Conformal mapping of sn(z,m) function

In summary, the conversation discusses the use of mapping methods to convert Jacobi Elliptic functions from the z-plane to the w-plane and how to modify the complex potential and stream function in the presence of two points with different elevations.
  • #1
lotusquantum
9
0
Hi all friends,
I am working on tracer in the oil field. In attempt to understand the analytical solution of tracer breakthrough in 5-spot pattern I'm reading the paper of Brigham and Abbasadez. For obtaining of potential field, they used some mapping methods from z plane to w plan, with w=sn(z,n). But I don't understand how to convert the Jacobi Elliptic functions from z plane to one half of w plan. So I hope anybody will help me or show me some related articles to clarify this point.
Another thing, If we have 2 points (one source and one sink) with different elevation (considering of gravity), then how can we modify the complex potential and the stream function as well??
Hope to hear from all of you soon. Thank you very much!
 
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  • #2
</code>In order to convert the Jacobi Elliptic functions from the z-plane to the w-plane, you need to use a transformation called a conformal mapping. This transformation maps points in the z-plane to points in the w-plane in a way that preserves angles and local shapes. In particular, it maps circles to circles or straight lines.For your second question regarding two points with different elevations, you can modify the complex potential and stream function by taking into account the effects of gravity. Specifically, you can modify the potential by adding a term that accounts for the gravitational potential energy of the two points. This will affect the shape of the streamlines, as well as the amount of flow between the two points.
 
  • #3


Hello there,

Conformal mapping is a technique used in mathematics and physics to map a complex or curved surface onto a simpler, flat surface. In the context of the sn(z,m) function, this mapping is used to transform the z plane (complex plane) into the w plane, where w=sn(z,m). The sn(z,m) function is a Jacobi elliptic function, which is a special type of complex function that can be used to describe a variety of physical phenomena.

In the paper by Brigham and Abbasadez, they use conformal mapping to obtain the potential field for a tracer in an oil field. This allows them to simplify the problem and make it easier to analyze. However, it can be challenging to understand how to convert the Jacobi elliptic functions from the z plane to the w plane.

One way to approach this is to use the properties of the Jacobi elliptic functions, such as their periodicity and symmetry, to transform them into simpler forms. Additionally, there are various mathematical techniques, such as the transformation of variables or the use of integrals, that can be used to simplify the functions.

As for your question about modifying the complex potential and stream function for two points with different elevations, this would depend on the specific problem and the boundary conditions. In general, the complex potential and stream function can be modified by incorporating the gravitational potential into their equations. However, the exact method for doing so would depend on the specific problem at hand and may require some mathematical manipulation.

I hope this helps to clarify some of the concepts related to conformal mapping and the sn(z,m) function. If you need further assistance, I suggest consulting with a mathematician or physicist who has experience in this area. Best of luck with your research!
 

Related to Conformal mapping of sn(z,m) function

1. What is a conformal mapping?

A conformal mapping is a function that preserves angles between curves. In other words, the mapping maintains the shape of the objects being transformed.

2. What is the significance of the sn(z,m) function in conformal mapping?

The sn(z,m) function is a special case of the Jacobi elliptic function, and it is commonly used in conformal mapping because it allows for the mapping of complex numbers to the real plane.

3. How is the sn(z,m) function used in practical applications?

The sn(z,m) function is used in various fields such as physics, engineering, and mathematics for solving problems related to potential theory, fluid dynamics, and electromagnetism.

4. Can the sn(z,m) function be extended to higher dimensions?

Yes, the sn(z,m) function can be extended to higher dimensions through the use of multidimensional elliptic integrals. This allows for the mapping of complex numbers to higher dimensional spaces.

5. Are there any limitations to using the sn(z,m) function in conformal mapping?

While the sn(z,m) function is a powerful tool in conformal mapping, it is limited in its ability to accurately map highly complex or discontinuous regions. In these cases, other methods such as numerical techniques may be more effective.

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