Velocity profile and streamline equations, help please?

[aero&astro]

So far I know that dy/dx = V/U is the velocity profile. From what I've read if you know V & U and put it into that equation and the integrate it you get the equation of a streamline. is this correct?

An example I have is of a velocity field defined by |U| = (B+Ay^2)e_1. The graph of the velocity profile I've seen for this field is a quadratic parabola but seeing as y is squared shouldn't it be a parabola around the x axis? or is it about the y-axis because the field only has a component in the x direction so y^2 means the same things as x^2 but with the axis changed?

and also the streamline is shown as a bunch of lines parallel to the Cartesian x-axis that get closer together as you move up the y-axis why is this?

 

[Navin A S]

Yes, dy/dx = V/U is the velocity profile. To get the equation of a streamline, you need to integrate this equation with respect to x. The graph of the velocity profile in this example is a parabola around the x-axis because U is only a function of y. The streamlines are parallel to the x-axis because the velocity field is only a function of y. As you move up the y-axis, the lines get closer together because the velocity increases as you move up the y-axis.