  Numerical simulation of moving immersed solid bodies in fluids is now
practiced routinely following pioneering work of Peskin and co-workers on
immersed boundary method (IBM), Glowinski and co-workers on fictitious domain
method (FDM), and others on related methods. A variety of variants of IBM and
FDM approaches have been published, most of which rely on using a background
mesh for the fluid equations and tracking the solid body using Lagrangian
points. The key idea that is common to these methods is to assume that the
entire fluid-solid domain is a fluid and then to constrain the fluid within the
solid domain to move in accordance with the solid governing equations. The
immersed solid body can be rigid or deforming. Thus, in all these methods the
fluid domain is extended into the solid domain. In this review, we provide a
mathemarical perspective of various immersed methods by recasting the governing
equations in an extended domain form for the fluid. The solid equations are
used to impose appropriate constraints on the fluid that is extended into the
solid domain. This leads to extended domain constrained fluid-solid governing
equations that provide a unified framework for various immersed body
techniques. The unified constrained governing equations in the strong form are
independent of the temporal or spatial discretization schemes. We show that
particular choices of time stepping and spatial discretization lead to
different techniques reported in literature ranging from freely moving rigid to
elastic self-propelling bodies. These techniques have wide ranging applications
including aquatic locomotion, underwater vehicles, car aerodynamics, and organ
physiology (e.g. cardiac flow, esophageal transport, respiratory flows), wave
energy convertors, among others. We conclude with comments on outstanding
challenges and future directions.
