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.
