Immersed boundary method

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 on an Eulerian coordinate and the structure is represented on a Lagrangian coordinate. For Newtonian fluids governed by the incompressible Navier–Stokes equations, the fluid equations are

and in case of incompressible fluids (assuming constant density) we have the condition

The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by ${\displaystyle \Gamma }$. Each fiber can be viewed as a parametric curve ${\displaystyle X(s,t)}$ where ${\displaystyle s}$ is the parameter and ${\displaystyle t}$ is time. Physics of the fiber is represented via the fiber force distribution ${\displaystyle F(s,t)}$. 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

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