Immersed boundary method

Definition

Let $\Omega \subset \R^d$ be a fluid domain with density $\rho > 0$ and viscosity $\mu > 0$. Let a structure be a curve or surface $X(s, t) \in \Omega$, with parameter $s$ in a set $S$. Let $F(s, t)$ be the elastic force density of the structure, a given function of the configuration $X(\cdot, t)$. The immersed boundary formulation couples the incompressible Navier–Stokes equations $$\rho (u_t + u \cdot \nabla u) = -\nabla p + \mu \Delta u + f, \qquad \nabla \cdot u = 0,$$ with two interaction equations, written with the Dirac delta function $\delta$: $$f(x, t) = \int_S F(s, t) \, \delta(x - X(s, t)) \, ds, \qquad \frac{\partial X}{\partial t}(s, t) = \int_\Omega u(x, t) \, \delta(x - X(s, t)) \, dx.$$ The immersed boundary method discretizes it in three parts:

1. the fluid equations on a fixed Cartesian grid of width $h$, by finite differences or another grid method;
2. the structure by a finite set of points $X_k$;
3. $\delta$ by a smooth discrete delta function $\delta_h$, supported on a few grid cells.