# Navier-Stokes equations

In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example, they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.

Although the full, unsteady Navier-Stokes equations correctly describe nearly all flows of practical interest, they are too complex for practical solution in many cases and a special "reduced" form of the full equations is often used instead — these are the Reynolds-averaged Navier-Stokes (RANS) equations. The solution of the full steady Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows the Reynolds-averaged form of the equations are most commonly used. The RANS form of the equations introduce new terms that reflect the additional modeling of the small turbulent motions.

Solution of flow equations by numerical methods is called computational fluid dynamics.

It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a \$1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.

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## Basic assumptions

The equations are derived by considering the mass, momentum, and energy balances for an infinitesimal control volume.

Indeed NS equations assume that the fluid is a continuum. They are conservation equations over a control volume which we will call [itex]\Omega[itex], which keeps the portion of the fluid which is studied. Unfortunately, [itex]\Omega[itex] depends on time, and the conservation laws need then to be modified so as to be valid on arbitrary control volumes.

### The substantive derivative

main article: substantive derivative.

Before going into the details of the Navier-Stokes equations, one must first define an operator:

[itex]\frac{D}{Dt}(\cdot)[itex]

This is the usual time derivative, but when following a particle of the fluid. The full form is:

[itex]\frac{\partial(\cdot)}{\partial t} + \mathbf{v}\nabla(\cdot)[itex]

The integral form of a conservation law L over the control volume [itex]\Omega[itex] is:

[itex]\frac{d}{dt}\int_\Omega \mathbf{L}\; d\Omega = 0[itex]

Using the particular derivative, one can swap the [itex]\frac{d}{dt}[itex] and [itex]\int[itex] operators:

[itex]\int_\Omega \frac{D}{Dt}\mathbf{L}\; d\Omega = 0\quad \forall \Omega[itex]

As this expression is valid for all [itex]\Omega[itex], it simplifies to:

[itex]\frac{D}{Dt}\mathbf{L} = 0[itex]

### Conservation laws

Main article: conservation laws.

the NS equations are derived from conservation laws using the transformation described above and need to be closed using state laws.

Over a control volume, using the transformation described above, the following quantities are deemed conserved:

#### Equation of continuity

Conservation of mass is written:

[itex]\frac{D\rho}{D t} + \rho\nabla\mathbf{v}= 0[itex]

where

[itex]\rho[itex] is the density of the fluid.

In the case of an incompressible fluid [itex]\rho[itex] is not a function of time; the equation is reduced to:

[itex]\nabla\mathbf{v} = 0[itex]

#### Conservation of momentum

Conservation of momentum is written:

[itex]\frac{\partial\rho\mathbf{v}}{\partial t} + \nabla(\rho\mathbf{v}\otimes\mathbf{v}) = \sum\mathbf{F}[itex]

Note that [itex]\mathbf{v}\otimes\mathbf{v}[itex] is a tensor, the [itex]\otimes[itex] representing the tensor product.

We can simplify it further, this becomes:

[itex]\rho\frac{D\mathbf{v}}{D t}=\sum\mathbf{F}[itex]

In which we recognise the usual F=ma.

## The equations

### General form

#### The form of the equations

The general form of the Navier-Stokes equations is:

[itex]\rho\frac{D\mathbf{v}}{D t} = \nabla \mathbb{P} + \rho\mathbf{f}[itex]

For the conservation of momentum. The tensor [itex]\mathbb{P}[itex] represents the surface forces applied on a fluid particle. In general, we have the form:

[itex]\nabla \mathbb{P} = \begin{pmatrix}

\frac{\partial \sigma_{xx}}{\partial x} & \frac{\partial\tau_{yx}}{\partial y} & \frac{\partial\tau_{zx}}{\partial z} \\ \\ \frac{\partial\tau_{xy}}{\partial x} & \frac{\partial \sigma_{yy}}{\partial y} & \frac{\partial\tau_{zy}}{\partial z} \\ \\ \frac{\partial\tau_{xz}}{\partial x} & \frac{\partial\tau_{yz}}{\partial y} & \frac{\partial\sigma_{zz}}{\partial z}\end{pmatrix} [itex]

To which we add the continuity equation:

[itex]\frac{D\rho}{Dt} + \rho\nabla\mathbf{v} = 0[itex]

The nature of the diagonal of [itex]\mathbb{P}[itex] is known, it is the gradient of pressure, thus:

[itex]\nabla \mathbf{\sigma} = \nabla p[itex]

where

p is the pressure

finally, we have:

[itex]\rho\frac{D\mathbf{v}}{D t} = \nabla p + \nabla \mathbb{T} + \rho\mathbf{f}[itex]

where the components of [itex]\mathbb{T}[itex] are the [itex]\tau[itex] of [itex]\mathbb{P}[itex].

#### The closure problem

Those equations are incomplete. To complete them, one must make hypotheses on the form of [itex]\mathbb{P}[itex]. In the case of a perfect fluid [itex]\tau[itex] components are nil, for example. Those equations used to complete the set are equations of state.

The variables to be solved for are the velocity components, the fluid density, static pressure, and temperature. The flow is assumed to be differentiable and continuous, allowing these balances to be expressed as partial differential equations. The equations can be converted to Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study.

The components of [itex]\mathbb{P}[itex] are the constraints on an infinitesimal element of fluid. They represent the normal and shear constraints. [itex]\mathbb{P}[itex] is therefore by definition antisymmetric.

So-called non-Newtonian fluids are simply fluids where this tensor has no special properties allowing for special solutions of the equations.

### Special forms

Those are certain usual simplifications of the problem for which sometimes solutions are known.

#### Newtonian fluids

Main article Newtonian fluids.

In Newtonian fluids the following assumption holds:

[itex]\tau_{ij}=\mu\frac{\partial v_i}{\partial x_j}[itex]

where

[itex]\mu[itex] is the viscosity of the fluid.

#### Bingham fluids

Main article Bingham plastic.

In Bingham fluids, we have something slightly different:

[itex]\tau_{ij}=\tau_0 + \mu\frac{\partial v_i}{\partial x_j},\;\frac{\partial v_i}{\partial x_j}>0 [itex]

Those are fluids capable of bearing some shear before they start flowing. An example is tooth paste.

#### Power-law fluid

Main article Power-law fluid.

It is an idealised fluid for which the shear stress, [itex]\tau[itex], is given by

[itex]\tau = K \left( \frac {\partial u} {\partial y} \right)^n [itex]

This form is useful for approximating all sorts of general fluids.

#### Incompressible fluids

Main article Incompressible fluids.

The Navier-Stokes equations are

[itex]

\rho\frac{Du_i}{Dt}=\rho f_i-\frac{\partial p}{\partial x_i}+\frac{\partial}{\partial x_j}\left[ 2\mu\left(e_{ij}-\Delta\delta_{ij}/3\right)\right][itex] for momentum conservation and

[itex]\frac{\partial\rho u_i}{\partial x_i}=0[itex]

where

[itex]\rho[itex] is the density,
[itex]u_i[itex] ([itex]i=1,2,3[itex]) the three components of velocity,
[itex]f_i[itex] body forces (such as gravity),
[itex]p[itex] the pressure,
[itex]\mu[itex] the dynamic viscosity, of the fluid at a point;
[itex]e_{ij}=\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)[itex];
[itex]\Delta=e_{ii}[itex] is the divergence,
[itex]\delta_{ij}[itex] is the Kronecker delta.

If [itex]\mu[itex] is uniform over the fluid, the momentum equation above simplifies to

[itex]

\rho\frac{Du_i}{Dt}=\rho f_i-\frac{\partial p}{\partial x_i} +\mu \left(

 \frac{\partial^2u_i}{\partial x_i\partial x_j}+                                                                                                                                         \frac{1}{3}\frac{\partial\Delta}{\partial x_i}\right)


[itex]

(if [itex]\mu=0[itex] the resulting equations are known as the Euler equations; there, the emphasis is on compressible flow and shock waves).

If now in addition [itex]\rho[itex] is assumed to be constant we obtain the following system:

[itex] \rho \left({\partial v_x \over \partial t}+ v_x {\partial v_x \over \partial x}+ v_y {\partial v_x \over \partial y}+ v_z {\partial v_x \over \partial z}\right)= \mu \left[{\partial^2 v_x \over \partial x^2}+{\partial^2 v_x \over \partial y^2}+{\partial^2 v_x \over \partial z^2}\right]-{\partial p \over \partial x} +\rho g_x[itex]
[itex] \rho \left({\partial v_y \over \partial t}+ v_x {\partial v_y \over \partial x}+ v_y {\partial v_y \over \partial y}+ v_z {\partial v_y \over \partial z}\right)= \mu \left[{\partial^2 v_y \over \partial x^2}+{\partial^2 v_y \over \partial y^2}+{\partial^2 v_y \over \partial z^2}\right]-{\partial p \over \partial y} +\rho g_y[itex]
[itex] \rho \left({\partial v_z \over \partial t}+ v_x {\partial v_z \over \partial x}+ v_y {\partial v_z \over \partial y}+ v_z {\partial v_z \over \partial z}\right)= \mu \left[{\partial^2 v_z \over \partial x^2}+{\partial^2 v_z \over \partial y^2}+{\partial^2 v_z \over \partial z^2}\right]-{\partial p \over \partial z} +\rho g_z[itex]

Continuity equation (assuming incompressibility):

[itex] {\partial v_x \over \partial x}+{\partial v_y \over \partial y}+{\partial v_z \over \partial z}=0 [itex]
Simplified version of the N-S equations. Adapted from Incompressible Flow, second edition by Ronald Panton

Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics may be a more appropriate approach. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.

## References

• Inge L. Rhyming Dynamique des fluides, 1991 PPUR

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