12 SARAH D. OLSON AND ANITA T. LAYTON

that G is independent of θ in 2-d. Similarly, we also define B to be a radially

symmetric function satisfying ΔB = G . Applying the divergence operation to

Eq. (2.35a)- (2.35b), we can simplify Eq. (2.35a) to get a solution for the pressure.

Once the pressure is found, we can plug back into Eq. (2.35a) to solve the resulting

velocity. This results in the following solution,

μu(x) = Uo + (go · ∇)∇Bε(x − Xo) − goGε(x − Xo), (2.36a)

p(x) = go · ∇Gε(x − Xo) (2.36b)

where Uo = go/8π is a constant chosen to eliminate constant flow [29]. We refer to

Eq. (2.36a) as the regularized Stokeslet velocity. Note that these solutions are exact

for a regularized point force and are everywhere incompressible. Additionally, since

the singularity has been removed, these equations are valid for any point in the fluid

domain as well as all points on the structure. This makes the MRS a Lagrangian

method; one only needs a Lagrangian discretization of the structure and does not

need an underlying Cartesian grid for the fluid since the fluid velocity and pressure

can be evaluated at any time or point using the same equation. Note that one can

easily look at 3-d fluid-structure interactions by just changing the regularization

function ψ , and hence the kernel, G . The 3-d method of regularized Stokeslets

has been derived and analyzed previously [29,31].

We can extend the solution given in Eq. (2.36a)-(2.36b) for any number of point

forces that can be disjoint or connected. This is via a superposition of regularized

fundamental solutions. To outline the MRS in 2-d, let the structure X be immersed

in a viscous, incompressible fluid. This structure is discretized as Xk, n where k =

1,...,m are the immersed boundary points and the time steps are denoted by n.

The forces are given by a function to determine gk n. The algorithm is as follows:

(1) Evaluate gk

n

based on the configuration of the structure

Xkn

(2) Evaluate the regularized Stokeslet, determining the velocity at the point

Xi

n,

accounting for the contribution from each of the point forces gk

n

as

follows:

(2.37)

μun(Xi n

) = Uo+

m

k=1

gk

n

Bε(rk)

rk

− Gε(rk) + [gk

n

· (x − Xk)](x − Xk)

rkBε (rk) − Bε(rk)

rk3

where i = 1,...,m.

(3) Update the location of the structure using the no-slip condition given in

Eq. (2.4a) (similar to IB method in §2.1).

(4) Repeat, marching forward in time.

Again, in comparison to the IB method in §2.1, we can use forces defined from the

same constitutive laws as for the IB method. A few of these were outlined in §2.1.3.

The forces in the IB method and the MRS both are defined on the structure in

Lagrangian coordinates. In the IB method, forces are represented in the Eulerian

coordinate system by convolution against a regularized delta function. Similarly, in

the MRS, when forces are applied on a curve, we have a force that is a convolution

against a regularized blob function. The structure is discretized and by choosing

a numerical quadrature, we arrive at a sum of point forces that are a product of

force densities and quadrature weights.

The MRS is a Lagrangian method. Due to this fact, we avoid the spreading

and interpolation of forces in order to solve the fluid velocity on a Cartesian grid

and determine velocity on the Lagrangian structure. At each time step, the fluid