What Are the Navier-Stokes Equations? (2023)

Fluid Flow: Conservation of Momentum, Mass, and Energy Navier-Stokes Equations

What Are the Navier-Stokes Equations?

The Navier-Stokes equations govern the motion of fluids and can be seen as Newton's second law of motion for fluids. In the case of a compressible Newtonian fluid, this yields

What Are the Navier-Stokes Equations? (1)

where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, and μ is the fluid dynamic viscosity. The different terms correspond to the inertial forces (1), pressure forces (2), viscous forces (3), and the external forces applied to the fluid (4). The Navier-Stokes equations were derived by Navier, Poisson, Saint-Venant, and Stokes between 1827 and 1845.

These equations are always solved together with the continuity equation:

What Are the Navier-Stokes Equations? (2)

The Navier-Stokes equations represent the conservation of momentum, while the continuity equation represents the conservation of mass.

How Do They Apply to Simulation and Modeling?

These equations are at the heart of fluid flow modeling. Solving them, for a particular set of boundary conditions (such as inlets, outlets, and walls), predicts the fluid velocity and its pressure in a given geometry. Because of their complexity, these equations only admit a limited number of analytical solutions. It is relatively easy, for instance, to solve these equations for a flow between two parallel plates or for the flow in a circular pipe. For more complex geometries, however, the equations need to be solved.

Example: Laminar Flow Past a Backstep

In the following example, we numerically solve the Navier-Stokes equations (hereon also referred to as "NS equations") and the mass conservation equation in a computational domain. These equations need to be solved with a set of boundary conditions:

(Video) The million dollar equation (Navier-Stokes equations)

The fluid velocity is specified at the inlet and pressure prescribed at the outlet. A no-slip boundary condition (i.e., the velocity is set to zero) is specified at the walls. The numerical solution of the steady-state NS (the time-dependent derivative in (1) is set to zero) and continuity equations in the laminar regime and for constant boundary conditions is as follows:

Velocity magnitude profile and streamlines. Velocity magnitude profile and streamlines.

Pressure field. Pressure field.

Different Flavors of the Navier-Stokes Equations

Depending on the flow regime of interest, it is often possible to simplify these equations. In other cases, additional equations may be required. In the field of fluid dynamics, the different flow regimes are categorized using a nondimensional number, such as the Reynolds number and the Mach number.

About the Reynolds and Mach Numbers

The Reynolds number, Re=ρUL/μ, corresponds to the ratio of inertial forces (1) to viscous forces (3). It measures how turbulent the flow is. Low Reynolds number flows are laminar, while higher Reynolds number flows are turbulent.

The Mach number, M=U/c, corresponds to the ratio of the fluid velocity, U, to the speed of sound in that fluid, c. The Mach number measures the flow compressibility.

In the flow past a backstep example, Re = 100 and M = 0.001, which means that the flow is laminar and nearly incompressible. For incompressible flows, the continuity equation yields:

What Are the Navier-Stokes Equations? (6)

Because the divergence of the velocity is equal to zero, we can remove the term:

What Are the Navier-Stokes Equations? (7)

(Video) Navier-Stokes Equations - Numberphile

from the viscous force term in the NS equations in the case of incompressible flow.

In the following section, we examine some particular flow regimes.

Low Reynolds Number/Creeping Flow

When the Reynolds number is very small (Re1) , the inertial forces (1) are very small compared to the viscous forces (3) and they can be neglected when solving the NS equations. To illustrate this flow regime, we will look at pore-scale flow experiments conducted by Arturo Keller, Maria Auset, and Sanya Sirivithayapakorn of the University of California, Santa Barbara.

Graphic showing the boundary conditions in the pore-scale flow experiment. Graphic showing the boundary conditions in the pore-scale flow experiment.

About the Experiment

The domain of interest covers 640 μm by 320 μm. Water moves from right to left across the geometry. The flow in the pores does not penetrate the solid part (gray area in the figure above). The inlet and outlet fluid pressures are known. Since the channels are at most 0.1 millimeters in width and the maximum velocity is lower than 10-4 m/s, the maximum Reynolds number is less than 0.01. Because there are no external forces (gravity is neglected), the force term (4) is also equal to zero.

Therefore, the NS equations reduce to:

What Are the Navier-Stokes Equations? (9)

Modeling the Experiment

The below plot shows the resulting velocity contours and pressure field (height).

The flow is driven by a higher pressure at the inlet than at the outlet. These results show the balance between the pressure force (2) and the viscous forces (3) in the NS equations. Along the thinner channels, the impact of viscous diffusion is larger, which leads to higher pressure drops.

(Video) Description and Derivation of the Navier-Stokes Equations

High Reynolds Number/Turbulent Flow

In engineering applications where the Reynolds number is very high, the inertial forces (1) are much larger than the viscous forces (3). Such turbulent flow problems are transient in nature; a mesh that is fine enough to resolve the size of the smallest eddies in the flow needs to be used.

Running such simulations using the NS equations is often beyond the computational power of most of today's computers and supercomputers. Instead, we can use a Reynolds-Averaged Navier-Stokes (RANS) formulation of the Navier-Stokes equations, which averages the velocity and pressure fields in time.

These time-averaged equations can then be computed in a stationary way on a relatively coarse mesh, thus drastically reducing the computing power and time required for such simulations (typically a few minutes for two-dimensional flow and a few minutes to a few days for three-dimensional flow).

The Reynolds-Averaged Navier-Stokes (RANS) formulation is as follows:

What Are the Navier-Stokes Equations? (11)

Here, U and P are the time-averaged velocity and pressure, respectively. The term μT represents the turbulent viscosity, i.e., the effects of the small-scale time-dependent velocity fluctuations that are not solved for by the RANS equations.

The turbulent viscosity, μT, is evaluated using turbulence models. The most common one is the k-ε turbulence model (one of many RANS turbulence models). This model is often used in industrial applications because it is both robust and computationally inexpensive. It consists of solving two additional equations for the transport of turbulent kinetic energy k and turbulent dissipation ϵ.

To illustrate this flow regime, let us look at the flow in a much larger geometry than the por-scale flow: a typical ozone purification reactor. The reactor is about 40 meters long and looks like a maze with partial walls or baffles that divide the space into room-sized compartments. Based on the inlet velocity and diameter, which in this case correspond to 0.1 m/s and 0.4 meters respectively, the Reynolds number is 400,000. This model is solved for the time-averaged velocity, U; pressure, P; turbulent kinetic energy, k; and turbulent dissipation, ϵ:

The results show the flow patterns, flow velocity, and turbulent viscosity μT. The results show the flow patterns, flow velocity, and turbulent viscosity μT.

Flow Compressibility

The flow compressibility is measured by the Mach number. All the previous examples are weakly compressible, meaning that the Mach number is lower than 0.3.

(Video) A Brief History of the Navier-Stokes Equations

Incompressible Flow

When the Mach number is very low, it is OK to assume that the flow is incompressible. This is often a good approximation for liquids, which are much less compressible than gases. In that case, the density is assumed to be constant and the continuity equation reduces to ∇⋅u=0. The creeping flow example showing water flowing at a low speed through the porous media is a good example of incompressible flow.

Compressible Flow

In some cases, the flow velocity is large enough to introduce significant changes in the density and temperature of the fluid. These changes can be neglected for M<0.3. For M>0.3, however, the coupling between the velocity, pressure, and temperature field becomes so strong that the NS and continuity equations need to be solved together with the energy equation (the equation for heat transfer in fluids). The energy equation predicts the temperature in the fluid, which is needed to compute its temperature-dependent material properties.

Compressible flow can be laminar or turbulent. In the next example, we look at a high-speed turbulent gas flow in a diffuser (a converging and diverging nozzle).

The diffuser is transonic in the sense that the flow at the inlet is subsonic, but due to the contraction and the low outlet pressure, the flow accelerates and becomes sonic (M = 1) in the throat of the nozzle.

The results in these three plots show strong similarities, which confirms the strong coupling between the velocity, pressure, and temperature fields. After a short region of supersonic flow (M > 1), a normal shock wave brings the flow back to subsonic flow. This set-up has been studied in a number of experiments and numerical simulations by M. Sajben et. al. [1-6].

What Flow Regimes Cannot Be Solved by the Navier-Stokes Equations?

The Navier-Stokes equations are only valid as long as the representative physical length scale of the system is much larger than the mean free path of the molecules that make up the fluid. In that case, the fluid is referred to as a continuum. The ratio of the mean free path, λ, and the representative length scale, L, is called the Knudsen number, Kn=λ/L

(Video) Derivation of the Navier-Stokes Equations

The NS equations are valid for Kn<0.01. For 0.01<Kn<0.1, these equations can still be used, but they require special boundary conditions. For Kn>0.1, they are not valid. At the ambient pressure of 1 atm – for instance, the mean free path of air molecules – is 68 nanometers. The characteristic length of your model should therefore be larger than 6.8 μm for the NS equations to be valid.

Published: January 15, 2015
Last modified: February 22, 2017


  1. M. Sajben, J.C. Kroutil, and C.P. Chen, “A High-Speed Schlieren Investigation of Diffuser Flows with Dynamic Distortion”, AIAA Paper 77-875, 1977.
  2. T.J. Bogar, M. Sajben, and J.C. Kroutil, “Characteristic Frequencies of Transonic Diffuser Flow Oscillations,” AIAA Journal, vol. 21, no. 9, pp. 1232–1240, 1983.
  3. J.T. Salmon, T.J. Bogar, and M. Sajben, “Laser Doppler Velocimetry in Unsteady, Separated, Transonic Flow”, AIAA Journal, vol. 21, no. 12, pp. 1690–1697, 1983.
  4. T. Hsieh, A.B. Wardlaw Jr., T.J. Bogar, P. Collins, and T. Coakley, “Numerical Investigation of Unsteady Inlet Flowfields,” AIAA Journal, vol. 25, no. 1, pp. 75–81, 1987.
  5. http://www.grc.nasa.gov/WWW/wind/valid/transdif/transdif01/transdif01.html
  6. http://www.grc.nasa.gov/WWW/wind/valid/transdif/transdif02/transdif02.html


What are the Navier-Stokes equations used for? ›

The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing.

What are the terms in the Navier-Stokes equation? ›

where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, and μ is the fluid dynamic viscosity. The different terms correspond to the inertial forces (1), pressure forces (2), viscous forces (3), and the external forces applied to the fluid (4).

What type of equation is the Navier-Stokes equation? ›

Navier-Stokes equation, in fluid mechanics, a partial differential equation that describes the flow of incompressible fluids. The equation is a generalization of the equation devised by Swiss mathematician Leonhard Euler in the 18th century to describe the flow of incompressible and frictionless fluids.

Is the momentum equation the Navier-Stokes equation? ›

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum equation.

Who actually solved the Navier-Stokes? ›

Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.

Has anyone solved the Navier-Stokes equation? ›

Due to their complexity, it is natural to wonder how they can be solved. The reality is that no analytical solutions exist to the Navier-Stokes equations in their most general form.

What are the 3 laws that are based on the Navier-Stokes equation? ›

This area of study is called Computational Fluid Dynamics or CFD. The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation.

Why hasn't Navier-Stokes been solved? ›

The Navier-Stokes equation is difficult to solve because it is nonlinear.

How many unknowns are in Navier-Stokes? ›

1.8 Navier-Stokes equations
Number of EquationsNumber of Unknowns
Navier-Stokes3 (symmetry)3

Is Navier-Stokes equation hyperbolic or parabolic? ›

The Navier-Stokes equations are the fundamental equations governing the motion of viscous fluid. Among the versions of these equations, we consider here the nonstationary Navier-Stokes equations for viscous incompressible fluid. The system of equations is a nonlinear parabolic equation.

Does Navier-Stokes have a solution? ›

It was concluded that there exist no smooth and physically reasonable solutions of the Navier-Stokes equation at a high Reynolds number (beyond laminar flow) [9]. As is well known, the flow of viscous incompressible fluid is governed by the Navier-Stokes equation, which is a Poisson equation.

What is not considered in Navier-Stokes equation? ›

Turbulence force is not considered in order to obtain Navier – Stokes equation from Reynolds equation. While all other given forces are considered.

Is the energy equation Navier-Stokes? ›

The Navier-Stokes equations follow the principle of conservation of the energy, momentum, and mass of a fluid flow. The energy equation of the Navier-Stokes system follows the energy conservation law, which equates the total energy of a system to the sum of work and heat added to the system.

Why is the Navier-Stokes problem important? ›

The Navier-Stokes equations are a family of equations that fundamentally describe how a fluid flows through its environment. Biomedical researchers use the equations to model how blood flows through the body, while petroleum engineers use them to reveal how oil is expected to flow through a well or pipeline.

Is Navier-Stokes equation a millennium problem? ›

The Navier-Stokes equations are among the Clay Mathematics Institute Millennium Prize problems, seven problems judged to be among the most important open questions in mathematics.

What are the 7 unsolved math problems called? ›

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

What's the hardest math equation? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

What is the hardest math problem in the universe? ›

Today's mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It's one of the seven Millennium Prize Problems, with $1 million reward for its solution.

What is the biggest math problem ever solved? ›

Mathematicians worldwide hold the Riemann Hypothesis of 1859 (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

What is the oldest unsolved math problem? ›

But he doesn't feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

What are the six unsolved math problems? ›

The problems consist of the Riemann hypothesis, Poincaré conjecture, Hodge conjecture, Swinnerton-Dyer Conjecture, solution of the Navier-Stokes equations, formulation of Yang-Mills theory, and determination of whether NP-problems are actually P-problems.

Can we use Navier-Stokes equation for turbulent flow? ›

This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon.

Which millennium problem is closest to solved? ›

To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton, upon whose work Perelman built.

What is the newest discovery in math? ›

New, Highly Tunable Composite Materials--With a Twist

June 14, 2022 — Mathematicians have found that they can design a range of composite materials from moiré patterns created by rotating and stretching one lattice relative to another. Their electrical and other ...

Is Navier-Stokes Eulerian or Lagrangian? ›

Thus, the Navier-Stokes equations should be and indeed they are always used in the Eulerian perspective.

Why is Navier-Stokes equation nonlinear? ›

The nonlinear term in Navier–Stokes equations of Equation (1.17) is the convection term, and most of the numerical difficulties and stability issues for fluid flow are caused by this term.

Why is Navier-Stokes pressure negative? ›

This is because the pressure gradient enters the Navier-Stokes equations and so it is pressure differences that drive the flow. So, in regions of separated flow, the low pressure inside that region will be relative to the lowest fixed pressure in your system and may well go negative.

What are the difficulties in solving of Navier Stokes equation? ›

The principal difficulty in solving the Navier–Stokes equations (a set of nonlinear partial differential equations) arises from the presence of the nonlinear convective term (V ·∇)V.

Are any of the Millennium problems been solved? ›

The only Millennium Problem that has been solved to date is the Poincare conjecture, a problem posed in 1904 about the topology of objects called manifolds.

Why is the Navier-Stokes equation so hard to solve? ›

The Navier-Stokes equation is difficult to solve because it is nonlinear. This word is thrown around quite a bit, but here it means something specific. You can build up a complicated solution to a linear equation by adding up many simple solutions.

How are Navier-Stokes equations used in CFD? ›

The energy equation of the Navier-Stokes system follows the energy conservation law, which equates the total energy of a system to the sum of work and heat added to the system. In CFD simulations, the Navier-Stokes energy equation provides the basic explanation of energy associated with the flow behavior.

Is Navier-Stokes equation valid for turbulent flow? ›

So, the equations themselves are absolutely valid for turbulent flows, because the equations themselves are continuous.

Which is the toughest equation in the world? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."


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