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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^3$ be an open set, let $\\nu \u003e 0$ be a constant, and let $f \\colon \\Omega \\to \\R^3$ be a given force.\nConsider a [[vector field|0299VCW1TH]] $u \\colon \\Omega \\to \\R^3$, the velocity, and a function $p \\colon \\Omega \\to \\R$, the pressure.\nThe pair $(u, p)$ is a *Stokes flow* in $\\Omega$ if it satisfies the *Stokes equations*\n$$- \\nu \\Delta u + \\nabla p = f, \\qquad \\nabla \\cdot u = 0 \\quad \\text{in } \\Omega.$$\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] acting on each component, $\\nabla p$ is the [[gradient|0ABK9V9Y64]] and $\\nabla \\cdot u$ is the [[divergence|0MWV6ZTK6S]].\n\n## Remarks\n\nExample: in the channel $\\Omega = \\set{(x, y, z) : -1 \u003c y \u003c 1}$ with $f = 0$, the Poiseuille flow $u = (1 - y^2, 0, 0)$ with $p = -2\\nu x$ is a Stokes flow.\nIndeed $\\Delta u = (-2, 0, 0)$, so $-\\nu \\Delta u + \\nabla p = (2\\nu, 0, 0) + (-2\\nu, 0, 0) = 0$, and $\\nabla \\cdot u = 0$.\nIt also vanishes on the walls $y = \\pm 1$.\n\nNon-example: the same $u$ with constant pressure is not a Stokes flow when $f = 0$, because then $-\\nu \\Delta u + \\nabla p = (2\\nu, 0, 0) \\ne 0$.\n\nThe Stokes equations arise from the [[Navier–Stokes equations|18KXZHVF4M]] when the Reynolds number is small, so that the term $(u \\cdot \\nabla) u$ can be neglected, and the flow is steady.\nThe time-dependent version $\\partial_t u - \\nu \\Delta u + \\nabla p = f$ is also called the Stokes equations.\n",
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