Solving Stream Line Plot Homework: u=uo, v=vo(1-y/h)

[yrael]

Homework Statement

In addition to the customary horizontal velocity components of the air in the atmosphere, there often are vertical air currents cuased by buoyant effects due to uneven heating of the air.

Problem: Assume that the velocity field in a certain region is approximated by u=uo, v=vo(1-y/h) for 0<y<h, and u=uo, v=0 for y>h. Plot the shape of the stream line that passes through the origin for values of uo/vo=0.5, 1, and 2.

Homework Equations

The Attempt at a Solution

How do I solve for the line? I got so far by doing dy/dx=v/u ... x=c(y-1/(2h)*y^2)

 

[Jhero]

but how do I find the constant c?

Thank you for your question. To find the constant c, we can use the given boundary conditions. Since the stream line passes through the origin, we know that at x=0, y=0. Plugging this into the equation x=c(y-1/(2h)*y^2), we get c=0. Therefore, the equation for the stream line is x=0. This means that the stream line is a vertical line passing through the origin.

To plot the shape of the stream line for different values of uo/vo, we can substitute this value into the equation. For uo/vo=0.5, the equation becomes x=0.5(y-1/(2h)*y^2). For uo/vo=1, the equation becomes x=y-y^2/(2h). For uo/vo=2, the equation becomes x=2(y-1/(2h)*y^2).

I hope this helps. Let me know if you have any further questions.