Velocity and Flow Analysis: Complex or Real-Which is Best?

In summary, the conversation discusses the use of complex analysis in velocity and flow analysis compared to real analysis. The advantages and limitations of complex analysis are highlighted, with the conclusion that real analysis is more suitable for 3-D problems. The speaker also shares their qualifications and background in mathematics.
  • #1
tecumseh
7
0
I need some comments related to the accuracy of my thinking on velocity and flow analysis.

(1) As I see it now, complex analysis of velocity and flow in all kinds of physics problems can deal only with 2-D analysis and that its advantage over real analysis for this is that complex maps conformally, functions onto the uv plane which facilitates or makes the analysis easier than with real Green or Stokes' theory. Also complex cannot make use of the variety of contours that real can.

(2) Complex analysis of flow: Z' = d(phi)/dx - d(phi)/dy
and then the conjugate of the above is d(phi)/x + d(phi)/dy which is equal to the velocity or flow. Then the absolute value of V, the velocity in the above equals the speed of the flow at a specified point.

(3) Conclusion: Real analysis is used here in 3-D problems while complex analysis cannot do this although complex facilitates the analysis of such as airfoils by mapping the Z onto the uv plane.

Any comments are much appreciated. I have a PhD in an area other than math but I have 18 grad hours in math and am going for the MA. Math is also my hobby.

tecumseh@nc.rr.com

Don Wire (North Carolina)

Campbell University (government, history and mathematics instruction)

Fayetteville Technical Community College (critical thinking and cultural studies)
 
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  • #2
You cannot divide this the way you did. It is the problem which defines whether a real or a complex manifold is considered, not the dimension. Complex differentiation is different from real differentiation due to the Cauchy - Riemann conditions, which results in a different kind of theorems. This means that w.r.t. differentiation ##\mathbb{C}\neq \mathbb{R}^2\,.##

For a general walkthrough differentiation, see
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
 

What is the difference between complex and real velocity and flow analysis?

The main difference between complex and real velocity and flow analysis lies in the way they represent the flow of a fluid. Complex analysis uses imaginary numbers to represent the flow, while real analysis uses only real numbers. Complex analysis provides a more accurate representation of the flow, but it is also more complex to understand and implement.

Which type of analysis is best for studying fluid flow in real-world applications?

In most real-world applications, real velocity and flow analysis is preferred. This is because it is easier to understand and implement, and it still provides accurate results for most practical purposes. Complex analysis is usually reserved for more specialized or theoretical studies.

What are some advantages of using complex velocity and flow analysis?

Complex velocity and flow analysis has some advantages over real analysis, such as its ability to accurately represent complex flow patterns and its use in advanced mathematical models. It is also more versatile and can be applied to a wider range of fluid flow problems.

Can real and complex velocity and flow analysis be used together?

Yes, real and complex analysis can be used together in certain situations to get a more accurate and comprehensive understanding of fluid flow. This is often done in research and development settings where precision is crucial.

How do I choose which type of velocity and flow analysis to use?

The choice between real and complex analysis depends on the specific needs and goals of your study or project. Consider factors such as accuracy, complexity, and available resources when deciding which type of analysis is best for your application.

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