Find the equation of 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.

Asked on 26th October 2019 in Aerodynamics.
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1 Answer(s)

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.

Answered on 26th October 2019.
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