cpo

2*c

)aCu95.104.5(Ld

]mg])/(6.0exp[Nfa[aCu

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ν+πρ

+∆−π=

.)kCdu95.18.5(Ld

]mg])D/(6.0exp[Nfa[kCu

2/1

*c

2cpo

2*c

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ν+πρ

+−π=

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RollingRolling

SlidingSliding

( )2/1

2/1*c

*c

*c2

2cpo

2*c

)Lu2.072.1(aCu95.104.5)Lu1.072.1(Ld

mg])/(6.0exp[NfaaCu

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ν

+ν

+πν

+ρ

+∆−π=

2/1

2/1*c

*c

*c

2cpo

2*c

)Lu2.072.1(kCdu95.18.5)Lu1.072.1(dL

]mg])/(6.0exp[Nfa[kCu

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ν

+ν

+πν

+ρ

+∆−π=

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Rolling detachment is the dominant mechanism Rolling detachment is the dominant mechanism for detachment of rough spherical particles.for detachment of rough spherical particles.

Roughness significantly reduces the adhesion Roughness significantly reduces the adhesion pullpull--off force.off force.

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.

Accounting for surface roughness improves the Accounting for surface roughness improves the agreement between the model prediction and agreement between the model prediction and experimental dataexperimental data

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