tsuji lecture 9

7/27/2019 Tsuji Lecture 9

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particle-wall colllision



Particle - wall interaction

Particle - wall interaction

lift force,

lubrication

collision (massive particle)

van der Waals force (small particle)

Mechanical interaction

Hydrodynamic interaction



Particle - wall collision

Particle - wall collision

( )ω 2

( ) ( ) ( ) ( )( )r

V V V V X Y Z

2 2 2 2= , ,

x

y

z

1) Particle deformation is neglected so, throughout the collision process, the

distance between the particle center of mass and the contact point is constant and

equal to the particle radius.

2) Coulomb’s friction law applies to particles sliding along the wall.

3) Once a particle stops sliding, there is no further sliding.

Assumptions

( )ω 0

( ) ( ) ( ) ( )( )r

V V V V X Y Z

0 0 0 0= , ,



( )2Ω

( ) ( ) ( ) ( )2222 , , Z Y X vvvv =r

x

y

z

( )0Ω

( ) ( ) ( ) ( )0000 , , Z Y X vvvv =r



Case I : the particle stops sliding in the compression period

Case II : the particle stops sliding in the recovery period

Case III : the particle continues to slide throughout the

compression and recovery period



sliding period

Post-collision

Pre-collision

Compression period (1) Recovery period (2)

Impulse( )

r

J s ( )

r

J r ( )

r

J 2

Trans.

velocity ( )r

V 0 ( )r

V s ( )r

V 1 ( )r

V 2

Angular.

velocity( )r

ω 0 ( )r

ω s ( )r

ω 1 ( )r

ω 2

( ) ( ) ( )r r r

J J J s r 1 = +


m V V J s s( )( ) ( ) ( )

r r r

− =0 m V V J s r ( )( ) ( ) ( )

r r r

1 − = m V V J ( )( ) ( ) ( )r r r

2 1 2− =

I r J s s

( )( ) ( ) ( )r r r

r

ω ω − = − ×0 I r J s r ( )( ) ( ) ( )r r r

r

ω ω 1 − = − × I r J ( )( ) ( )) ( )r r r

r

ω ω 2 1 2− = − ×

Impulsive equations :



Case II : the particle stops sliding in the recovery period

Impulse ( )r

J 1 ( )

r

J s ( )

r

J r

( ) ( ) ( )r r r

J J J s r 2 = +

sliding period

Post-

collision

Pre-

collisionCompression period (1) Recovery period (2)

Angular.

velocity( )r

ω 0 ( )r

ω 1 ( )r

ω s ( )r

ω 2

Trans.

velocity( )

r

V 0 ( )

r

V 1 ( )

r

V s ( )

r

V 2

m V V J ( )( ) ( ) ( )r r r

1 0 1− = m V V J s r ( )( ) ( ) ( )r r r

2 − =

I r J ( )( ) ( ) ( )r r r

r

ω ω 1 0 1− = − ×

m V V J s s( )( ) ( ) ( )

r r r

− =1

I r J s s( )( ) ( ) ( )r r r

r

ω ω − = − ×1 I r J s r ( )( ) ( )) ( )r r r

r

ω ω 2 − = − ×




Post-

collision

Pre-

collisionCompression period (1) Recovery period (2)

Impulse ( )r

J 1 ( )

r

J 2

Trans.

velocity( )

r

V 0 ( )

r

V 1 ( )

r

V 2

Angular.

velocity

( )r

ω 0 ( )r

ω 1 ( )r

ω 2


m V V J ( )( ) ( ) ( )r r r

1 0 1− = m V V J ( )( ) ( ) ( )r r r

2 1 2− =

I r J ( )( ) ( ) ( )r r r

r

ω ω 1 0 1− = − × I r J ( )( ) ( ) ( )r r r

r

ω ω 2 1 2− = − ×

Case III : the particle continues to slide throughout the

compression and recovery period



particle mass m a p= ( / )3 4 3π ρ

( )moment of inertia about the axis of the diameter I ma= 2 5 2/

surface velocityr r

r rr r

r

U V r V a i V j V a k X Z Y Z X = + × = + + + −ω ω ω ( ) ( )

( ) ( )

( ) ( ) ( ) ( )V a i V a k

X

s

Z

s

Z

s

X

s+ + − =

ω ω

rr

r

0

( ) ( )( ) ( ) ( ) ( ) ( )

V a i V j V a k X Z Y Z X

1 1 1 1 10+ + + − =ω ω

r rr

( ) ( )( ) ( ) ( ) ( )

V a i V a k X Z Z X

2 2 2 20+ + − =ω ω

rr

coefficient of restitution eJ

J

Y

Y

=( )

( )

2

1


at the end of the sliding period

at the end of the compression period

at the end of the recovery period

J i J k f J i f J k X

s

Z

s

X Y

s

Z Y

s( ) ( ) ( ) ( )r

rr

r

+ = − ⋅ − ⋅ε ε

ε ε X Z

2 21+ =

Coulomb’s law applied to the

sliding period

friction coefficient

：factors indicating the proportion of the velocity in each component direction, that

is, the direction cosines of the approaching velocity in the x- and z plane.

Z X ε ε ,



V V f eY

( )

( )

0

27 1

r < − + −+

< <27 1

00

f eV

V

Y

( )

( )

rCondition

V V a

X X Z

( ) ( ) ( )( )

2 0 05

7

2

5=

⎛ ⎝ ⎜

⎞ ⎠⎟ − ω

V eV Y Y ( ) ( )2 0

= −

V V

a Z Z X

( ) ( ) ( )

( )

2 0 05

7

2

5=

⎛

⎝ ⎜

⎞

⎠⎟ + ω

V V f e V X X X Y

( ) ( ) ( )( )

2 0 01= + +ε

V V f e V Z Z Z Y

( ) ( ) ( )

( )

2 0 0

1= + +ε

V eV Y Y ( ) ( )2 0

= −

Trans.

velocity

ω X Z V

a

( )2 =

ω ω Y Y

( ) ( )2 0=

ω Z X V

a

( )2 = −

ω ω ε X X Z Y a

f e V ( ) ( ) ( )

( )2 0 05

21= − +

ω ω ε Z Z X Y

a

f e V ( ) ( ) ( )

( )2 0 05

2

1= + +

ω ω Y Y

( ) ( )2 0=

Angular

velocity

Relation between pre-and post-collisional velocitiesRelation between pre-and post-collisional velocities



Irregular bouncingIrregular bouncing

Regular bouncing e < 1

Irregular bouncing

Roughness of wall

Nonspherical particle



Particle collision with a rough wallParticle collision with a rough wall

Nonspherical particle Nonspherical particle

O

C

ω

V r O

C

V

ω

θ

r

C

O

V

ω

θ

r



O

C

ω

V r

O

C

V

ω

θ

r

C

O

V

ω

θ

r



O

r 1

r 12

r 11

r 10

r 8r 7 r 6

r 5

r 3r 2

r 4

r 9

θ 1 θ 2θ 3

θ 5

θ 6

θ 7θ 8

θ 9

θ 10

θ 11

θ 12

θ 4

1

5

4

3

2

9

8 7

6

12

11

10



O

C H

O

C H

O

C H

O

C H

O

C H

O

C H



45 0.2m

(a)(b)

tsuji lecture 9

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