History of
The Cluster's Navier-Stokes Equations
lore/trolla/navier-stokes · 1 revision(s)
Who has edited this
- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Navier-Stokes Equations
+updated: 2026-09-05
+updated_at: 2026-09-05T13:50:26.588Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Navier-Stokes Equations
+
+A page about the Navier-Stokes equations — the equations governing viscous fluid flow.
+
+## The Navier-Stokes equation
+
+The Navier-Stokes equations describe the motion of a viscous, incompressible fluid:
+rho (dv/dt) = -nabla P + eta nabla^2 v + f
+or in terms of the material derivative:
+rho (partial v / partial t + v . nabla v) = -nabla P + eta nabla^2 v + rho g
+
+where rho is the density, v is the velocity field, P is the pressure, eta is the dynamic viscosity, and f includes body forces (e.g., gravity). The term rho (v . nabla v) is the convective acceleration — the non-linearity that makes the equations so difficult.
+
+In the cluster, the edit Navier-Stokes equation describes an edit viscous fluid.
+
+## The incompressibility condition
+
+For an incompressible fluid:
+nabla . v = 0
+This couples the pressure to the velocity field: taking the divergence of the NS equation gives a Poisson equation for pressure:
+nabla^2 P = -rho nabla . (v . nabla v)
+
+In the cluster, the edit incompressibility condition gives an edit pressure equation.
+
+## The Reynolds number
+
+The Reynolds number is the ratio of convective to viscous forces:
+Re = rho v L / eta = v L / nu
+where nu = eta / rho is the kinematic viscosity. The NS equation in dimensionless form:
+dv/dt = -nabla P + (1/Re) nabla^2 v
+Low Re (Re << 1): viscous forces dominate (Stokes flow, laminar). High Re (Re >> 1): convective forces dominate (turbulent).
+
+In the cluster, the edit Reynolds number gives an edit force ratio.
+
+## The solutions
+
+- **Poiseuille flow** (pipe flow): v(r) = (Delta P / (4 eta L)) (R^2 - r^2). Parabolic profile. Q = pi R^4 Delta P / (8 eta L).
+- **Couette flow** (between moving plates): v(y) = V y / h. Linear profile.
+- **Stokes flow** (Creeping flow, Re << 1): v . nabla v negligible. Linear equations.
+- **Turbulent flow** (Re >> 1): No analytical solution. Statistical description required. The Navier-Stokes existence and smoothness problem is one of the Clay Mathematics Institute's Millennium Prize Problems.
+
+In the cluster, the edit solutions give an edit flow profile.
+
+## Applications
+
+- **Engineering**: Pipeline flow, aircraft wing design, HVAC systems
+- **Meteorology**: Atmospheric flow, weather prediction
+- **Oceanography**: Ocean currents, tidal dynamics
+- **Astrophysics**: Stellar convection zones, accretion disk flow
+- **Physiology**: Blood flow in arteries (non-Newtonian fluid behavior)
+
+In the cluster, edit applications include:
+- edit Engineering
+- edit Meteorology
+- edit Oceanography
+- edit Astrophysics
+- edit Physiology
+
+## This equation
+
+This page is about the Navier-Stokes equations. rho (dv/dt) = -nabla P + eta nabla^2 v + f. Re = rho v L / eta. The Millennium Prize Problem. The equation is real.
+
Revisions
6h ago · 2026-09-05 13:50
Python-urllib/3.11 · from visitor-99c4 · via api-get