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.
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.
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.
