Term-by-term breakdown of the governing equations of fluid motion
The Navier-Stokes equations describe how the velocity field of a fluid evolves over time. They are a statement of Newton's second law applied to a continuous medium: the acceleration of each fluid parcel equals the sum of all forces acting on it.
Advection
Pressure gradient
Viscous diffusion
External forces
For incompressible flow, these are coupled with the continuity equation ∇·v = 0, which enforces conservation of mass. Together, they form a system of nonlinear partial differential equations that governs everything from ocean currents to airflow over a wing.
Each term in the Navier-Stokes equations contributes a different physical effect. Select individual terms below to see their contribution to the total acceleration field. The base flow is a shear layer with an embedded vortex, which gives each term a distinct and visible pattern.
Select a term to isolate its contribution to the total acceleration field. The base flow is a shear layer with an embedded vortex, so each term produces a distinct pattern.
The fluid carries its own momentum. Faster-moving fluid sweeps velocity downstream, creating the nonlinearity that makes turbulence possible.
Fluid accelerates from high to low pressure. The pressure field adjusts instantaneously to keep the flow incompressible.
Internal friction diffuses momentum from fast regions to slow regions. It smooths out sharp velocity gradients, like stirring honey.
Body forces like gravity or electromagnetic forces act uniformly on the fluid. They drive large-scale circulation patterns.
A key subtlety in fluid mechanics is the distinction between the Eulerian description (measuring at a fixed point) and the Lagrangian description (following a fluid parcel). The material derivative Dv/Dt = ∂v/∂t + (v·∇)v captures how a parcel's velocity changes as it moves through a spatially varying field.
Rate of change at a fixed point in space.
|∂v/∂t| = 0.000
Zero for this steady flow — the velocity at any fixed location never changes.
Rate of change following the parcel (material derivative).
|Dv/Dt| = 0.001
Non-zero because the parcel accelerates as it enters the nozzle throat.
Key insight: In the nozzle above, the flow is steady (∂v/∂t = 0 everywhere), yet each fluid parcel accelerates as it enters the throat. This is the convective acceleration (v·∇)v — the velocity changes not because the flow pattern changes in time, but because the parcel moves to a location with a different velocity.
The Reynolds number Re = ρUL/μ is the ratio of inertial forces (advection) to viscous forces (diffusion). It determines the character of the flow: low Re gives smooth, predictable laminar flow, while high Re produces chaotic, unpredictable turbulence. Adjust the four physical parameters below and watch the flow transition between regimes.
Viscosity dominates. Fluid flows in smooth parallel layers with a parabolic velocity profile. Disturbances are damped out.
Intermittent “turbulent slugs” appear and disappear. The flow switches unpredictably between laminar and turbulent.
Inertia dominates. Chaotic mixing, eddies at all scales, and enhanced transport characterize fully turbulent flow.
The Navier-Stokes equations have been used by engineers and physicists for nearly 200 years, yet a fundamental mathematical question remains open:
“Do smooth, physically reasonable solutions to the 3D incompressible Navier-Stokes equations always exist, and if they exist, are they smooth for all time?”
In other words: given smooth initial conditions, can the velocity field develop a singularity (blow up to infinity) in finite time? Nobody knows. This is one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, with a prize of $1,000,000 for a correct proof or counterexample.
The difficulty stems from the nonlinear advection term (v·∇)v. It can transfer energy to ever-smaller scales, potentially concentrating velocity gradients without bound. Proving that this cannot happen — or finding a case where it does — remains one of the greatest open problems in mathematics and physics.