Where the equations come from: a step-by-step derivation from Newton's second law to the incompressible system behind the Millennium Problem
Newton's second law for a fluid
Every Navier-Stokes derivation starts with the same idea: apply Newton's second law to a tiny parcel of moving fluid. Force equals mass times acceleration. That is the basic starting point.
Pick a small blob of water, air, or any other fluid. It has some mass. Forces act on it: pressure squeezes it from all sides, internal friction tugs on it, gravity pulls it down. Newton says the net force determines how the blob accelerates.
But a fluid parcel isn't a billiard ball. It deforms as it moves. It stretches, twists, distorts as it rides the flow. So "acceleration" isn't as simple as tracking a single object. We need to follow the blob through all of that.
Think of it this way: imagine sitting in a canoe on a river. Your acceleration depends on how the current changes in time at your location, and on the fact that the current is carrying you into regions where the flow is faster or slower. Both effects contribute to how your velocity changes.
Writing this balance for every point in the fluid simultaneously gives us the starting point of the derivation of the Navier-Stokes equations.
The forces on a fluid parcel
Three kinds of forces show up in the derivation of the Navier-Stokes equation:
1. Pressure forces. Fluid pushes on the parcel from every direction. If pressure is higher on one side than the other, the parcel gets shoved toward the low-pressure side. This imbalance is captured by the pressure gradient $-\nabla p$: a vector pointing from high pressure to low, telling the fluid which way to go.
2. Viscous forces. Adjacent layers of fluid moving at different speeds drag on each other. Faster layers pull slower neighbors along; slower layers hold faster ones back. This internal friction smooths out velocity differences. For a simple fluid like water, the strength of this friction comes down to a single number: the viscosity $\mu$.
3. External forces. Anything acting on the fluid from outside, most commonly gravity. These are body forces: they act on every bit of fluid in the volume, not just at the surface.
The Navier-Stokes equations are what you get when you write "mass times acceleration = pressure force + viscous force + external force" at every point.
The Newtonian fluid assumption
Not all fluids behave the same way under stress. Honey resists motion differently than water. Ketchup gets runnier when you shake it. Cornstarch mixed with water gets stiffer when you hit it.
The Navier-Stokes equations make a specific assumption: the fluid is Newtonian. This means internal friction is directly proportional to how fast the fluid is being deformed. Double the rate of deformation, double the stress. It's a linear relationship.
Water and air are very well modeled as Newtonian. But this is an assumption, not a consequence of Newton's laws. The derivation requires it. Without it, you get a different class of equations entirely (non-Newtonian fluid models).
The proportionality constant is the dynamic viscosity $\mu$. It's a material property measuring how much a fluid resists shearing. Water: low viscosity. Honey: high viscosity.
There's also a second viscosity parameter $\lambda$, sometimes called the second viscosity coefficient, which matters when the fluid compresses or expands. A common further simplification, Stokes' hypothesis, sets $\lambda = -\frac{2}{3}\mu$. This is an additional assumption, not a theorem.
Assembling the momentum equation
Now we put the pieces together. We have:- Mass times acceleration on the left (the material derivative of velocity)- Pressure, viscous friction, and gravity on the right- The Newtonian-fluid rule connecting friction to the rate of deformationSubstituting and simplifying gives the compressible Navier-Stokes momentum equation:
$$\rho\Big(\frac{\partial u}{\partial t} + (u \cdot \nabla) u\Big) = -\nabla p + \mu\, \Delta u + (\mu + \lambda)\,\nabla(\nabla \cdot u) + \rho\, f$$
Every term has a physical meaning:- $\rho\,\partial_t u$: how velocity at a fixed point changes over time- $\rho(u \cdot \nabla) u$: the fluid carrying its own velocity from place to place (advection)- $-\nabla p$: pressure pushing from high to low- $\mu\,\Delta u$: viscosity smoothing out velocity differences- $(\mu + \lambda)\nabla(\nabla \cdot u)$: an extra viscous term that only matters when the fluid compresses or expands- $\rho f$: external forces like gravityThat's the full momentum equation. Pair it with conservation of mass and an equation of state (linking pressure to density), and you have the compressible Navier-Stokes system.
The incompressible specialization
Water in a pipe. Slow air currents. Ocean circulation. These flows involve fluids whose density stays essentially constant. Making that simplification transforms the general equations into something much cleaner.
Constant density means $\rho$ doesn't change, anywhere, ever. Conservation of mass then forces $\nabla \cdot u = 0$: the fluid can't compress or expand. This is the incompressibility constraint.
This simplification removes one term entirely. The extra viscous term $(\mu + \lambda)\nabla(\nabla \cdot u)$ vanishes entirely. The second viscosity coefficient $\lambda$ drops out. It simply doesn't matter when the fluid can't compress. The momentum equation becomes:
$$\rho\Big(\frac{\partial u}{\partial t} + (u \cdot \nabla)u\Big) = -\nabla p + \mu\,\Delta u + \rho f$$
Dividing both sides by $\rho$, writing $\nu = \mu / \rho$ (the kinematic viscosity), and redefining pressure to absorb the factor $1/\rho$ gives the standard form:
$$\frac{\partial u}{\partial t} + (u \cdot \nabla)u = -\nabla p + \nu\,\Delta u + f$$
$$\nabla \cdot u = 0$$
This is the system studied in the Clay Millennium Problem. It's the version you'll encounter throughout this site and in most mathematical treatments of Navier-Stokes. For a thorough comparison with the compressible system, see Incompressible vs. Compressible Navier-Stokes.
Set $\nu = 0$ (no viscosity at all) and you get the Euler equations, a related but importantly different system.
What the derivation does and does not tell us
The derivation gives us the Navier-Stokes equations. It tells us what they are and why they take the form they do. Every term traces back to a physical principle or an explicit assumption.
But deriving the equations isn't the same as understanding their solutions. The derivation doesn't answer:- Do solutions always exist for all time?- If they start smooth, do they stay smooth?- Can the velocity blow up to infinity in finite time?These are questions about the mathematical behavior of the equations, not their physical origin. In three dimensions, they're still open. That gap is the Clay Millennium Problem: whether smooth 3D incompressible Navier-Stokes data always produce global smooth solutions, or whether singularities can form in finite time.
The equations date to the 19th century, and the Clay Mathematics Institute has offered a $1 million prize since 2000 for a resolution of the modern regularity problem.
What to read next
Now that you've seen where the equations come from:- What Are the Navier-Stokes Equations? for the equations themselves, with term-by-term explanation- Incompressible vs. Compressible Navier-Stokes for what changes when density varies, and why the incompressible case is special- Euler vs. Navier-Stokes for what happens when you remove viscosity entirely- The Millennium Problem for the open question that makes these equations famous
Originally published on navier-stokes.org. The site covers the Navier-Stokes existence and smoothness problem with toggleable simple/rigorous explanations.
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