The Navier-Stokes Equations

Prerequisites: Vector Calculus, Newton's Second Law. Full path (4 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

The Navier-Stokes equations are Newton's second law applied to a fluid. We imagine the fluid as made of many small parcels, each one small compared to the flow we care about but large compared to a molecule. Each parcel speeds up, slows down or turns according to the forces acting on it. Three kinds of force matter. Differences in pressure push a parcel from regions of high pressure towards regions of low pressure. Friction between neighbouring layers of fluid that slide past each other, called viscosity, evens out differences in velocity. And external forces like gravity act on the whole parcel.

What makes the equations hard is that the parcel's acceleration depends on the flow itself. A parcel is carried along by the velocity field while the velocity field is shaped by the motion of all parcels. This feedback makes the equations nonlinear. When the flow is slow or the fluid is very viscous, like honey, viscosity wins and the flow is smooth and orderly. When the flow is fast and the viscosity small, like air around a car, the nonlinear term wins and the flow becomes turbulent, with eddies of many sizes inside each other. A single number, the Reynolds number, measures which effect dominates.

Most fluids we deal with, such as water, change their density very little. Then the equations also contain the condition that the fluid cannot be compressed: as much fluid flows into any small volume as flows out of it.

• What Makes the Hardest Equations in Physics So Difficult? (Quanta Magazine)
• Mathematicians Find Wrinkle in Famed Fluid Equations (Quanta Magazine)

Concrete

The fluid is described by a velocity field $\vec u(\vec x,t)$ and a pressure field $p(\vec x,t)$. For an incompressible fluid with constant density $\rho$ and constant dynamic viscosity $\mu$, the Navier-Stokes equations read

$$\rho\left(\frac{\partial \vec u}{\partial t} + (\vec u\cdot\nabla)\vec u\right) = -\nabla p + \mu\nabla^2\vec u + \vec f, \qquad \nabla\cdot\vec u = 0 ,$$

where $\vec f$ is an external force per unit volume, for example $\rho\vec g$ for gravity. Dividing by $\rho$ introduces the kinematic viscosity $\nu = \mu/\rho$.

Where each term comes from. The left side is mass per volume times acceleration of a fluid parcel. The parcel at $\vec x$ at time $t$ is at $\vec x + \vec u\,dt$ at time $t + dt$, so its change of velocity is $\vec u(\vec x + \vec u\,dt, t + dt) - \vec u(\vec x,t) = \left(\partial_t\vec u + (\vec u\cdot\nabla)\vec u\right)dt$. The combination $\partial_t + \vec u\cdot\nabla$ is called the material derivative. The term $(\vec u\cdot\nabla)\vec u$ is the source of the nonlinearity. On the right side, $-\nabla p$ is the net pressure force on a small volume, and $\mu\nabla^2\vec u$ is the net viscous force for a Newtonian fluid, one in which the viscous stress is proportional to the rate of strain. The condition $\nabla\cdot\vec u = 0$ is the continuity equation $\partial_t\rho + \nabla\cdot(\rho\vec u) = 0$ for constant $\rho$.

There is no separate evolution equation for $p$. Taking the divergence of the momentum equation with $\vec f = 0$ and using $\nabla\cdot\vec u = 0$ gives

$$\nabla^2 p = -\rho\,\partial_i u_j\,\partial_j u_i ,$$

so the pressure is determined at each instant by the velocity field and acts to keep the flow divergence free.

Reynolds number. If $U$ is a typical velocity and $L$ a typical length of the flow, the ratio of the inertial term to the viscous term is of order

$$\mathrm{Re} = \frac{UL}{\nu} .$$

For small $\mathrm{Re}$ the nonlinear term can be dropped (Stokes flow). For large $\mathrm{Re}$ flows typically become turbulent.

Example: flow through a pipe. Consider steady flow along a straight pipe of radius $R$ in the $z$ direction, driven by a constant pressure gradient $\partial_z p = -G$. We try $\vec u = u(r)\,\hat z$. Then $\nabla\cdot\vec u = 0$ and $(\vec u\cdot\nabla)\vec u = u\,\partial_z u\,\hat z = 0$, so the nonlinear term vanishes exactly. The $z$ component of the equation is

$$\mu\,\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right) = -G .$$

With $u$ finite at $r = 0$ and the no-slip condition $u(R) = 0$ at the wall, the solution is $u(r) = \frac{G}{4\mu}(R^2 - r^2)$. Integrating over the cross section gives the volume flow rate $Q = \frac{\pi G R^4}{8\mu}$ (Hagen-Poiseuille law). This laminar solution is observed at low and moderate $\mathrm{Re}$; at large $\mathrm{Re}$ the flow in a pipe becomes turbulent.

Compressible fluids. If the density varies, the continuity equation $\partial_t\rho + \nabla\cdot(\rho\vec u) = 0$ is kept in full, the viscous stress acquires a term proportional to $\nabla\cdot\vec u$, and an equation of state and an energy equation are needed to close the system.

Abstract

We consider the incompressible equations on $\mathbb R^3$ or on the torus $\mathbb T^3$, with $\nu > 0$ and $\vec f = 0$. Let $P$ be the Leray projector, the orthogonal projection of $L^2$ vector fields onto divergence-free ones. Applying $P$ removes the pressure and turns the system into an evolution equation for $u$ alone,

$$\partial_t u + P\left[(u\cdot\nabla)u\right] = \nu\Delta u, \qquad \nabla\cdot u = 0, \qquad u(0) = u_0 .$$

For smooth solutions that decay fast enough, multiplying by $u$ and integrating by parts gives the energy identity

$$\frac12\|u(t)\|_{L^2}^2 + \nu\int_0^t\|\nabla u(s)\|_{L^2}^2\,ds = \frac12\|u_0\|_{L^2}^2 ,$$

because $\int u\cdot(u\cdot\nabla)u\,d^3x = 0$ for divergence-free $u$. For large data in three dimensions this is essentially the only global a priori bound available.

The equations are invariant under the scaling $u_\lambda(x,t) = \lambda u(\lambda x,\lambda^2 t)$, $p_\lambda(x,t) = \lambda^2 p(\lambda x,\lambda^2 t)$. A norm is called critical if it is invariant under this scaling. In three dimensions $\|u\|_{L^3}$ and the $\dot H^{1/2}$ norm are critical, while the energy controls only a supercritical quantity. This mismatch is the basic reason the regularity problem is open: the known bound becomes weaker at small scales, where a singularity would form.

What is known:

• Leray proved that for every divergence-free $u_0 \in L^2$ there exists a global weak solution satisfying the energy inequality (with $\le$ in place of $=$). Whether these Leray-Hopf solutions are unique and smooth is open.
• For initial data in a critical space such as $\dot H^{1/2}$ there is a unique solution, smooth for $t > 0$, on a short time interval, and for all time if the data are small in that space.
• In two dimensions smooth solutions exist for all time, because the vorticity is a scalar that satisfies a maximum principle.
• Buckmaster and Vicol showed that weak solutions in a class weaker than Leray-Hopf are not unique.

The Millennium Prize problem asks whether, for every smooth divergence-free $u_0$ with suitable decay (or on $\mathbb T^3$), a smooth solution exists for all $t > 0$, or whether a solution can blow up in finite time.

Setting $\nu = 0$ gives the Euler equations. Arnold showed that these are the geodesic equations of the $L^2$ metric on the group of volume-preserving diffeomorphisms of the fluid domain, so ideal fluid flow is geodesic motion on an infinite-dimensional Lie group. The viscous term breaks this Hamiltonian structure and makes the system dissipative.

Why is it interesting?

The Navier-Stokes equations describe one of the most ubiquitous features of our world: the flow of fluids like water. They are used to study ocean currents, the turbulence in the wake of an airplane or the flow of blood.

In addition, they are mathematically extremely interesting and there is a Millennium Prize Problem with a $1 million reward for the first person who can prove rigorously that smooth solutions exist for all times in three dimensions, or that they can break down.

The Navier-Stokes equations are extremely important for modern transport. Not only designers of ships use them, but also aircraft and car engineers use it to make computer simulations to test the aerodynamics of objects.

In addition, the Navier-Stokes equation is used in medical research to calculate blood flow. This is essential, for example, during a coronary bypass operation, where the blood flow must be precisely controlled.

Another application is in climate models. These depend crucially on the flow of oceans and the atmosphere, which both can be modeled thanks to the Navier-Stokes equation.

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus