Flow Mapping Theorem and Obstacles

In summary, the conversation is about a person struggling with fluid flow problems and needing help with using the Inverse Function Rule to solve a function. They mention having trouble with part (iii) and provide their progress on parts (i) and (ii). They also mention needing clarification on a function with alpha as 2i and provide a formula for Jalpha. Another person offers a solution using the standard equation of an ellipse.
  • #1
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4
0
Hi All

I have one final question that's related to flow problems with obstacles.

Any help would be greatly appreciated as I am finding fluid flows extremely difficult.

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"Examinations are formidable even to the best prepared, for
the greatest fool may ask more than the wisest man can answer".
Charles Caleb Colton, 1825
 
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  • #2
Hi All

Think I've cracked part (i) and (ii). However, I'm stuch on part (iii). I think I need to use the Inverse Function Rule somehow to get the function stated but when I do, it doesn't give the answer. Can someone please help? The inverse function rule states that f (z)'=1/1-1/(f(z))^2.

Dont know if it will help but for part (ii) I got z+(4/z)-(16/z^3)...

Please please help...
 
  • #3
There is something missing! You say "let [itex]J_{2i} be the function with [itex]\alpha= 2i[/itex]". I presume that somewhere earlier they defined another function [itex]J_{\alpha}[/itex] but you don't include that information.
 
  • #4
The function you specift is given by the following formula:


Jalpha=w+(alpha)^2/w.

Hence j2i=3+4/3 and j2i = 3-4/3. Substituting this into the standard equation of an ellipse for a^2 and b^2 and rearranging gives the required formula. Does this help?
 

FAQ: Flow Mapping Theorem and Obstacles

What is the Flow Mapping Theorem?

The Flow Mapping Theorem is a mathematical theorem that states that a continuous vector field on a compact, oriented manifold has a non-zero flow if and only if it does not contain any fixed points. In other words, if a vector field has no points where the vector is equal to zero, it will have a flow that continuously moves along the vector field.

What is a compact manifold?

A compact manifold is a mathematical space that is both closed and bounded. This means that it contains all of its limit points and is contained within a finite region. In the context of the Flow Mapping Theorem, a compact manifold refers to the space in which the vector field exists.

What is an oriented manifold?

An oriented manifold is a mathematical space that has a consistent notion of direction. This means that for any point on the manifold, there is a well-defined "right-hand rule" that determines the direction of the tangent vectors at that point. In the context of the Flow Mapping Theorem, this orientation is important in determining the direction of the flow along the vector field.

What is an obstacle in the context of the Flow Mapping Theorem?

An obstacle in the context of the Flow Mapping Theorem refers to a point or region in the manifold where the vector field becomes discontinuous. This can occur at points where the vector field is equal to zero or when there is a sudden change in the direction of the vector field. These obstacles can affect the flow of the vector field and may result in the flow being unable to continue along certain paths.

How is the Flow Mapping Theorem used in real-world applications?

The Flow Mapping Theorem has many applications in various fields of science and engineering, including fluid dynamics, weather forecasting, and robotics. It is used to study the behavior of vector fields and their flows, which can help in understanding complex physical phenomena and solving practical problems. For example, it can be used to model the flow of air around an airplane wing or the flow of blood through the human body.

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