For a certain two-dimensional incompressible flow- velocity field is given by 2xy-y2- The streamlines for this flow are given by the family of curves

V⃗=2xyî y^2ĵ where u = 2xy and v = y^2 Equation of stream line, dxu=dyv dx2xy=dy y^2 or dx2x= dyy On integrating, we get 12log_ex= log_ey+log_ec log_ex^ ...

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Fluid Mechanics

Streamline , Streakline and Pathline

For a certain...

Question

For a certain two-dimensional incompressible flow, velocity field is given by $2 x y^i - y^{2}^j$ . The streamlines for this flow are given by the family of curves

x^{2} y^{2}

= constant

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x y^{2}

= constant

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2 x y - y^{2}

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xy = constant

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Solution

The correct option is B $x y^{2}$ = constant
$\to V = 2 x y^i - y^{2}^j$
where u = 2xy and $v = - y^{2}$
Equation of stream line,

$\frac{d x}{u} = \frac{d y}{v}$

$\frac{d x}{2 x y} = \frac{d y}{- y^{2}}$

$o r \frac{d x}{2 x} = - \frac{d y}{y}$

On integrating, we get

$\frac{1}{2} l o g_{e} x = - l o g_{e} y + l o g_{e} c$

$l o g_{e} x^{1 / 2} + l o g_{e} y = l o g_{e} c$

$l o g_{e} x^{1 / 2} y = l o g_{e} c$

$x^{1 / 2} y = c$

$o r x y^{2} = c_{1} = c o n s t a n t$

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For a certain two-dimensional incompressible flow, velocity field is given by 2xy^i−y2^j. The streamlines for this flow are given by the family of curves

The correct option is B xy2 = constant→ V =2xy^i−y2^j where u = 2xy and v=−y2 Equation of stream line, dxu=dyv dx2xy=dy−y2 or dx2x=−dyy On integrating, we get 12logex=−logey+logec logex1/2+logey=logec logex1/2y=logec x1/2y=c or xy2=c1 = constant

For a certain two-dimensional incompressible flow, velocity field is given by $2 x y^i - y^{2}^j$ . The streamlines for this flow are given by the family of curves