Contemporary Mathematics
Volume 628, 2014
http://dx.doi.org/10.1090/conm/628/12545
Simulating Biofluid-Structure Interactions with an
Immersed Boundary Framework A Review
Sarah D. Olson and Anita T. Layton
Abstract. This review focuses on biofluid-structure interactions that are
modeled using an immersed boundary framework. We consider elastic struc-
tures immersed in a viscous, incompressible fluid that interact with the fluid
via a forcing term in the momentum equation. The standard immersed bound-
ary (IB) method of Peskin is reviewed in terms of numerical implementation,
force derivation, and choice of compactly supported delta function. We then
review related methods including the immersed interface method, generalized
IB method, and regularized Stokeslets methods. Several advances in numerical
methods are detailed, including porous boundaries, multi-fluids, time stepping
strategies, and the incorporation of viscoelasticity. The review ends with a
discussion of advantages of several methods and avenues of future research.
1. Introduction
The interaction between fluid flows and immersed structures are nonlinear
multi-physics phenomena, and their applications can be found in a wide range of
scientific and engineering disciplines. In biology, many applications can be found,
including dynamics of insect wings, flagellated or ciliated organisms, suspensions
of blood cells and other synthetic particles, parachute dynamics, and many more.
This is an active area of research in terms of development of new numerical methods
as well as model development for the structure.
The IB method was originally developed by Peskin [137, 138], for studying
blood flow through a beating heart [134]. In this method, a dynamic elastic struc-
ture is immersed in a viscous, incompressible fluid. This mathematical formulation
and numerical method is a framework to model fluid-structure interaction prob-
lems by mechanically coupling the fluid to forces in a support region around the
structure. This is a fully coupled system since the structure is able to alter the
fluid velocity via time and spatially dependent forces exerted on the surrounding
fluid, and in turn, the movement of the structure is determined by the local fluid
velocity.
We will keep with the theme of this volume and focus on aspects of the IB
method of Peskin (1972) and related methods to model biological elastic structures
interacting with a fluid. Since the initial development of the IB method, several
different extensions and variations have been developed. This review will focus on
2010 Mathematics Subject Classification. Primary 76M25, 76Z05, 76Z05.
NSF DMS 1122461.
c 2014 American Mathematical Society
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