Consider a velocity field where radial and tangential components of velocity are Vr=0 and Vθ=4r. Obtain the equation of the streamlines.
A velocity field has radial and tangential components of velocity as \({V_r} = 0\) and \({V_\theta } = 4r\).Obtain equation of streamlines.
Since \({V_r} = 0,\)so there is no radial component of velocity.So the streamlines will be circular, with centres at origin.
\[\begin{array}{l}u = – {v_\theta }\sin \theta \\ = – 4r\sin \theta \\ = – 4r\frac{y}{r} = – 4y\\v = {v_{_\theta }}\cos \theta = 4r\cos \theta = 4r\frac{x}{r} = 4x\end{array}\] therefore equation of streamline\[\begin{array}{l}\left( {\frac{{dy}}{{dx}}} \right) = \frac{v}{u} = \left( {\frac{{ – x}}{y}} \right)\\ \Rightarrow ydy = – xdx\\ \Rightarrow \frac{{{y^2}}}{2} + \frac{{{x^2}}}{2} = c\\ \Rightarrow {x^2} + {y^2} = {\rm{constant}}\end{array}\] This is equation of circle,with center at origin.
