## PHY454H1S Continuum Mechanics. Lecture 10: Navier-Stokes equation. Taught by Prof. K. Das.

Posted by peeterjoot on February 11, 2012

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Review. Newtonian fluid.

Reading: section 6.* from [1].

We stated the model for a newtonian fluid

and started considering conservation of mass with a volume through an area element . For the rate of change of mass *flowing out of the volume * is

Application of Green’s theorem, for a fixed (in time) volume produces

or in differential form for an infinitesimal volume

Expanding out the divergence term using

For an incompressible fluid

so the conservation of mass equality relation takes the form

# Conservation of momentum.

In classical mechanics we have

our analogue here is found in terms of the stress tensor

Here is the force per unit volume. With body forces we have

where is an external force per unit volume. Observe that , through the constituative relation, includes both contributions of linear displacement and the vorticity component.

From the constitutive relation 2.1, we have

Observe that the term

is the component of , whereas

is the component of .

We have therefore that

or in vector notation

We can expand this a bit more writing our velocity differential

Considering rates

In vector notation we have

Newton’s second law 3.13 now becomes

This is the Navier-Stokes equation. Observe that we have an explicitly non-linear term

something we don’t encounter in most classical mechanics. The impacts of this non-linear term are very significant and produce some interesting effects.

## Incompressible fluids.

Incompressibility was the condition

so the Navier-Stokes equation takes the form

## Boundary value conditions.

In order to solve any sort of PDE we need to consider the boundary value conditions. Consider the interface between two layers of liquids as in figure (\ref{fig:continuumL9:continuumL10fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL10fig1}

\caption{Rocker tank with two viscosity fluids.}

\end{figure}

Also found an illustration of this in fig 1.13 of white’s text online.

We see the fluids sticking together at the boundary. This is due to matching of the tangential velocities at the interface.

# References

[1] D.J. Acheson. *Elementary fluid dynamics*. Oxford University Press, USA, 1990.

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