18 worked lessons · Fluid mathematics
From flowing water
to Navier–Stokes.
Start with velocity, pressure, and viscosity. Follow exact examples into energy estimates and the careful reading of a new research result.
Choose a starting point ↓Read the equation in pieces
A balance of momentum.
∂u/∂t + (u · ∇)u = −∇p/ρ + νΔu + f, ∇ · u = 0
Here u is velocity, p is pressure, ρ is constant density, ν is kinematic viscosity, and f is force per unit mass. This form models an incompressible Newtonian fluid with constant viscosity. Initial conditions and boundary conditions are also part of a problem.
Motion
The left side follows how velocity changes with time and as a particle moves through the field.
Forces
Pressure, viscosity, and the specified external force determine the momentum balance.
Incompressibility
Zero divergence expresses local volume preservation; it does not mean that fluid is stationary.
The calculations use vectors and derivatives. If those are new, begin with the undergraduate library and read each lesson’s prerequisites.
Follow the ideas.
Build the fluid foundations
Undergraduate · 16 min
Test incompressibility without requiring zero velocity
Compute divergence to distinguish volume preservation from motionlessness.
Work through the lesson →Undergraduate · 16 min
Separate change at a point from change along a moving particle
Calculate a material derivative by adding local and advective change.
Work through the lesson →Undergraduate · 16 min
Connect a velocity gradient with Newtonian shear stress
Distinguish dynamic viscosity, kinematic viscosity, and shear rate.
Work through the lesson →Undergraduate · 16 min
Compare inertial and viscous scales without declaring a regime automatically
Form a dimensionless Reynolds number from characteristic scales.
Work through the lesson →Undergraduate · 16 min
Verify an exact flow between moving plates
Check velocity, boundary conditions, and every term of a simple steady Navier–Stokes solution.
Work through the lesson →Undergraduate · 16 min
Balance a pressure drop with viscous curvature
Derive a parabolic velocity profile between stationary plates.
Work through the lesson →Undergraduate · 16 min
Check viscous energy loss in an exact periodic flow
Relate a decaying Fourier shear mode to its kinetic-energy dissipation.
Work through the lesson →Undergraduate · 16 min
Build incompressibility into a two-dimensional velocity field
Generate divergence-free planar motion from a streamfunction.
Work through the lesson →
Explore the analysis and research
Graduate · 20 min
Move derivatives onto test functions
Derive a divergence-free weak formulation of incompressible flow.
Work through the lesson →Graduate · 20 min
Separate a smooth energy identity from a weak energy inequality
Derive the energy balance and understand what lower-regularity limits can retain.
Work through the lesson →Graduate · 20 min
Identify the three-dimensional vorticity stretching term
Distinguish vorticity transport, stretching, and diffusion.
Work through the lesson →Graduate · 20 min
Find which velocity norm survives Navier–Stokes scaling
Compute norm scaling in three spatial dimensions.
Work through the lesson →Graduate · 20 min
Remove the longitudinal component of a Fourier velocity
Compute the divergence-free projection of a single nonzero Fourier mode.
Work through the lesson →Graduate · 20 min
Recover pressure as a constraint-enforcing field
Derive the pressure Poisson equation by taking divergence.
Work through the lesson →Graduate · 20 min
Compare forced and unforced equations without changing the theorem
Keep the forcing class explicit when interpreting existence or breakdown claims.
Work through the lesson →Graduate · 20 min
Distinguish finite energy from bounded amplitude
Construct concentrating functions whose L² norms stay controlled while maxima grow.
Work through the lesson →Graduate · 20 min
See how one regular solution can control a comparison
Recognize the energy estimate behind a weak–strong uniqueness argument.
Work through the lesson →Graduate · 20 min
Read the September 2026 Navier–Stokes announcement precisely
Separate theorem hypotheses, formal verification, and institutional evaluation when reading a research announcement.
Work through the lesson →