The x and y components of a velocity field are given as u=4x/(x^2+y^2) and v=4y/(x^2+y^2), what is the equation of streamlines.

x and y components of a velocity field are given as \(u = \frac{{4x}}{{\left( {{x^2} + {y^2}} \right)}}\) and \(v = \frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}}\), what is the equation of streamlines.

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    Equation of streamline is given as

    \(\left( {\frac{{dy}}{{dx}}} \right) = \left( {\frac{v}{u}} \right)\) therefore,

    \(\left( {\frac{{dy}}{{dx}}} \right) = \frac{{\frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}}}}{{\frac{{4x}}{{\left( {{x^2} + {y^2}} \right)}}}}\)

    \(\left( {\frac{{dy}}{{dx}}} \right) = \frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}} \times \frac{{\left( {{x^2} + {y^2}} \right)}}{{4x}}\)

    \( = \left( {\frac{y}{x}} \right)\)

    \(\frac{{dy}}{{dx}} = \frac{y}{x}\)

    \(\frac{{dy}}{y} = \frac{{dx}}{x}\)

    \(\ln y = \ln x + c\)

    \(\ln y – \ln x = c\)

    \(\ln \left( {\frac{y}{x}} \right) = \ln c\)

    \(\frac{y}{x} = \ln c\)

    \(y = {c_1}x\)

    The streamlines are straight lines from a source.

    techAir Answered on 4th January 2019.
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