SIMULATING IMMERSED BIOLOGICAL STRUCTURES 3

The incompressible Stokes equations, given in Eqs. (2.1b) and (2.2), correspond to

the case of zero Reynolds number, where viscous forces dominate and inertial forces

can be neglected.

The structure is parameterized by s and we denote the position of the elastic

structure at time t by X(s, t). In this simplified representation, one can model

elastic structures corresponding to open or closed curves. We assume that the

structure is neutrally buoyant and massless. The Lagrangian domain of the im-

mersed structure is B. The elastic structure exerts a Lagrangian force density on

the surrounding fluid and is given by F(X,t). In order to determine the Eulerian

force density f(x,t), we need to spread the Lagrangian force density F(X,t) to the

Cartesian grid via the following interaction equation,

(2.3) f(x,t) =

B

F(X,t)δc(x − X)ds,

where δc is a compactly supported smooth approximation to a δ distribution. This

distributes the singular force layer F(X,t) to the surrounding fluid such that f(x,t)

is mainly zero except in a small region around the structure. Examples of compactly

supported delta functions and important properties are defined in §2.1.2 and a

description of the derivation of a Lagrangian force density F(X,t) is given in §2.1.3.

The motion of the fluid is coupled to the motion of the elastic structure, thus

we must also have a prescribed condition for the movement of the structure. Since

the structure is immersed in a viscous fluid, the velocity across the structure will

be continuous. Therefore, we can enforce a no-slip condition,

∂X

∂t

= U(X,t) (2.4a)

=

Ω

u(x,t)δc(x − X)dx, (2.4b)

where the elastic structure X(s, t) will move with the local fluid velocity at that

point, U(X,t). Since the fluid velocity is solved for on the Cartesian grid, we must

use Eq. (2.4b) to interpolate the velocity u(x,t) to get the velocity at the immersed

boundary points, U(X,t). Assuming we have the necessary boundary conditions for

the fluid flow and/or pressure on Ω, we can solve the incompressible fluid equations,

either Eq. (2.1a) or (2.1b) with Eq. (2.2), for a given immersed structure’s force

density F in Eq. (2.3) using a variety of methods including projection methods and

the use of FFTs on periodic domains [15, 62, 92, 118]. Sample results of the IB

method are given in Fig. 1, where the forces on the immersed boundary points of

the structure are proportional to curvature. The Lagrangian and Eulerian force

density are shown in Fig. 1(a) and (b), respectively.

2.1.1. Summary of Numerical Method. To simplify, let the fluid domain Ω be

a subset of

R2

that is discretized into a uniform Cartesian grid with mesh width

h such that xi = xi−1 + h, yj = yj−1 + h, and xij = (xi,yj ) for i = 1,...,q and

j = 1,...,r. The structure X = (Xk,Yk) will be discretized at time t = 0 to

have uniform spacing for k = 1,...,m immersed boundary points. At time step n,

assume we have a given discretized configuration of the structure Xk

n.

An outline

of the numerical algorithm is as follows:

(1) Evaluate the problem dependent elastic Lagrangian force density Fk

n

for

the structure at each of the immersed boundary points

Xkn