Navier-Stokes equations

In Mechanical of the fluids, the Navier-Stokes equations are partial derivative equations non-linear which describe the movement of the fluids in the Approximation of the continuous mediums. They control for example the movements of the air of the atmosphere, the currents oceanic, the water run-off in a pipe, and many other phenomena of flow of fluids. They are named according to two physicists of the 19th century, Claude Navier and George Stokes. Let us note that it is possible to show the Navier-Stokes equations starting from the equation of Boltzmann.

General formula for a fluid made up of only one chemical species

There exists many forms of the Navier-Stokes equations. We will present only some of them. These forms also depend on the notations used. Thus, there exist several equivalent ways to express the differential operators.

The differential formulation of these equations is the following one:

• Equation of continuity (or equation of weight breakdown)

  $\backslash \; frac\; \{\backslash \; partial\; \backslash \; rho\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; (\backslash \; rho\; \backslash \; vec\; \{v\})\; =\; 0$

• Equation of partial assessment of the quantity of mouvement

  $\backslash \; frac\; \{\backslash \; \backslash \; left\; (\backslash \; rho\; \backslash \; vec\; \{v\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; +\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; rho\; \backslash \; vec\; \{v\}\; \backslash \; otimes\; \backslash \; vec\; \{v\}\; \backslash \; right)\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; p\; +\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{\backslash \; tau\}\}\; +\; \backslash \; rho\; \backslash \; vec\; \{F\}$

• Equation of assessment of the partial énergie$\backslash \; frac\; \{\backslash \; \backslash \; left\; (\backslash \; rho\; E\; \backslash \; right)\}\{\backslash \; partial\; T\}\; +\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; left\; \backslash ;\; \backslash \; left\; (\backslash \; rho\; E\; +\; p\; \backslash \; right)\; \backslash \; vec\; \{v\}\; \backslash ;\; \backslash \; right\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{\backslash \; tau\}\}\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; \backslash \; right)\; +\; \backslash \; rho\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; -\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; dowry\; \{Q\}\}\; +\; r$

In these equations:

• $t$ represents the Temps (unit IF: $\backslash \; rm\; s$);
• $\backslash \; rho$ indicates the Density fluid (unit IF: $kg.m^\; \{-\; 3\}$);
• $\backslash \; vec\; \{v\}\; =\; (v\_1,\; v\_2,\; v\_3)$ indicates the speed eulérienne of a fluid particle (unit IF: $\backslash \; rm\; m.s^\; \{-\; 1\}$);
• $p$ indicates the Pression (unit IF: $\backslash \; rm\; Pa$);
• $\backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{\backslash \; tau\}\}\; =\; \backslash \; left\; (\backslash \; tau\_\; \{I,\; J\}\; \backslash \; right)\; \_\; \{I,\; J\}$ is the Tenseur viscous constraints (unit IF: $\backslash \; rm\; Pa$);
• $\backslash \; vec\; \{F\}$ indicates the resultant of the mass forces being exerted in the fluid (unit IF: $\backslash \; rm\; N.kg^\; \{-\; 1\}$);
• $e$ is total energy per unit of mass (unit IF: $\backslash \; rm\; J.kg^\; \{-\; 1\}$);
• $\backslash \; vec\; \{\backslash \; dowry\; \{Q\}\}$ is the Heat flow lost by thermal conduction (unit IF: $\backslash \; rm\; J.m^\; \{-\; 2\}\; .s^\; \{-\; 1\}$);
• $r$ represents the voluminal loss of heat due to the radiation (unit IF: $\backslash \; rm\; J.m^\; \{-\; 3\}\; .s^\; \{-\; 1\}$).

Note:

• total energy can break up into energy interns $u$ and in kinetic energy selon

  $e\; =\; U\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ;\; \backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; =\; U\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ;\; v^2$
• the operator Nabla, $\backslash \; overrightarrow\; \{\backslash \; nabla\}\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_1\},\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_2\},\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_3\}\; \backslash \; right)$ in Cartesian coordinates, is an operator of space derivation of the 1st order. The operators Gradient , divergence and Laplacien can be written using this operator:
• $\backslash mathrm\{div\}\; \backslash ;\; \backslash \; vec\; \{has\}\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; \{has\}$;
• $\backslash overrightarrow\{\backslash mathrm\{grad\}\}\; \backslash ;\; With\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; A$;
• $\backslash \; Delta\; \backslash ;\; With\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; nabla\}\; has\; \backslash \; right)\; =\; \backslash \; nabla^2\; A$.

Expression in Cartesian coordinates

In Cartesian coordinates $(x\_1,\; x\_2,\; x\_3)$, the Navier-Stokes equations is written:

• Equation of continuity:

  $\backslash \; frac\; \{\backslash \; partial\; \backslash \; rho\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_i\}\; (\backslash \; rho\; v\_i)\; =\; 0$

• Equation of assessment of the momentum ($j\; =\; 1,2,3$)

  $\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; rho\; v\_j\; \backslash \; right)\}\{\backslash \; partial\; T\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_i\}\; \backslash \; left\; (\backslash \; rho\; v\_i\; v\_j\; \backslash \; right)\; =\; -\; \backslash \; frac\; \{\backslash \; partial\; p\}\; \{\backslash \; partial\; x\_j\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; tau\_\; \{I,\; J\}\}\; \{\backslash \; partial\; x\_i\}\; +\; \backslash \; rho\; f\_j$

• Equation of assessment of the partial énergie

  $\backslash \; frac\; \{\backslash \; \backslash \; left\; (\backslash \; rho\; E\; \backslash \; right)\}\{\backslash \; partial\; T\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_i\}\; \backslash \; left\; \backslash ;\; \backslash \; left\; (\backslash \; rho\; E\; +\; p\; \backslash \; right)\; v\_i\; \backslash ;\; \backslash \; right\; =\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; sum\_\; \{j=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_i\}\; \backslash \; left\; (\backslash \; tau\_\; \{I,\; J\}\; v\_j\; \backslash \; right)\; +\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; rho\; f\_i\; v\_i\; -\; \backslash \; sum\_\; \{i=1\}\; ^3\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; dowry\; \{Q\}\; \_i\}\; \{\backslash \; partial\; x\_i\}\; +\; r$

Newtonian fluid, assumption of Stokes

At first approximation, for many usual fluids like water and the air, the Tenseur of the viscous constraints is proportional to the symmetrical part of the tensor of the rates of deformation (assumption of Newton) and the heat flow is proportional to the gradient of the temperature (Fourier analysis), i.e.

$\backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{\backslash \; tau\}\}\; =\; \backslash \; driven\; \backslash \; left\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; otimes\; \backslash \; vec\; \{v\}\; \backslash \; right)\; +\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; otimes\; \backslash \; vec\; \{v\}\; \backslash \; right)\; ^t\; \backslash \; right\; +\; \backslash \; eta\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; \backslash \; right)\; \backslash ;\; \backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{I\}\}$

$\backslash \; vec\; \{\backslash \; dowry\; \{Q\}\}\; =\; -\; \backslash \; lambda\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; T$

where:

• $\backslash \; mu$ indicates the dynamic Viscosité fluid (unit IF: $\backslash \; rm\; Po$ (One tenth of a poise), $\backslash \; rm\; 1\; Po=\; 1\; Pa.s$);
• $\backslash \; eta$ indicates the viscosity of volume of the fluid (unit IF: $\backslash \; rm\; Po$);
• $\backslash \; overrightarrow\; \{\backslash \; overrightarrow\; \{I\}\}$ indicates the tensor unit;
• $\backslash \; lambda$ indicates the thermal conductivity of the fluid (unit IF: $\backslash \; rm\; J.K^\; \{-\; 1\}\; .m^\; \{-\; 1\}\; .s^\; \{-\; 1\}$);
• $T$ indicates the temperature (unit IF: $\backslash \; rm\; K$).

The whole of the fluids for which this assumption is checked are called fluid Newtonian . One generally associates the to them assumption of Stokes :

$3\; \backslash \; eta\; +\; 2\; \backslash \; driven\; =\; 0~$.

This assumption appears completely false but is usually used in aeronautics.

Note::

Many fluids, such as polymers, the heavy hydrocarbons, honey, or the paste of toothpaste, do not check these assumptions. Science charged to study the relations enters constraint and deformation for such fluids are called the Rhéologie.

Expression for the compressible flows of fluids

The flow of a fluid is known as incompressible when one can neglect his variations of density during time. This assumption is checked when the Mach number $Ma$ is weak . In general, one considers the incompressible flow when . In the contrary case, i.e. for a compressible flow , one associates to close the system a equation of state of the fluid, form

$f\; (p,\; \backslash \; rho,\; T)\; =\; 0\; \backslash ,$

For a Perfect gas, this equation of state is written

$p\; =\; \backslash \; rho\; \backslash \; frac\; \{R\}\; \{M\}\; T$

where $R$ indicates the Constante perfect gases and $M$ the molar Masse of the fluid.

Expression for the incompressible flows of fluids

For a Newtonian viscous fluid and when the flow is incompressible, the equation of energy is uncoupled from the equations of continuity and momentum, i.e. one can determine the speed and the pressure independently of the equation of energy. The expression of the equations of continuity and momentum are simplified considerably. One obtains then

• Equation of continuity called then equation of incompressibilité

  $\backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; =\; 0$

• Equation of partial assessment of the quantity of mouvement

  $\backslash \; frac\; \{\backslash \; \backslash \; vec\; \{v\}\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; left\; (\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; right)\; \backslash \; vec\; \{v\}\; =\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; rho\}\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; p\; +\; \backslash \; naked\; \backslash \; nabla^2\; \backslash \; vec\; \{v\}\; +\; \backslash \; vec\; \{F\}$

where $\backslash \; naked\; =\; \backslash \; tfrac\; \{\backslash \; driven\}\; \{\backslash \; rho\}$ indicates the kinematic Viscosité fluid (unit IF: $\backslash \; rm\; m^2.s^\; \{-\; 1\}$)

Interpretation

The equation of momentum is the equivalent of the fundamental relation of dynamics (also called second law of Newton): $\backslash \; Sigma\; \backslash \; vec\; \{F\}\; =\; m\; \backslash \; vec\; \{has\}$.

In this formula, one sees appearing three types of forces :

• forces of pressure , specific of the mechanics of the fluids.
• forces of viscosity . Note that the second term containing the viscosity of volume disappears if the fluid is incompressible.
• Of others mass forces , which can be forces of Gravité ($\backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{G\}$) or electromagnetic ($\backslash \; scriptstyle\; \backslash \; vec\; \{F\}\; =\; \backslash \; frac\; \{Q\}\; \{\backslash \; rho\}\; (\backslash \; vec\; \{E\}\; +\; \backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})$). In the case of the Revolved, this term represents the weight of a fluid particle and represents the Poussée of Archimedes. Indeed, when the fluid is at rest, one finds immediately the equation of the Hydrostatique:

$\backslash \; vec\; \{\backslash \; nabla\}\; p=\; \backslash \; rho\; \backslash \; vec\; \{G\}$

The expression of the acceleration is more delicate and is expressed in two manners

• the Lagrangian Description consists in following the particles of fluids. Acceleration is the particulate derivative speed: $\backslash \; tfrac\; \{\backslash \; mathrm\; D\; \backslash \; vec\; \{v\}\}\; \{\backslash \; mathrm\; Dt\}$.
• the Description eulérienne consists in being placed in a fixed position. Acceleration is then the sum of derived the partial speed $\backslash \; tfrac\; \{\backslash \; partial\; \backslash \; vec\; \{v\}\}\; \{\backslash \; partial\; T\}$ (local acceleration) and a advectif term $(\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; nabla\})\; \backslash \; vec\; \{v\}$.

The resolution of the Navier-Stokes equation is extremely difficult. It remains one of the great enigmas mathematics unsolved to date. Besides the American foundation Clay Mathematics Institute offers 1 million dollars to that which will be able to tap all its secrecies to him.

To complexity inherent in the partial derivative equations are added that of the non-linearity introduced by the term of Advection of acceleration. Most of the time, one tries to solve a simplified version of the equation by eliminating one from these terms. For example, with weak Reynolds number, one can neglect the advectif term (flow of Stokes) and with strong Reynolds number, one frees oneself from viscosity (equation of Euler).

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