Plotting Streamlines: Origin at t=0,1,2

[pyroknife]

Homework Statement

The velocity field in a flow is given by V = (x^2)*yi + (x^2)*tj. (a) Plot the streamline through the origin at times t = 0, t = 1, and t = 2. (b) Do the streamlines plotted in part (a) coincide with the path of particles through the origin? Explain.

i&j are directional vectors.

Homework Equations

Providing this just in case you guys haven't heard of stream lines. "A streamline is a line everywhere tangent to the velocity vector at a given instant."

The Attempt at a Solution

dx/u=dy/v (equation from streamline from Fluid mechanics textbook)
where u=x^2 * y
v=(x^2)*t

After plugging those in and differentiating I get (y^2)/2=tx+C
where C is a constant.


My problem is I don't know how to go about plotting "the streamline through the origin at t=0,1,2."

Do I plug the 3 t's into the equation 3 times and plot those 3 equations I get? If so, how do I get C?

 

[vela]

Different values of C correspond go different streamlines. You have to choose C so that the equation holds for x=0, y=0.

 

[pyroknife]

Different values of C correspond go different streamlines. You have to choose C so that the equation holds for x=0, y=0.

Oh I see since the question is through the original that would make C=0 right for this equation?

Which would give me (y^2)/2=tx. Then I just plug in t=0,1,2 into this equation and plot the 3 equations?

 

[vela]

Yup.

 

[pyroknife]

That would give me
the equations
y=0
y=(2x)^.5
y=2*x^.5

The solutions seem to only have the streamlines plotted in the first quadrant. But doesn't the 2 square root functions exist in both the 1st and 4th quadrant? The first equation y=0 would also exist in all 4 quadrants.

 

[Chestermiller]

In fluid mechanics, the concept of streamlines only applies to steady state flow. This is a non-steadystate problem. In such problems, you can solve for the pathlines of particles, but the pathlines change with time. For your problem, the pathlines are determined by:

dx/dt = v_x = x²y

dy/dt = v_y= x²t

You need to solve this coupled set of ODEs for sets of initial values of x, y, and t.

Chet