A non-dimensionalization problem

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In summary, to non-dimensionalize f(x), you can use the substitution x=zL. Then, to differentiate it with respect to z, you can use the chain rule to get f'(z)=L^{2}g'(Lz). This method ensures that the resulting function is non-dimensional.
  • #1
jimmy1066
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Homework Statement


A function [tex]f(x)[/tex] has dimensions of length, [tex]L[/tex].
I need to non-dimensionalize [tex]f(x)[/tex] using the substitution [tex]x=zL[/tex] and then differentiate it with respect to [tex]z[/tex].

Homework Equations


[tex]x=zL[/tex]

The Attempt at a Solution


First I made the substitution:
[tex]f(x) \Rightarrow f(zL)[/tex]

[tex]\frac{\partial f(zL)}{\partial z}[/tex] = L f '(z L)

Is this correct? I'm not sure because this now has dimensions of length.
What happens if I write: f(x) = f(zL) = f(x(z)) = F(z) and then differentiate w.r.t. z?
 
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  • #2
I would go about it in the following way: define f(x)=Lg(x), then using the substitution for x as x=Lz, then:

[tex]
f(x)=Lg(Lz)
[/tex]
So

[tex]
f'(z)=Lg'(Lz)(Lz)'=L^{2}g'(Lz)
[/tex]
 

1. What is a non-dimensionalization problem?

A non-dimensionalization problem is a mathematical technique used to simplify complex systems by removing units and scaling variables to make them dimensionless. This allows for easier analysis and comparison of different systems.

2. Why is non-dimensionalization important in scientific research?

Non-dimensionalization is important because it helps to identify and focus on the key parameters and relationships within a system. It also allows for generalization and comparison between different systems, making it a powerful tool in scientific research.

3. What are the benefits of non-dimensionalization?

Non-dimensionalization allows for easier visualization and understanding of complex systems, as well as simplifying mathematical equations and reducing computational efforts. It also allows for the identification of universal patterns and relationships within different systems.

4. How is non-dimensionalization performed?

Non-dimensionalization is performed by choosing a set of base units and scaling variables according to these units. This involves defining dimensionless parameters and equations, and then using them to analyze the system in terms of these dimensionless quantities.

5. Can non-dimensionalization be applied to all systems?

Yes, non-dimensionalization can be applied to all systems, regardless of their complexity. It is a versatile technique that can be used in various fields of science and engineering, such as fluid mechanics, thermodynamics, and electromagnetism.

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