## PHY454H1S Continuum Mechanics. Lecture 14: Non-dimensionality and scaling. Taught by Prof. K. Das.

Posted by peeterjoot on March 7, 2012

# Disclaimer.

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

# Review. Surfaces

We are considering a surface as depicted in (\ref{fig:continuumL14:continuumL14Fig13})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL13Fig13}

\caption{Variable surface geometries}

\end{figure}

With the surface height given by

where this describes the interface. Taking the difference

we define a surface. We considered a small displacement as in (\ref{fig:continuumL14:continuumL14Fig14}).

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL13Fig14}

\caption{A vector differential element}

\end{figure}

Recall that if is a constant, then is a normal to the surface. We showed this by considering the differential

We can construct the unit normal by scaling. For our 1D example we have

so that our unit normal is

A unit tangent can also be constructed by inspection

# Traction vector at the interface.

Recall that our stress tensor has the form

(here we are switching notations for the stress since we will be using for surface tension in this section)

The traction vector components are

Considering a control volume as illustrated in we can arrive at what we call the jump stress balance equation

figure (\ref{fig:continuumL14:continuumL14fig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL14fig3}

\caption{Control volume for liquid air interface}

\end{figure}

where

and the suffix and prefix indicates that we are considering the interface between fluids labelled and (liquid and air respectively in the diagram).

For a derivation see Prof after class?

Force balance along the normal direction gives

If you do this calculation, you will get

I think this was called the Laplace equation?

Question: How was defined? A: Energy per unit area.

Figure (\ref{fig:continuumL14:continuumL14fig4}) was given as part of an explaination of surface tension and curvature, but I missed part of that discussion. Perhaps this is elaborated on in the class notes?

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL14fig4}

\caption{Molecular gas and liquid interactions at a surface.}

\end{figure}

# Non dimensionalization and scaling

## Motivation.

By scaling we mean how much detail do you want to look at in the analysis. Consider the figure (\ref{fig:continuumL14:continuumL14fig5a}) where we imagine that we zoom in on something that appears smooth from a distance, but perhaps grandular close up as in figure (\ref{fig:continuumL14:continuumL14fig5b}). Picking the length scale to be used in this case can be very important.

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL14fig5a}

\caption{Coarse scaling example.}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL14fig5b}

\caption{Fine grain scaling example (a zoom).}

\end{figure}

FIXME: prof wrote:

what was that about?

## Rescaling by characteristic length and velocity.

Suppose that a fluid is flowing with

\begin{itemize}

\item a characteristic velocity , with dimensions

\item a characteristic length scale

\end{itemize}

Considering the dimensions of the terms in the NS equation

so

Now let’s alter the NS equation using some scaling to put it into a dimensionless form

so that

Putting everything together, NS takes the form

or

Introducing the Reynold’s number

We have NS in dimensionless form

The implications of this will be discussed further in the next lecture.

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