(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A flow field is considered to be steady two-dimensional and can be described by the following

velocity components in the xy- or r[tex]\theta[/tex]-plane at the front half of the cylinder:

[tex]

u_r=V\cos\theta\left(1-\frac{a^2}{r^2}\right)

[/tex]

[tex]

u_{\theta}=-V\sin\theta\left(1+\frac{a^2}{r^2}\right)

[/tex]

Question: Transform this system from cylindrical coordinates into cartesian coordinates and give [tex]u_x[/tex], [tex]u_y[/tex] and [tex]u_z[/tex]

2. Relevant equations

The link between cartesian and cylindrical coordinates is:

[tex] x = x [/tex]

[tex]y = r \cos \theta[/tex]

[tex]z = r \sin \theta[/tex]

or the other way around:

[tex] x = x [/tex]

[tex]r^2 = x^2+y^2[/tex]

[tex]\theta =tan^{-1}(y/x)[/tex]

3. The attempt at a solution

For [tex]u_r[/tex] I take 1 term r outside the brackets and then transform to get:

[tex]

u_r=V y \left(\frac{1}{x^2+y^2}-\frac{a^2}{\left(x^2+y^2\right)^3}\right)

[/tex]

And similarly for [tex]u_{\theta}[/tex] I get:

[tex]

u_{\theta}=-V x \left(\frac{1}{x^2+y^2}+\frac{a^2}{\left(x^2+y^2\right)^3}\right)

[/tex]

This is where I get stuck, because I don't know how to separate the x and y parts of [tex]u_{\theta}[/tex] and [tex]u_{r}[/tex] to find the velocities in cartesian coordinates. Does anyone have any hints?!

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# Homework Help: Transform flow around a cylinder to cartesian

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