Skip to content

The Navier–Stokes Equation

Equation

For an incompressible Newtonian fluid:

\[\boxed{\;\rho\left[\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v}\right] = -\nabla p + \mu\nabla^2\mathbf{v} + \mathbf{f}\;} \qquad \nabla\cdot\mathbf{v} = 0\]

Physical meaning — term by term

Term Reading
\(\rho\,\partial\mathbf{v}/\partial t\) unsteady acceleration
\(\rho(\mathbf{v}\cdot\nabla)\mathbf{v}\) convective acceleration — the nonlinearity
\(-\nabla p\) pressure-gradient force
\(\mu\nabla^2\mathbf{v}\) viscous diffusion of momentum
\(\mathbf{f}\) body forces (gravity, EM, …)

It is Euler's equation plus momentum diffusion: fast regions leak momentum to slow neighbors at rate set by viscosity.

Variables

\(\rho\) — density (kg m⁻³) · \(\mathbf{v}\) — velocity (m s⁻¹) · \(p\) — pressure (Pa) · \(\mu\) — dynamic viscosity (Pa·s) · \(\mathbf{f}\) — body force density (N m⁻³).

Assumptions

Continuum · Newtonian (stress ∝ strain rate) · incompressible in this form · constant \(\mu\). Compressible and non-Newtonian generalizations exist but are beyond PC316.

Derivation sketch

Apply the Reynolds transport theorem to momentum with the full stress tensor \(T_{ij} = -p\,\delta_{ij} + \mu\left(\partial_i v_j + \partial_j v_i\right)\) (Newtonian constitutive law). The divergence of the viscous part, with \(\nabla\cdot\mathbf{v} = 0\), collapses to \(\mu\nabla^2\mathbf{v}\).

Exact solutions

Rare and precious — found by symmetry + assumptions + boundary conditions:

Almost everything else — including whether smooth 3-D solutions always exist — is open. That existence-and-smoothness question is one of the Clay Millennium Prize Problems (US$1M).

Applications

All of real fluid dynamics: aerodynamics, hemodynamics, meteorology, oceanography, lubrication, microfluidics, turbulence (by direct simulation at modest \(Re\), by RANS modeling beyond).

Limitations

Continuum only (no rarefied gases); Newtonian only; numerically brutal at high Reynolds number (resolution cost \(\sim Re^{9/4}\) — see the energy cascade).

Quiz

Q1 (conceptual). Nondimensionalize Navier–Stokes with scales \(U, L\). What single parameter remains, and where?

Answer

\(\frac{\partial\mathbf{v}^*}{\partial t^*} + (\mathbf{v}^*\cdot\nabla^*)\mathbf{v}^* = -\nabla^* p^* + \frac{1}{Re}\nabla^{*2}\mathbf{v}^*\) — the Reynolds number, multiplying (inversely) the viscous term. Dynamic similarity follows immediately.

Q2 (multiple choice). Which term makes superposition of solutions fail?

  • (a) \(-\nabla p\) (b) \(\mu\nabla^2\mathbf{v}\) (c) \((\mathbf{v}\cdot\nabla)\mathbf{v}\) (d) \(\mathbf{f}\)
Answer

(c) — quadratic in \(\mathbf{v}\); all other terms are linear.

Q3 (computational). For fully developed flow between fixed plates driven by \(dp/dx = -G\) (gap \(2h\), centered at \(y=0\)), find \(u(y)\).

Answer

\(\mu\,u'' = -G \Rightarrow u = \frac{G}{2\mu}(h^2 - y^2)\) — the plane-Poiseuille parabola, by direct double integration with no-slip at \(y = \pm h\).