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The velocity field of water waves near the surface is given by

$\begin{displaymath} u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t) \end{displaymath}$

where $\epsilon$ , , and $\omega$ are all positive constants. Find and draw the streamlines of the flow. You can assume that the water surface is approximately at and the water is below that surface.
The pathlines for water waves are more difficult to find. Therefor, assume that $\epsilon$ is small. In that case the particle displacements are small, and that allows you to approximate in the sine and cosine by the -value of the initial particle position which is constant:

$\begin{displaymath} u = \epsilon \sin(kx_0+\omega t) \qquad v = - \epsilon \cos(kx_0+\omega t) \end{displaymath}$

Find and draw the particle paths under that assumption.
The velocity field in Couette flow is given by

$\begin{displaymath} u=cy \qquad v = 0 \qquad w = 0 \end{displaymath}$

Draw the streamlines of this flow and a couple of velocity profiles. Find the velocity derivative tensor , the strain rate tensor , and the matrix .
Find the three eigenvalues and eigenvectors of the strain rate tensor for Couette flow. Normalize them to length one, because these give you the unit vectors ${\widehat \imath} '$ , ${\widehat \jmath} '$ , and ${\widehat k}'$ of the principal axis system. (If you did it right, they are automatically orthogonal). Draw the principal axis system in your previous plot of the streamlines.
For each principal axis, describe whether the fluid is strained in that direction, compressed, or neither.
In Poiseuille flow (laminar flow through a pipe), the velocity field is in cylindrical coordinates given by

$\begin{displaymath} \vec v = {\widehat \imath}_z v_{\rm max} \left(1 -\frac{r^2}{R^2}\right) \end{displaymath}$

where $v_{\rm {max}}$ is the velocity on the centerline of the pipe and the pipe radius. Use Appendix B to find the velocity derivative and strain rate tensors of this flow. Evaluate the strain rate tensor at , $\frac12R$ and . What can you say about the straining of small fluid particles on the axis?