Consider a velocity field where the radial and tangential components of velocity are \(V_{r} = 0\) and \(V_{θ} = cr\), respectively, where \(c\) is a constant. Obtain the equations of the streamlines.
Consider a velocity field where the radial and tangential components of velocity are \(V_{r} = 0\) and \(V_{θ} = cr\), respectively, where \(c\) is a constant. Obtain the equations of the streamlines.
Here \(V_{r}=0\) and \(V_{\theta}=cr\) , \(x\) component of velocity is given as \[u=-V_{\theta}sin\theta=-cr\frac{y}{r}=-cy\] and \(y\) component of velocity is \[V=V_{\theta}cos\theta=cr\frac{x}{r}=cx\]
\[Vdx-udy=0
\\\Rightarrow \frac{dy}{dx}=\frac{V}{u}=\frac{-x}{y}\] On integrating
\[x^{2}+y^{2}=C\]
This equation represents a circle with centre at origin.
