## PHY454H1S Continuum Mechanics. Lecture 17: Impulsive flow. Boundary layers. Oscillatory driven flow. Taught by Prof. K. Das.

Posted by peeterjoot on March 16, 2012

# Disclaimer.

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

# Review. Impulsively started flow.

Were looking at flow driven by an impulse, a sudden motion of the plate, as in figure (\ref{fig:continuumL17:continuumL17Fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{continuumL17Fig1}

\caption{Impulsively driven time dependent fluid flow.}

\end{figure}

where the fluid at the origin is pushed so that it is given the velocity

latex t < 0$} \\ U(t) & \quad \mbox{for } \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.1)$

where as .

Navier-Stokes takes the form

With a similarity variable

and

we found that we needed to solve

where

with solution

Here, we’ve used the error function

as plotted in figure (\ref{fig:continuumL17:continuumL17Fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{continuumL17Fig2}

\caption{Error function.}

\end{figure}

# Boundary layers.

Let’s look at spacetime points which are constant in

so that the speed at equals the speed at . This is illustrated in figure (\ref{fig:continuumL17:continuumL17Fig3})

\begin{figure}[htp]

\centering

\def\svgwidth{0.6\columnwidth}

\caption{Velocity profiles at different times.}

\end{figure}

## Universal behavior

Looking at a plot with different viscosities for position vs time scaled as as in figure (\ref{fig:continuumL17:continuumL17Fig4}) we see a sort of universal behavior

\begin{figure}[htp]

\centering

\def\svgwidth{0.6\columnwidth}

\caption{Universal behaviour.}

\end{figure}

Characterizing this we introduce the concept of boundary layer thickness

\begin{definition}

\emph{(Boundary layer thickness)}

The length scale over which viscosity is dominant. This is the viscous length scale.

\end{definition}

This is similar to what we have in the heat equation

where the time scale for the diffusion can be expressed as

We could consider a scenario such as a heated plate in a cavity of height as in figure (\ref{fig:continuumL17:continuumL17Fig5})

\begin{figure}[htp]

\centering

\def\svgwidth{0.5\columnwidth}

\caption{Characteristic distances in heat flow problems.}

\end{figure}

with a temperature on the bottom plate. We can ask how fast the heat propagates through the medium.

# Another worked problem.

Consider an oscillating plate, driving the motion of the fluid, as in figure (\ref{fig:continuumL17:continuumL17Fig6})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{continuumL17Fig6}

\caption{Time dependent fluid motion due to oscillating plate.}

\end{figure}

(we are thinking here about the always oscillating case, and not an impulsive plate motion).

We write

with substitution into

we have

or

This is an equation of the form

where

or

where

check:

Considering the boundary value constraints we have

Since we must have

so we must kill off the exponentially increasing (albeit also oscillating) term by setting . Also, since

we must have

or

so

and

or

This is a damped transverse wave function

where

is the wave speed.

Since we have an exponential damping here, the flow of fluid will essentially be confined to a boundary layer, where after distance , the oscillation falls off as

We can find a nice illustration of such a flow in [1].

# Flow over static object.

Next time we’ll start considering fluid flow around a fixed object as in figure (\ref{fig:continuumL17:continuumL17Fig7})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.3\textheight]{continuumL17Fig7}

\caption{Fluid flow around fixed object.}

\end{figure}

# References

[1] Wikipedia. Stokes boundary layer — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 16-March-2012]. http://en.wikipedia.org/w/index.php?title=Stokes_boundary_layer&oldid=466077125.

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