Chapter 7

# Navier-Stokes equation

## Complément :

A video (in french) about Navier - Stokes equation (with some fun science experiments)

## Fondamental : Surface Forces of viscosity

Studying a simple case in which the planes parallel to (Oxz) slide on each other (see figures).

This case is a good approximation of reality when the dimensions of the flow along (Ox) and (Oz) are big compared to the thickness of the fluid layer.

It is assumed that the velocity vector can be written as (see figure) :

Since , this flow can be that of an incompressible fluid.

Surface Forces of viscosity

Consider two fluid elements and , separated by the surface , of area and perpendicular to (Oy).

The shear force exerted by on , is tangent to .

In the case of the figure, the surface is faster than the surface and thus takes the fluid with it.

This force is :

• Proportional to the area of the surface .

• The same direction as if is an increasing function of .

• To a fluid called "Newtonian", the shearing force is a linear function of the derivative .

Finally, for unidirectional flow, such that , the force of tangential surface , known as shear force or viscosity force, which is exerted through a normal area of surface to (it is the force exerted by the upper layer on the lower layer) is expressed as :

The viscosity has the effect, in a unidirectional flow, to accelerate the slow elements and to slowdown fast elements.

It is therefore an internal transfer of momentum, which has the characteristics of diffusion of momentum.

The coefficient , called coefficient of viscosity of the fluid may be, with a good approximation, considered as a constant characteristic of the fluid at a given temperature.

The unit for viscosity coefficient is the Poiseuille (symbol , equal to ).

Some examples (in normal conditions) :

 Corps pur Eau Air Glycérine Viscosité (Pl)

## Attention : Surface Forces of viscosity

For unidirectional flow, such that :

The force of a tangential surface , known as shear force or viscosity force, which is exerted through a normal area of surface to (it is the force exerted by the upper layer on the lower layer) is expressed so :

## Fondamental : Volume equivalence of viscous forces

Consider a "pad" of fluid of volume .

Volume equivalence of viscous forces

It is assumed that the velocity field can also be written as :

For such a field, the viscous forces are on the (Ox) axis and only the faces and are concerned.

We can write, applying the law of action and reaction, the resultant viscosity forces on the pad of fluid :

Either :

For a field of any speed but corresponding to an incompressible flow, we assume that this result can be generalized in the form :

Where we have introduced the laplacian of vector field .

is the volumetric density of the viscous forces ; it is only a mathematical equivalent because these forces apply only to the surface of a system.

## Attention : Volume equivalence of viscous forces

For a field of any speed but corresponding to an incompressible flow, the volume density of viscous forces is :

## Fondamental : Navier-Stokes equation

The Newton's second law applied to a particle of fluid, taking into account the viscosity force leads to the Navier – Stokes equation :

It is the Navier – Stokes' equation.

Or :

Let's recall that :

• Incompressible fluid :

• The condition for the speed in the fluid - solid interface is :

## Attention : Navier-Stokes equation

It is the Navier - Stokes.

Or :

## Complément : Newtonian liquids and non-Newtonian liquids

The flow of honey and of toothpaste :

The flow of honey and of toothpaste
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