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
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