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Transform flow around a cylinder to cartesian

  1. Apr 13, 2010 #1
    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?!
     
    Last edited: Apr 13, 2010
  2. jcsd
  3. Apr 13, 2010 #2

    Cyosis

    User Avatar
    Homework Helper

    You want to find an expression of some vector u in Cartesian coordinates. In other words you're interested in [itex]\mathbf{u}=u_x \mathbf{\hat{x}}+u_y \mathbf{\hat{y}}+u_z \mathbf{\hat{z}}[/itex]. However you have the vector u as [itex]\mathbf{u}=u_x \mathbf{\hat{x}}+u_r \mathbf{\hat{r}}+u_\theta \mathbf{\hat{\theta}}[/itex]. 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.
     
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