The Navier–Stokes Equations
The Navier–Stokes equations are Newton's second law written for a viscous fluid: they govern every flow from blood in capillaries to weather systems. Existence and smoothness of their general 3-D solutions remains a Millennium Prize problem.
Variables
| Symbol | Meaning | Constraints / unit |
|---|---|---|
| u | Velocity field (vector) | m/s |
| ρ | Fluid density | kg/m³ |
| p | Pressure field | Pa |
| μ | Dynamic viscosity | Pa·s |
| f | Body force per unit volume (gravity, magnetic…) | N/m³ |
Reading the equation term by term
- — local acceleration: how the velocity changes in time at a fixed point.
- — convective acceleration: the nonlinear term responsible for turbulence.
- — pressure gradient: fluid accelerates from high to low pressure.
- — viscous diffusion: momentum smearing between adjacent layers.
- — body forces: gravity, buoyancy, electromagnetic forces.
For incompressible flow the momentum equation is paired with the continuity constraint:
The Reynolds number connection
Non-dimensionalising the equation makes a single parameter appear in front of the viscous term — the Reynolds number . Low Re means viscosity wins (laminar); high Re means the nonlinear convective term wins (turbulent).
MHD extension — magnetohydrodynamic flow
When the fluid conducts electricity and moves through a magnetic field , the Lorentz force enters as the body force:
For boundary-layer flow over a stretching sheet under a transverse field , this reduces to a drag term — the setting of MHD nanofluid research, where similarity transformations reduce the PDEs to a nonlinear ODE system solved numerically (e.g. with MATLAB's bvp4c).
Special cases you already know
- Drop viscosity (μ = 0) and integrate along a streamline → Bernoulli's equation.
- Steady, unidirectional flow in a circular pipe → the Hagen–Poiseuille law.