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Cell Adhesion and Detachment in Shear Flow 217
⎧ ∫
u ⎪⎨
n+1/6 − u n ∫
ρ f · v dx − p n+1/6 ∇ · v dx =0, ∀v ∈ W 0,h ,
∫ Ω △t
Ω
⎪⎩ q∇ · u n+1/6 dx =0, ∀q ∈ L 2 h; u n+1/6 ∈ W n+1
g , 0,h pn+1/6 ∈ L 2 0,h.
Ω
(21)
Step 3. Compute u n+2/6 via the solution of
⎧∫
∂u
⎪⎨
Ω ∂t · v dx + (u
∫Ω
n+1/6 · ∇)u · v dx =0,
∀v ∈ W 0,h , on (t
⎪⎩
n ,t n+1 ),
(22)
u(t n )=u n+1/6 ; u(t) ∈ W n+1
g , 0,h
u n+2/6 = u(t n+1 ). (23)
Step 4. Compute u n+3/6 via the solution of
⎧ ∫
⎨ u n+3/6 − u n+2/6
∫
ρ f · v dx + αµ f ∇u n+3/6 · ∇v dx =0,
Ω △t
Ω
⎩
∀v ∈ W 0,h ; u n+3/6 ∈ W n+1
g . 0,h
(24)
Step 5. Predict the position and the translation velocity of the center of mass
of the particles as follows: Take V n+ 4 6 ,0
G
= V n G and Gn+ 4 6 ,0 = G n .
Then predict the new position of the particle via the following subcycling
and predicting-correcting technique:
For k =1,...,N,
Call Adhesive Dynamics Algorithm,
̂V n+ 4 6 ,k
G
= V n+ 4 6 ,k−1
G
+ F r (G n+ 4 6 ,k−1 )△t/2N, (25)
Ĝ n+ 4 6 ,k = G n+ 4 6 ,k−1 +(̂V n+ 4 6 ,k
G
V n+ 4 6 ,k
G
+ V n+ 4 6 ,k−1
G
)△t/4N, (26)
= V n+ 4 6 ,k−1
G
+(F r 4 (Ĝn+ 6 ,k )
+ F r (G n+ 4 6 ,k−1 ))△t/4N, (27)
G n+ 4 6 ,k = G n+ 4 6 ,k−1 +(V n+ 4 6 ,k
G
+ V n+ 4 6 ,k−1
G
)△t/4N, (28)
enddo;
and let V n+ 4 6
G = 4 Vn+ 6 ,N
G
, G n+ 4 6 = G n+ 4 6 ,N .
Step 6. Now, compute u n+5/6 , λ n+5/6 , V n+5/6 ,andω n+5/6 via the solution
G
of
⎧
⎪ ⎨
⎪ ⎩
ρ f
∫
Ω
u n+5/6 − u n+3/6
∫
· v dx + βµ f
△t
, ∀v ∈ W 0,h ,
=0,
= 〈λ, v〉 B
n+4/6
h
〈µ, u n+5/6 〉 B
n+4/6
h
Ω
∇u n+5/6 · ∇v dx
∀µ ∈ Λ n+4/6
0,h
; u n+5/6 ∈ W n+1
g , 0,h λn+5/6 ∈ Λ n+4/6
0,h
,
(29)