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.
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.
