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

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:
$$u_r=V y \left(\frac{1}{x^2+y^2}-\frac{a^2}{\left(x^2+y^2\right)^3}\right)$$
And similarly for $$u_{\theta}$$ I get:
$$u_{\theta}=-V x \left(\frac{1}{x^2+y^2}+\frac{a^2}{\left(x^2+y^2\right)^3}\right)$$

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.