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) =
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,
= U(X,t) (2.4a)
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
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
An outline
of the numerical algorithm is as follows:
(1) Evaluate the problem dependent elastic Lagrangian force density Fk
the structure at each of the immersed boundary points
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