Transform flow around a cylinder to cartesian

[MichielM]

Homework Statement

A flow field is considered to be steady two-dimensional and can be described by the following
velocity components in the xy- or r\theta-plane at the front half of the cylinder:
<br /> u_r=V\cos\theta\left(1-\frac{a^2}{r^2}\right)<br />
<br /> u_{\theta}=-V\sin\theta\left(1+\frac{a^2}{r^2}\right)<br />

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

Homework Equations

The link between cartesian and cylindrical coordinates is:
x = x
y = r \cos \theta
z = r \sin \theta

or the other way around:
x = x
r^2 = x^2+y^2
\theta =tan^{-1}(y/x)

The Attempt at a Solution

For u_r I take 1 term r outside the brackets and then transform to get:
<br /> u_r=V y \left(\frac{1}{x^2+y^2}-\frac{a^2}{\left(x^2+y^2\right)^3}\right)<br />
And similarly for u_{\theta} I get:
<br /> u_{\theta}=-V x \left(\frac{1}{x^2+y^2}+\frac{a^2}{\left(x^2+y^2\right)^3}\right)<br />

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

 

[Cyosis]

You want to find an expression of some vector u in Cartesian coordinates. In other words you're interested in \mathbf{u}=u_x \mathbf{\hat{x}}+u_y \mathbf{\hat{y}}+u_z \mathbf{\hat{z}}. However you have the vector u as \mathbf{u}=u_x \mathbf{\hat{x}}+u_r \mathbf{\hat{r}}+u_\theta \mathbf{\hat{\theta}}. To get your answer you need to find a relation between the Cartesian and cylindrical unit vectors. You can find this relation in your book or derive it yourself.