GATE Papers >> Mechanical >> 2016 >> Question No 224

Question No. 224 Mechanical | GATE 2016
For a certain two-dimensional incompressible flow, velocity field is given by $2xy\;\widehat i-y^2\widehat j$ . The streamlines for this flow are given by the family of curves

Answer : (B) $xy^2$ = constant


Solution of Question No 224 of GATE 2016 Mechanical Paper

Given velocity field: $ \overrightarrow{\mathrm V}=2\mathrm{xy}\widehat{\mathrm i}-\mathrm y^2\widehat{\mathrm j} $

X - Component of velocity $ \left(\mathrm u\right)=2\mathrm{xy} $

Y - Component of velocity $ \left(\mathrm v\right)=-\mathrm y^2 $

Equation of stream line:

$ \begin{array}{l}\frac{\mathrm{dx}}{\mathrm u}=\frac{\mathrm{dy}}{\mathrm v}=\frac{\mathrm{dz}}{\mathrm\omega}\\\frac{\mathrm{dx}}{2\mathrm{xy}}=\frac{\mathrm{dy}}{-\mathrm y^2}\\\frac{\mathrm{dx}}{2\mathrm x}=\frac{-\mathrm{dy}}{\mathrm y}\\\frac{\mathrm{dx}}{2\mathrm x}+\frac{\mathrm{dy}}{\mathrm y}=0\;\mathrm{Integrating}\;\mathrm{both}\;\mathrm{sides},\\\int\frac{\mathrm{dx}}{2\mathrm x}+\int\frac{\displaystyle\mathrm{dy}}{\displaystyle\mathrm y}=\int0\\\frac12\mathcal l\mathrm{nx}+\mathcal l\mathrm{ny}=\mathcal l\mathrm{nc}\\\mathrm{Where},\;\mathcal l\mathrm{nc}\;=\;\mathrm{sum}\;\mathrm{constant}\\\mathcal l\mathrm{nx}+2\mathcal l\mathrm{ny}=\mathcal l\mathrm{nc}'\\\mathcal l\mathrm{nx}+\mathcal l\mathrm{ny}^2=\mathcal l\mathrm{nc}'\\\mathrm{xy}^2=\mathrm{Constant}\\\downarrow\end{array} $

Equation of streamline

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