# Find the equation of streamlines.

Consider a velocity field where the $$x$$ and $$y$$ components of velocity are
given by $$u = cx/(x^{2} + y^{2})$$ and $$v = cy/(x^{2} + y^{2})$$ where $$c$$ is a constant.
Obtain the equations of the streamlines.

Asked on 26th October 2019 in

Here $$x$$ component of velocity is $$u=\frac{cx}{x^{2}+y^{2}}$$ and $$y$$ component of velocity is $$v=\frac{cy}{x^{2}+y^{2}}$$ $vdx-udy=0$
$$\Rightarrow vdx=udy \\\Rightarrow \frac{dy}{dx}=\frac{v}{u}=\frac{\frac{cy}{x^{2}+y^{2}}}{\frac{cx}{x^{2}+y^{2}}}=\frac{y}{x} \\\Rightarrow \frac{dy}{dx}=\frac{y}{x}=\frac{dy}{y}=\frac{dx}{x}$$

On integrating

$$ln(y)=ln(x)+C \\\Rightarrow ln\left ( \frac{y}{x} \right )=C \\\Rightarrow y=e^{C}x \\\Rightarrow y=xC_{1}$$

Therefore streamlines are straight lines originating from the origin.