Section
Fluid dynamics
Velocity, transport, incompressible momentum equations, and cylindrical fluid geometry.
Core idea
These entries introduce incompressible fluid motion and its geometric and analytic quantities.
Definitions and equations
- Affine velocity field
- Conservative incompressible momentum equation
- Viscous diffusion time scale
- Incompressible Euler equations
- External force in a normalized fluid equation
- Incompressible velocity field
- Kinetic energy of an incompressible velocity field
- Material derivative
- Momentum flux tensor in an incompressible fluid
- Incompressible Navier–Stokes equations
- Particle trajectory of a velocity field
- Pressure in incompressible flow
- Quadratic advection of a velocity field
- Rate-of-strain tensor
- Reynolds number
- Parallel shear flow
- Advective transport time scale
- Fluid velocity field
- Viscosity in the incompressible Newtonian model
- Vorticity of a velocity field
- Vorticity equation for incompressible flow
- Axisymmetric vector field
- Radial velocity in cylindrical coordinates
- Axial velocity
- Meridional velocity
- Swirl component of a cylindrical velocity
- Angular velocity about an axis
- Specific angular momentum about an axis
- Centripetal acceleration for circular motion
- Rotating flow about a fixed axis
- Axisymmetric Navier–Stokes component equations
- Angular-velocity equation in axisymmetric flow
- Vortex line
- Transport operator along a vector field
Scaling and perturbations
Angular averages and oscillatory perturbations
Cones, averages and moment constraints
Heat flow and integral profiles
Sobolev spaces and pressure analysis
Energy, continuation and periodization
- Uniformly bounded kinetic energy
- Changing viscosity by spatial rescaling
- Classical uniqueness from difference energy
- Cutoff pressure-flux estimate for a velocity difference
- Delayed parabolic rescaling at a fixed terminal time
- Energy bound from a time-integrable L2 force
- Global kinetic-energy identity
- Kinetic-energy inequality
- Local kinetic-energy balance for smooth incompressible flow
- Localized difference-energy identity
- Difference equation for two incompressible flows
- Periodization of separated incompressible flows
- Identifying pressure gradients without a pressure growth assumption
- Relative kinetic energy
- Integration cancellation for divergence-free transport and pressure
- Uniqueness against a smooth compactly supported reference flow
- Viscous dissipation in the incompressible energy balance
Weak equations, regularity, and relaxed models
- Active scalar equation
- Averaged Navier–Stokes model
- Caffarelli–Kohn–Nirenberg partial regularity theorem
- Endpoint L-infinity-in-time L3 regularity criterion
- Euler subsolution with prescribed energy
- Hypodissipative Navier–Stokes equations
- Incompressible porous media equation
- Leray–Hopf solution
- Leray projection on Euclidean space
- Local energy inequality
- Navier–Stokes energy class
- Reynolds averaging stress
- Reynolds defect system
- Singular set of a weak velocity
- Suitable weak solution
- Weak Navier–Stokes solution
Prerequisite collection
Navier–Stokes: 433 new knowls grouped by batch and subject brings together the foundations, analysis, geometry, and fluid equations developed in batches B00–B14.