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

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In summary, the conversation discusses the concept of streamlines in fluid mechanics and how to plot them through the origin at different times using the given velocity field. It is determined that the equations for streamlines in this case only exist in the first quadrant due to the non-steady state nature of the problem. The conversation also mentions the difference between streamlines and pathlines, and the need to solve for sets of initial values to find pathlines.
  • #1
pyroknife
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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?
 
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  • #2
Different values of C correspond go different streamlines. You have to choose C so that the equation holds for x=0, y=0.
 
  • #3
vela said:
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?
 
  • #4
Yup.
 
  • #5
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.
 
  • #6
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 = vx = x2y

dy/dt = vy= x2t

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

Chet
 

1. What is the purpose of plotting streamlines with origin at t=0,1,2?

The purpose of plotting streamlines with origin at t=0,1,2 is to visualize the flow of a fluid over time. This allows scientists to better understand the behavior and patterns of fluid motion, and can be useful in various fields such as meteorology, oceanography, and engineering.

2. How are streamlines plotted with origin at t=0,1,2?

Streamlines with origin at t=0,1,2 are plotted by first determining the velocity field of the fluid, which describes the direction and magnitude of the fluid's movement at each point in space. Then, starting at the specified points (t=0,1,2), the path of the fluid particles is traced by following the velocity vectors at each point in time.

3. What information can be obtained from plotting streamlines at multiple origins?

Plotting streamlines at multiple origins allows scientists to see how the fluid flow changes and evolves over time. This can reveal important information about the behavior and characteristics of the fluid, such as areas of turbulence, eddies, or convergence/divergence zones.

4. Can streamlines at different origins intersect?

Yes, streamlines at different origins can intersect. This can occur when the fluid flow changes direction or when there is a point of convergence or divergence in the flow. The intersection of streamlines can provide insight into the movement and behavior of the fluid at that particular point.

5. How can plotting streamlines at t=0,1,2 be applied in real-world situations?

Plotting streamlines at t=0,1,2 can have various real-world applications, such as predicting weather patterns, understanding ocean currents, and studying the flow of air or water in engineering systems. It can also be used to analyze and improve the aerodynamics of vehicles, turbines, and other structures that interact with fluid flows.

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