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

ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Polystyrene on Polyurethane High elastic High elastic ! Static Equilibrium ! Static Equilibrium modulus spherical modulus spherical particles on particles on ! ! Hydrodynamic Hydrodynamic Forces and Toque Forces and Toque elastomeric elastomeric substrates. substrates. ! van ! van der der Waals Waals/Pull /Pull- -off Force off Force ! ! Rolling and Sliding Removal Rolling and Sliding Removal ! Critical Shear Velocity for ! Critical Shear Velocity for Detachment Detachment ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 1

Polystyrene particles on a silicon wafer F l M t d F t O a F Po ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi π µ 3 f dC MOMENT DETACHMENT MOMENT DETACHMENT = d F V Drag Force Drag Force t C c d + − α + ≥ M F ( ) F a F a = + 0 . 687 t t 0 L P 0 C 1 0 . 15 Re 2 d dV SLIDING DETACHMENT SLIDING DETACHMENT ( ) Lift Force dy Lift Force = ρµ 1 2 F 1 . 61 d V 2 l 1 2 F ≥ dV kF t P 0 dy ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 2

Contact Radius πµ Contact Radius 2 2 f d V = Hydrodynamic Hydrodynamic m M ⎡ ⎤ t Torque π ⎛ π 2 Torque C ⎞ d 3 W d 3 W d ⎢ ⎥ = + + π + ⎜ ⎟ 3 A A c a P 3 W dP ⎢ A ⎝ ⎠ ⎥ 2 K 2 2 ⎣ ⎦ ( ) ( ) − ⎡ ⎤ 1 − ν − ν 2 2 4 1 1 = + K ⎢ 1 2 ⎥ + = ⎣ ⎦ 3 E E + 1 2 Near Wall Near Wall u M 1 . 72 y 3 Peak Velocity Peak Velocity = π Poll- -Off Force Off Force JKR Poll F W d P o A 4 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 3 = π JKR F W d Pull- -Off Force Off Force Pull po A 4 1 ⎛ π ⎞ 2 3 Contact Radius Contact Radius 3 W d = ⎜ ⎟ A a ⎜ ⎟ ⎝ ⎠ 8 K at Separation at Separation ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 3

+ + = u y + + + = − β 2 ≤ v y , y 1 . 85 o + + + = β w 2 y z o β o = 0 . 01085 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Λ + =100 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 4

1 / 3 ⎛ ⎞ π 3 / 2 2 1 / 2 C W = ⎜ ⎟ * A Rolling Rolling u 0 . 46 ⎜ ⎟ ρ c 3 / 2 2 1 / 2 ⎝ ⎠ d K 1 / 2 ⎛ ⎞ CkW = ⎜ ⎟ * Sliding Sliding A u 0 . 5 ⎜ ⎟ ρ c ⎝ ⎠ d No lift, α α =0 =0 No lift, ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 5

ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi � 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 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 6