2 SARAH D. OLSON AND ANITA T. LAYTON

related methods including the immersed interface method, a sharp interface method

with potentially higher order accuracy, and the method of regularized Stokeslets for

zero Reynolds number applications. We refer the interested reader to reviews on

IB methods for solids [121], engineering applications of fluid-structure interaction

[44,122], and numerical methods for fluid-structure interaction [20,71].

In this review, we will focus on the IB method, its applications, and additional

extensions. A general overview of the IB method and its implementation will be

detailed in §2.1, and applications will be discussed in §4. The IB method has

also motivated the development of the immersed interface method and the method

of regularized Stokeslets, which will be detailed in §2.2 and §2.3 . A few recent

advances for modeling biological structures, e.g. porous boundaries and multi-fluid

domains, will also be highlighted in §3. We will end this review with a discussion

of advantages of particular methods and future avenues of research.

2. Numerical formulations

2.1. Immersed boundary (IB) method. This is a non-conforming method

as two different grids will be used for the structure and the fluid. The fluid motion

will be described by a set of Eulerian variables defined on a Cartesian grid that

does not conform to the geometry of the elastic structure. The motion of the elas-

tic structure will be described using Lagrangian variables defined on a curvilinear

mesh. The IB method employs these two different grids and sets of variables that

communicate with each other via the forcing term of the structure. This allows for

a straightforward implementation of complicated fluid-structure interactions since

the underlying Cartesian grid for the fluid domain is not required to coincide with

the Lagrangian structure. With an evolving structure, the computational com-

plexity is greatly reduced when a stationary, non-deforming Cartesian grid is used

versus remeshing at each time step to have the structure conform to the fluid grid.

Let Ω be the fluid domain, which can be a subset of

R2,

a subset of

R3,

or an

infinite fluid domain (all of

R2

or

R3).

For this discussion, we will restrict Ω as

a subset of

R2.

In Ω, points within the fluid that lie on the Cartesian grid of the

fluid domain will be represented as x, where x = (x1,x2) in 2-d. The velocity field

u(x,t) and pressure p(x,t) are Eulerian variables that are defined at each point on

the Cartesian grid, corresponding to the fluid domain Ω. In the classical IB method,

we assume the Newtonian fluid flow is governed by either the Navier-Stokes (NS)

or Stokes (St) equation,

ρ

∂u(x,t)

∂t

+ u(x,t) · ∇u(x,t) = −∇p(x,t) +

μ∇2u(x,t)

+ f(x,t), (NS)

(2.1a)

0 = −∇p(x,t) +

μ∇2u(x,t)

+ f(x,t), (St) (2.1b)

where f(x,t) is the Eulerian force density on the Cartesian grid and μ and ρ are

the constant fluid viscosity and density, respectively. Both fluids are assumed to

be incompressible and therefore satisfy

(2.2) ∇ · u(x,t) = 0 .

When Eq. (2.1a) is nondimensionalized, the Reynolds number Re = ρV L/μ is a

nondimensional ratio corresponding to the relative contributions of inertial forces

to viscous forces where V is a characteristic velocity and L is a characteristic length.