Consider a velocity field where the \(x\) and \(y\) components of velocity are given by \(u = cy/(x^{2} + y^{2})\) and \(v = −cx/(x^{2 }+ y^{2})\), where \(c\) is a constant. Obtain the equations of the streamlines.
Consider a velocity field where the \(x\) and \(y\) components of velocity are given by \(u = cy/(x^{2} + y^{2})\) and \(v = −cx/(x^{2 }+ y^{2})\), where \(c\) is a constant. Obtain the equations of the streamlines.
Here \(x\) and \(y\) components of velocities are given.We need to find equation of streamlines. \[vdx-udy=0
\\\frac{dy}{dx}=\frac{v}{u}=\frac{-cx}{x^{2}+y^{2}}\times \frac{x^{2}+y^{2}}{cy}=\frac{-x}{y}
\\\Rightarrow \frac{dy}{dx}=\frac{-x}{y}
\\\Rightarrow ydy=-xdx\]
On integrating we get \[\Rightarrow \frac{y^{2}}{2}=\frac{-x^{2}}{2}+C
\\\Rightarrow x^{2}+y^{2}=4C
\\\Rightarrow x^{2}+y^{2}=C_{1}\]
Therefore streamlines are concentric with their centres at the origin.
