Navier-Stokes equationsequation
French engineer Claude-Louis Navier (1821) and British physicist George Stokes (1845) combined into a system equations expressing the conservation of mass and momentum in a viscous fluid. The momentum equation: ρ (∂v/∂t + v·∇v) = -∇p + μ ∇²v + f. The left side is mass times acceleration (including convective transport), the right side includes pressure forces, viscous friction, and external influences. The incompressibility condition ∇·v = 0 is added. Their genius was accounting for internal friction (the term with viscosity μ).
How it works
With them, engineers calculate how air flows around a car, and meteorologists predict cyclone movement. Viscosity (flow resistance) calms the flow, and if viscosity is low, turbulence arises — chaotic mixing.
💡 The Clay Mathematics Institute has announced a $$1 million prize for proving that solutions to the equations always exist and do not 'blow up' (remain smooth) in the three-dimensional case — it is one of the Millennium Prize problems.
\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}
ρ is fluid density, ∂v/∂t is local acceleration, v·∇v is convective acceleration, ∇p is pressure gradient, μ is dynamic viscosity, ∇²v is the Laplacian of velocity (viscous dissipation), f is external body forces (e.g., gravity)
\nabla \cdot \mathbf{v} = 0
∇·v is the divergence of the velocity vector; for an incompressible fluid, it is zero, meaning no sources or sinks