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Cell Adhesion and Detachment in Shear Flow 213
Fig. 2. An example of two-dimensional flow region with one rigid body.
W g0,p = {v | v ∈ (H 1 (Ω)) 2 , v = g 0 (t) on the top and bottom of Ω and
v is periodic in the x 1 -direction},
W 0,p = {v | v ∈ (H 1 (Ω)) 2 , v = 0 on the top and bottom of Ω and
v is periodic in the x 1 -direction},
{
∫ }
L 2 0 = q | q ∈ L 2 (Ω), qdx =0 ,
Ω
Λ 0 (t) ={µ | µ ∈ (H 1 (B(t))) 2 , 〈µ, e i 〉 B(t) =0, i =1, 2, 〈µ, −→ Gx ⊥ 〉 B(t) =0}
with e 1 = {1, 0} t , e 2 = {0, 1} t , −→ Gx ⊥ = {−(x 2 − G 2 ),x 1 − G 1 } t and 〈·, ·〉 B(t)
an inner product on Λ 0 (t) which can be the standard inner product on
(H 1 (B(t))) 2 (see [GPH + 01, Section 5] for further information on the choice of
〈·, ·〉 B(t) ). Then the fictitious domain formulation with distributed Lagrange
multipliers for flow around a freely moving neutrally buoyant particle (see
[GPHJ99, GPH + 01] for detailed discussion of non-neutrally buoyant cases) is
as follows:
For a.e. t>0, find u(t) ∈ W g0,p, p(t) ∈ L 2 0, V G (t) ∈ R 2 , G(t) ∈ R 2 ,
ω(t) ∈ R, λ(t) ∈ Λ 0 (t) such that
∫ [ ]
∫
∫
∂u
ρ f
Ω ∂t +(u · ∇)u · v dx +2µ f D(u) :D(v) dx − p∇ · v dx
Ω
Ω
∫
∫
−〈λ, v〉 B(t) = ρ f g · v dx + F · v dx, ∀v ∈ W 0,p , (1)
Ω
Ω
∫
q∇ · u(t)dx =0, ∀q ∈ L 2 (Ω), (2)
Ω
〈µ, u(t)〉 B(t) =0, ∀µ ∈ Λ 0 (t), (3)
dG
dt = V G, (4)
V G (0) = V 0 G, ω(0) = ω 0 , G(0) = G 0 = {G 0 1,G 0 2} t , (5)
{
u 0 (x),
∀x ∈ Ω \ B(0),
u(x, 0) = u 0 (x) =
V 0 G + ω0 {−(x 2 − G 0 2),x 1 − G 0 1} t (6)
, ∀x ∈ B(0),