Finding Streamlines: How to Use the Streamline Equation for Homework?

  • Thread starter CMJ96
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In summary, the conversation discusses solving a differential equation involving velocity values and integrating to find an expression for the pathlines. The solution also requires solving a system of equations and including an integration constant to account for initial conditions.
  • #1
CMJ96
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Homework Statement


w1CNc3i.png


Homework Equations


$$\frac{dy}{dx}=\frac{v_y}{v_x}$$[/B]

The Attempt at a Solution


I have subbed the given values for $$v_y$$ $$v_x$$ into the equation above, and integrated, i got the following expression
$$y=\left(1+t \right) lnx $$
I'm not sure where to go next, do I sub in $$x_0$$ $$y_0$$ ?
 

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  • #2
You cannot solve it like that. Your differential equation depends on time ##t##, which is not constant. You need to solve the system
$$
\frac{dx}{dt}= v_x, \quad \frac{dy}{dt}= v_y.
$$

Edit: Sorry, missed the title. Those are the pathlines. For the streamlines yes, but you are missing an integration constant that you will need to adapt your solution to the initial condition. Otherwise you would have only one streamline.
 
  • #3
So I should sub in limits of x=x_0 and y=y_0 up to y=y and x=x and add in a constant of integration?
 
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Related to Finding Streamlines: How to Use the Streamline Equation for Homework?

1. What is the purpose of determining streamlines?

Determining streamlines is important in fluid mechanics and aerodynamics as it allows us to visualize the flow of a fluid and understand its behavior. It can also help us analyze the efficiency and performance of a fluid system.

2. How do you determine streamlines experimentally?

Streamlines can be determined experimentally by using tracer particles in a fluid and tracking their movement. This can be done using techniques such as dye injection, smoke visualization, or particle image velocimetry (PIV).

3. Can streamlines be calculated mathematically?

Yes, streamlines can also be calculated mathematically using equations that describe the velocity and pressure fields of a fluid. This is often done using computational fluid dynamics (CFD) techniques.

4. What is the difference between a streamline and a pathline?

A streamline represents the instantaneous direction of fluid flow at a given point, while a pathline represents the actual path a fluid particle takes over time. Streamlines can change with time, while pathlines are fixed once the fluid has passed through a point.

5. How can streamlines be useful in practical applications?

Streamlines are commonly used in the design and analysis of aerodynamic systems, such as aircraft and cars. They can also be used in predicting the behavior of fluids in pipes, pumps, and other industrial systems.

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