The Immersed Boundary (IB) method is a
computational technique that was introduced by Peskin in the early
1970s as a method for studying a particular biofluid dynamics problem,
namely the coupled motion of the blood filling the ventricles of the
heart, the muscles of the heart wall, and the leaflets of the cardiac
valves. Since that time, the method has been extended and applied to
a wide variety of biofluid dynamics problems. Among others, these
include platelet aggregation during blood clotting, swimming of
aquatic flagellates, motion of the basilar membrane in the inner ear,
the formation of bacterial biofilms, fluid motion in the afferent
arterioles of the kidney, and pumping by ciliated arrays of tentacles
on marine organisms. What these problems have in common with the
heart problem is that they involve the motion of a viscous
incompressible fluid, the motion of one or more deformable elastic
objects immersed in the fluid, and/or the motion of one or more
deformable elastic surfaces that bound the fluid. Because the objects
or surfaces are deformable and elastic their motion is coupled to the
fluid motion, i.e., the motion of each affects the other.
These, in fact, are features of essentially all biofluid dynamics
problems. This situation is different from the traditional
engineering fluid dynamics problem which involves fluid motion through
or around a rigid object of specified geometry. The immersed boundary
method has proved itself to be well suited for studying biofluid
dynamics problems on scales ranging from subcellular to organ-size to
organism-size. It has been used to look at hydrodynamic interactions
among multiple organisms moving in the same fluid. The IB method
deals primarily with time-dependent problems, and it takes into
account both fluid viscosity and fluid inertia.
In designing the original IB method, Peskin realized the advantages of describing the fluid variables and the immersed objects in different ways. The fluid variables (velocity, pressure, force density) are described in what is known as an Eulerian manner. One focuses on each point in space and asks how a quantity, like the fluid velocity, changes with time at that point in space. The objects are described in what is called a Lagrangian manner. Each material point on an object is tagged with a unique label, and one then tracks how the location of the material point with a given label changes as time advances and the system evolves. The state of the system at any time is given by the fluid velocity and pressure fields, and by the locations of the Lagrangian points which constitute the immersed objects. The essence of how the Immersed Boundary method computes the change in the system over a short time interval (or timestep) is as follows: From the Lagrangian description of each immersed object, it is straightforward to determine how much the object has been stretched and deformed at the beginning of the timestep, and from this to calculate the elastic forces generated within the object. Since the object is in direct contact with the fluid, these forces affect the fluid motion. To calculate the effect of these forces on the fluid, they are transmitted from the elastic objects to the fluid which immediately surrounds them, and thus contribute to the force density term in the fluid dynamics equations which drives the fluid motion. In fact, within the IB method, the fluid dynamics equations are solved everywhere (including inside the elastic objects) and the only way that the fluid `feels' the presence of the elastic objects is through the force density just described. Contributions to the force density from the immersed objects are localized to regions immediately adjacent to the objects, and it in this way that the geometry of the objects makes itself felt. Once the fluid force density is known, the partial differential equations which describe the fluid motion are solved to determine the new fluid velocity and pressure at each point of space. Finally, the fact that there is `no-slip' between a point of a viscous fluid and a point of an object immediately adjacent to it gives an equation of motion for each immersed boundary point: The velocity of the immersed boundary point is just the velocity of the fluid at the same location. The immersed boundary point location is therefore changed by an amount equal to this velocity multiplied by the duration of the timestep.
The beauty and power of the IB method is that, despite the existence of irregular immersed boundaries, the fluid dynamics equations are solved on a regular finite difference grid. This allows an IB code to utilize fast numerical techniques to calculate the fluid velocities. By contrast, for the competing finite element method, a regular grid cannot be used to update the fluid velocities, and therefore, fast numerical solvers cannot be used. Furthermore, the motion of the immersed boundary would require that the shape of the finite elements change from time step to time step, and accuracy considerations would require that the entire domain be regridded periodically. The irregular grids require cumbersome data structures and the regridding is expensive. Similar drawbacks hold for finite difference methods which use boundary-fitted coordinates.