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The Navier-Stokes equations describe fluid flows in continuum mechanics.

## When to Use the Tag and aims of this description

Use the tag when asking questions about fluid flows as modeled by the Navier-Stokes equations. Hopefully, this description may give a basis for unified notation in discussions. Perhaps it will even help people to formulate questions in a clearer way.

## Introduction

The Navier-Stokes equations model fluid flows based on the hypothesis: the molecular nature of matter is ignored. This motivates the use of differential equations to express basic mechanical principles. When reasoning in terms of "particles" in this context, one should understand "a small amount" of matter (much larger than the size of molecules, but still small enough for them to be "infinitesimal" with respect to the differential mathematics involved), not molecules.

In what follows, bold symbols denote vectors (2D or 3D) and the usual differential operators are used without explanation. The following quantities are used throughout:

• $\boldsymbol{u}$: the velocity field,
• $p$: the pressure field,
• $\rho$: the density of the fluid,
• $E$: the total energy of the flow,
• $\boldsymbol{q}$: the heat flux,
• $\sigma$: the Cauchy stress tensor, expressing the internal forces that neighbouring particles of fluid exert upon each other,
• $\mu$: the dynamic viscosity (assumed constant throughout),
• $\boldsymbol{f}$: external (volumic) forcing term (for example, the acceleration of gravity $\boldsymbol{g}$).

## General formulation

We need to respect three principles:

• Mass conservation: no matter is created nor destroyed,
• The rate of change of momentum of a fluid particle is equal to the force applied to it (Newton's second law)
• Energy conservation: it is neither created nor destroyed.

In all generality, these may be expressed as:

Mass conservation: $\partial_t \rho + \nabla \cdot (\rho \boldsymbol{u}) = 0$

Balance of momentum: $\rho (\partial_t \boldsymbol{u} + (\boldsymbol{u}\cdot\nabla) \boldsymbol{u}) = \nabla \cdot \sigma + \rho \boldsymbol{f}$

Energy conservation: $\partial_t E + \nabla \cdot (\boldsymbol{u} E + \sigma \boldsymbol{u} - \boldsymbol{q}) = 0$

## Incompressible fluids

In a context where we ignore heat phenomena (isothermal fluid), assume constant density of the fluid as well as a linear relation between stress and strain, $\sigma = -p \mathbb{I} + \mu ( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T)$ (Newtonian fluid), the important conservation laws are those for mass and momentum (i.e., Newton's second law), which are given by

Mass conservation: $\nabla \cdot \boldsymbol{u} = 0$

Balance of momentum: $\rho (\partial_t \boldsymbol{u} + (\boldsymbol{u}\cdot \nabla)\boldsymbol{u}) = -\nabla p + \mu \Delta \boldsymbol{u} + \rho \boldsymbol{f}$

The pressure acts as a means to enforce the incompressibility (divergence-free) condition represented by the mass conservation equation; it does not have the same meaning as in the compressible case. Energy is not conserved in this context, it is dissipated by the viscous nature of the fluid (internal friction) and lost as heat (which we don't "track" in this context).

These are perhaps the most commonly encountered form of the Navier-Stokes equations. They are famously the subject of a Clay Mathematics Institute Millennium Prize.

## Compressible fluids

When the fluid is compressible, the density becomes a field to be solved for, and we need an additional equation for the system, provided by the energy conservation principle. The exact form depends on the nature of the fluid, more exactly it's thermodynamic behaviour.

Mass conservation: $\partial_t \rho + \nabla \cdot (\rho \boldsymbol{u}) = 0$

Balance of momentum: $\rho (\partial_t \boldsymbol{u} + (\boldsymbol{u}\cdot \nabla)\boldsymbol{u}) = -\nabla p + \mu \Delta \boldsymbol{u} + (\mu/3 + \mu^v)\nabla(\nabla\cdot \boldsymbol{u})+ \rho \boldsymbol{f}$

where $\mu^v$ is the bulk viscosity coefficient.

Conservation of energy may be expressed in various ways depending on the fluid, a general discussion would require to delve into thermodynamic considerations which are outside the scope of this article. [Simple example?]

## Remarks

As in all physical problems, to obtain a unique and physically reasonable solution one must know the initial conditions and the conditions at all boundaries. An example of boundary conditions are the noslip boundary conditions, which require the fluid to adhere to the boundary: $\boldsymbol{u}\vert_{\text{boundary}} = \boldsymbol{v}_{\text{boundary}}$

The equations above are expressed in physical dimensions. It is possible to rescale time and space and normalize the velocity and pressure fields in a number of ways to get rid of the physical constants or to make a specific new one appear. The best known formulation involves the Reynolds number: $$\boldsymbol{u}(\boldsymbol{x},t), p(\boldsymbol{x},t) \mapsto U~\boldsymbol{u}(\boldsymbol{x}/L,~t~U/L),\rho U^2~p (\boldsymbol{x}/L,~t~U/L)$$ with $L, U$ a reference length and velocity (of a moving body, for example). This leads to writing the balance of momentum equation for incompressible flow (neglecting the forcing term) as: $$\partial_t \boldsymbol{u} + (\boldsymbol{u}\cdot \nabla)\boldsymbol{u} = -\nabla p + \frac{1}{\mathrm{Re}} \Delta \boldsymbol{u}$$ where $$\mathrm{Re} = \frac{\rho L U}{\mu}$$ is the Reynolds number, expressing the ratio of inertial to viscous effects. The higher the number, the bigger the influence of inertial effects. In the infinite Reynolds number limit, we recover the Euler equations. High Reynolds numbers flows tend to exhibit .

## Prerequisites to Navier-Stokes

Phys: Newtonian Mechanics; Classical Mechanics; Continuum Mechanics; ...

Math: Partial Differential Equations (PDE); ...

## Recommended books

Batchelor, G.K., An introduction to fluid dynamics, Cambridge University Press (1967)

Chorin, A.J. and Marsden, J.E., A mathematical introduction to fluid mechanics, Springer (1993)