SIMULATING IMMERSED BIOLOGICAL STRUCTURES 11

in §2.1, solving the incompresible Stokes equations in Eq. (2.1b)-(2.2) or solving

the incompressible Navier-Stokes equations with Re ≈

10−4

−

10−6

in Eq. (2.1a)

and (2.2). In the case of zero Reynolds number applications, other methods can be

used since the Stokes equations are linear, have no memory, and the only time de-

pendence will be from a forcing term due to the immersed structure. Fundamental

solutions exist for the Stokes equations; this allows for the use of Lagrangian meth-

ods such as boundary integral methods and the method of regularized Stokeslets

as an alternative to the IB method.

In a 2-d infinite fluid domain, the Stokeslet (or Green’s function) is the fluid

flow that results from a point force of strength go applied at the point Xo and is

given by

us(x) =

−go

4πμ

ln(r) +

[go · (x − Xo)](x − Xo)

4πμr2

(2-d) (2.33)

where r = ||x − Xo||. The velocity behaves like ln(r) in the 2-d Stokeslet. We will

primarily focus this discussion on 2-d fluids. However, there is also a fundamental

solution for a point force in a 3-d fluid that behaves like 1/r.

The structures can be closed or open curves, as well as sets of disconnected

points. The forces are then applied on the given structure. Due to the linearity

of the Stokes equation, we can write the resulting velocity as a superposition of

Stokeslets when there are several point forces. The method of regularized Stokeslets,

developed by Cortez et al. [29, 31], regularizes the singularity at x = Xo in the

denominator. The approach is similar to that of the IB method, in that the singular

force will be spread to the surrounding fluid.

In the IB method, a compactly supported smooth approximation to the δ func-

tion was used to spread forces to the Cartesian fluid grid via Eq. (2.3). Here, we

will use a blob or cutoff function ψ that is a radially symmetric approximation

to the δ function. The cutoff function can have compact or infinite support and

has that the property that the integral of ψ on the infinite fluid domain is equal

to 1. Note that since this a Lagrangian method, an infinite support blob does not

decrease computational eﬃciency as it would in the standard IB method introduced

in §2.1. If using a blob function with infinite support, the majority of the force will

be concentrated within a region around the point force and will then decay quickly.

An example blob function is,

ψ =

2

4

π(r2 + 2)3

(2-d). (2.34)

The form of the blob functions are derived in order to solve Stokes equations for

a regularized force as well as ensuring that when taking lim → 0, we recover a δ

function.

In the method of regularized Stokeslets (MRS), we wish to solve the incom-

pressible Stokes equations where the force density f is given as a regularized point

force go at Xo,

μΔu(x) = ∇p(x) − goψ (x − Xo) , (2.35a)

∇ · u(x) = 0 . (2.35b)

The exact solution is no longer the Stokeslet as given in Eq. (2.33). For a given

choice of regularization function ψ , we now need to derive a regularized Green’s

function G that satisfies ΔG = ψ . Since ψ is radially symmetric, we assume