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.
