Partial Differential Equations - Modelling and ... - ResearchGate

214 J. Hao et al.
where u and p denote velocity and pressure, respectively, the boundary conditions
for the velocity field g 0 (t) is0 at the bottom of Ω and (c, 0) t at the
top of Ω with a fixed speed c for shear flow, λ is a Lagrange multiplier,
D(v) =[∇v +(∇v) t ]/2, g is gravity, F is the pressure gradient pointing in
the x 1 -direction, V G is the translation velocity of the particle B, andω is the
angular velocity of B. We suppose that the no-slip condition holds on ∂B. We
also use, if necessary, the notation φ(t) for the function x → φ(x,t).
Remark 1. The hydrodynamical forces and torque imposed on the rigid body
by the fluid are built in (1)–(6) implicitly (see [GPHJ99, GPH + 01] for details),
thus we do not need to compute them explicitly in the simulation. Since in
(1)–(6) the flow field is defined on the entire domain Ω, it can be computed
with a simple structured grid.
The forces obtained from those Hookean springs in the model for cell
adhesion has been splitted from the above equations and will be used when
predicting and correcting the motion and positions of cells with the short
repulsion force as discussed in the next section. ⊓⊔
Remark 2. In (3), the rigid body motion in the region occupied by the particle
is enforced via Lagrange multipliers λ. To recover the translation velocity
V G (t) and the angular velocity ω(t), we solve the following equations:
{
〈ei , u(t) − V G (t) − ω(t) −→ Gx ⊥ 〉 B(t) =0, for i =1, 2,
〈 −→ Gx ⊥ , u(t) − V G (t) − ω(t) −→
(7)
Gx ⊥ 〉 B(t) =0.
⊓⊔
Remark 3. In (1), 2 ∫ Ω D(u) :D(v) dx can be replaced by ∫ ∇u : ∇v dx
Ω
since u is divergence free and in W 0,p . Also the gravity g in (1) can be absorbed
into the pressure term. ⊓⊔
3.2 Space Approximation and Time Discretization
Concerning the space approximation of the problem (1)–(6) by a finite element
method,wehavechosenP 1 -iso-P 2 and P 1 finite elements for the velocity field
and pressure, respectively (like in [BGP87]). More precisely, with h, aspace
discretization step, we introduce a finite element triangulation T h of Ω and
then T 2h a triangulation twice coarser. (In practice, we should construct T 2h
first and then T h by joining the midpoints of the edges of T 2h , dividing thus
each triangle of T 2h into four similar subtriangles as shown in Figure 3.)
We approximate then W g0,p, W 0,p , L 2 and L 2 0 by the following finite dimensional
spaces, respectively: