A 2D steady velocity field is given by V x2y2i 2xyj Derive

A 2-D steady velocity field is given by V = (x^2-y^2)i - 2xyj. Derive the streamline pattern and sketch a few streamlines in the upper half plane.

Solution

Solution

Here in this case

Vx = x² - y²

Vy = 2xy

Vx / Vy =( x² - y² ) / 2xy

Putting x = Vy

    This would give us

V + y dV / dy = o.5 ( 1 / v   - V)

     dy / y =(( 2V / 1 - 3V² )) dV

      Integrating gives

ln y =( 2 ln( ( 1 - 3v²) ) / -3 ) + ln c

    given ln ((y³) * (1 - 3( x / y)² )² = C

This is how differential equation is derived and solution is determined for strema lines.