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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^d$ be a fluid domain with density $\\rho \u003e 0$ and viscosity $\\mu \u003e 0$.\nLet a structure be a curve or surface $X(s, t) \\in \\Omega$, with parameter $s$ in a set $S$.\nLet $F(s, t)$ be the elastic force density of the structure, a given function of the configuration $X(\\cdot, t)$.\nThe *immersed boundary formulation* couples the incompressible [[Navier–Stokes equations|18KXZHVF4M]]\n$$\\rho (u_t + u \\cdot \\nabla u) = -\\nabla p + \\mu \\Delta u + f, \\qquad \\nabla \\cdot u = 0,$$\nwith two interaction equations, written with the [[Dirac delta function|0KPHENZ5S7]] $\\delta$:\n$$f(x, t) = \\int_S F(s, t) \\, \\delta(x - X(s, t)) \\, ds, \\qquad \\frac{\\partial X}{\\partial t}(s, t) = \\int_\\Omega u(x, t) \\, \\delta(x - X(s, t)) \\, dx.$$\nThe *immersed boundary method* discretizes it in three parts:\n\n1. the fluid equations on a fixed Cartesian grid of width $h$, by [[finite differences|X7HYEBA5SA]] or another grid method;\n2. the structure by a finite set of points $X_k$;\n3. $\\delta$ by a smooth discrete delta function $\\delta_h$, supported on a few grid cells.\n\n## Remarks\n\nThe second interaction equation says that the structure moves with the local fluid velocity, which is the no-slip condition.\nThe same $\\delta_h$ spreads forces and interpolates velocities, so spreading is the adjoint of interpolation.\nHence the power given by the structure equals the power received by the fluid.\nA standard $\\delta_h$ in $d$ dimensions is a product $\\prod_{i=1}^{d} h^{-1} \\phi(x_i / h)$ of one-dimensional kernels $\\phi$ supported in $[-2, 2]$.\n\nExample: a closed elastic membrane $X(s, t)$, $s \\in [0, 2\\pi)$, in a fluid at rest, with tension force $F = \\kappa \\, \\partial_s^2 X$ and $\\kappa \u003e 0$.\nIf the initial membrane is an ellipse, the forces drive the fluid, and the membrane relaxes towards a circle of the same enclosed area.\n\nNon-example: a method that meshes the fluid region outside the structure with a body-fitted mesh that moves with the structure fails clause 1.\nIts grid is not fixed, so it is not an immersed boundary method.\n",
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