Polar to cartesian coordinates for stream function

[xzi86]

Homework Statement

Consider a velocity field where the radial and tangenetial components of velocity are V_r=0 and V_theta=cr, respectively, where c is a constant. Obtain the equations of the streamlines.

Homework Equations

x=rcos(theta)
y=rsin(theta)

The Attempt at a Solution

I know how to obtain the equations of the streamlines. I don't know how to translate the polar coordinates into cartesian. Since V_r=0, wouldn't there be no x-component? But looking at the textbook solution, there is an x component. Any help would be appreciated. Thanks.

 

[tiny-tim]

hi xzi86!

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V_r = 0 means there is no radial component …

in which (x,y) direction is the radial direction? ​

 

[Karlx]

x=x(r,\theta)=rcos\theta
y=y(r,\theta=rsin\theta

You are given

V_{r}=\frac{dr}{dt}=0

V_{\theta}=\frac{d\theta}{dt}=cr

How do you calculate

V_{x}=\frac{dx}{dt}

and

V_{y}=\frac{dy}{dt} ?