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 Aerodynamics.
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1 Answer(s)

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

\\\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.

Answered on 26th October 2019.
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