In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented in an Eulerian coordinate system and the structure is represented in Lagrangian coordinates. For Newtonian fluids governed by the Navier–Stokes equations, the fluid equations are
ρ ( ∂ u ( x , t ) ∂ t + u ⋅ ∇ u ) = − ∇ p + μ Δ u ( x , t ) + f ( x , t ) {\displaystyle \rho \left({\frac {\partial {u}({x},t)}{\partial {t}}}+{u}\cdot \nabla {u}\right)=-\nabla p+\mu \,\Delta u(x,t)+f(x,t)}
and if the flow is incompressible, we have the further condition that
∇ ⋅ u = 0. {\displaystyle \nabla \cdot u=0.\,}
The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by Γ {\displaystyle \Gamma } . Each fiber can be viewed as a parametric curve X ( s , t ) {\displaystyle X(s,t)} where s {\displaystyle s} is the Lagrangian coordinate along the fiber and t {\displaystyle t} is time. The physics of the fiber is represented via a fiber force distribution function F ( s , t ) {\displaystyle F(s,t)} . Spring forces, bending resistance or any other type of behavior can be built into this term. The force exerted by the structure on the fluid is then interpolated as a source term in the momentum equation using
f ( x , t ) = ∫ Γ F ( s , t ) δ ( x − X ( s , t ) ) d s , {\displaystyle f(x,t)=\int _{\Gamma }F(s,t)\,\delta {\big (}x-X(s,t){\big )}\,ds,}
where δ {\displaystyle \delta } is the Dirac δ function. The forcing can be extended to multiple dimensions to model elastic surfaces or three-dimensional solids. Assuming a massless structure, the elastic fiber moves with the local fluid velocity and can be interpolated via the delta function
∂ X ( s , t ) ∂ t = u ( X , t ) = ∫ Ω u ( x , t ) δ ( x − X ( s , t ) ) d x , {\displaystyle {\frac {\partial X(s,t)}{\partial t}}=u(X,t)=\int _{\Omega }u(x,t)\,\delta {\big (}x-X(s,t){\big )}\,dx,}
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