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