## PHY454H1S Continuum Mechanics. Lecture 9: Newtonian fluids. Mass conservation. Constitutive relation. Incompressible fluids. Taught by Prof. K. Das.

Posted by peeterjoot on February 10, 2012

# Reading

section 1.4 from [1]. FIXME: Probably more elsewhere too.

# Disclaimer.

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

# Review: Relative motion near a point in a fluid

Referring to figure (\ref{fig:continuumL9:continuumL9fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL9fig1}

\caption{velocity displacements at a fluid point.}

\end{figure}

we write

or in coordinate form

We can now split the components of the gradient of into symmetric and antisymmetric parts in the normal way

## The antisymmetric term (name?)

With

we introduce the dual vector

defined according to

With

we can write

In matrix form this becomes

For the special case , our displacement equation in vector form becomes

Let’s do a quick verification that this is all kosher.

All’s good in the world of signs and indexes.

## The symmetric term (strain tensor).

Now let’s look at the symmetric term. With the initial volume

and the final volume written assuming that we are working in our principle strain basis, we have (very much like the solids case)

So much like we expressed the relative change of volume in solids, we now can express the relative change of volume per unit time as

or

We identify the divergence of the displacement as the relative change in volume per unit time.

# Newtonian Fluids.

\begin{definition}

\emph{(Newtonian Fluids)}

A fluid for which the rate of strain tensor is linearly related to stress tensor.

\end{definition}

For such a fluid, the constitutive relation takes the form

where is called the isotropic pressure, and is the viscosity of the fluid.

For comparison, in solids we had

While we are allowing for rotation in the fluids () that we did not consider for solids, we now impose a requirement that the strain tensor trace is not a function of the fluid displacements, with

What is the physical justification for this? In words this was explained after class as the effect of rotation invariance with an attempt to measure the pressure at a given point in the fluid. It doesn’t matter what direction we place our pressure measurement device at a given fixed location in the fluid. Note that this doesn’t mean the pressure itself is constant. For example with a gravitational body force applied, our pressure will increase with depth in the fluid. Noting this provides a nice physical interpretation of the trace of the strain tensor.

Can we mathematically justify this explanation? We see above that we have

so we are in effect making the identification

or

The relative change in a differential volume element changes exponentially.

## Dimensions

Some examples

\begin{itemize}

\item

\item

\item

\end{itemize}

# Conservation of mass in fluid.

Referring to figure (\ref{fig:continuumL9:continuumL9fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL9fig2}

\caption{FIXME: continuumL9fig2}

\end{figure}

we have a flow rate

or

per unit time. Here the velocity of fluid particle is .

we must have

\begin{itemize}

\item

positive if fluid is coming in.

\item

negative if fluid is going out.

\end{itemize}

By Green’s theorem

so we have

and must have

The total mass has to be conserved. The mass that is leaving the volume per unit time must move through the surface of the volume in that time. In differential form this is

Operating by chain rule we can write this as

To make sense of this, observe that we have for

so we have

or

## Incompressible fluid

When the density doesn’t change note that we have

which then implies

at all points in the fluid.

# References

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

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