10 SARAH D. OLSON AND ANITA T. LAYTON

boundary forces, expressed per unit current arclength α, then it has been shown

that for the Stokes equations [100]

[p] = fn, [pn] =

∂

∂α

fτ , (2.27)

[u] = [v] = 0, (2.28)

[μun] = fτ sin θ, [μvn] = −fτ cos θ (2.29)

where θ denotes the angle between the tangent line and the x-axis. Model con-

figuration for a immersed interface problem is illustrated in Fig. 2. Note that the

immersed boundary is a closed curve. This is no accident. The derivative of the

jump conditions (2.27)–(2.29) requires a closed curve (or closed surface in 3-d).

This is indeed a limitation of the immersed interface method that the IB method

does not share.

We will describe the procedures by which the steady-state Stokes equations can

be solved using the immersed interface method. The immersed interface method

can also be applied to the Navier-Stokes equations [103, 169], but as might be

expected, the procedures are more complicated. For a fluid with uniform viscosity,

the Stokes equations are (2.1b) and (2.2); that system can be solved simultaneously

as a coupled system. Alternatively, the system can be reduced to a sequence of

three Poisson problems, as described below. Applying the divergence operator to

Eq. (2.1b) yields

(2.30) Δp = ∇ · f

which we will solve by setting the right-hand-side to zero and by incorporating the

jump conditions (2.27). Consider the 2-d problem following a procedure similar to

§2.2.1, we discretize Eq. (2.30) to obtain the finite-difference equations

(2.31)

1

h2

(pi+1,j + pi−1,j − 4pi,j + pi,j−1 + pi,j+1) = Ci,j

where the correction terms Ci,j are zero except at irregular points.

Next we solve the Poisson equations (2.1b) for u and v. The finite difference

equation for u takes the form

(2.32)

1

h2

(ui+1,j + ui−1,j − 4ui,j + ui,j−1 + ui,j+1) =

1

2hμ

(

pxi+1,j − pxi−1,j

)

+

ˆ

C

i,j

where the correction term

ˆ

C

i,j

corrects for the approximation of px, and accounts

for the jump discontinuities in the derivative of u. The procedure for v is analogous.

To summarize, the steps in which the immersed interface method can be used

to simulate the interactions between a Stokes fluid and an immersed boundary are

as follows. At time tn, the boundary position Xn is known.

(1) From the boundary configuration

Xn,

compute boundary forces f

n.

(2) From the boundary forces f

n,

compute jump conditions (2.27)–(2.29).

(3) Form the correction terms, which are functions of the jump conditions

above, and solve Eqs. (2.31) and (2.32), plus analogous equation for v.

(4) Advance the boundary (see §2.1).

2.3. Regularized Stokeslet method. Many fluid-structure interaction prob-

lems involve small length scales and/or large viscosity, where the Reynolds number

is approximately zero. In these applications, one could use the IB method detailed